Three categories, one question

Every piece of equipment in the lab is Category 1, 2 or 3. The category answers one question: who has to be present before you switch it on?

Permission from an instructor is required in every case. What changes between categories is when that permission can be given, and who has to be in the room while the machine is running.

Category 1

Permission may be given in advance. Work alone once you have it.

Category 2

Permission at the time of use. A peer must be present throughout.

Category 3

Permission at the time of use. A peer and an instructor must be present throughout.

The category never removes a requirement. General safety procedures and PPE apply to every category, to every person present — see Equipment Operating Procedure and Personal Protective Equipment.

Category 1

• Can be used with instructor permission — may be given in advance or at the time of use.
• Can be operated independently, without peer or instructor supervision, once permission is granted.
• You must still follow all general safety procedures and PPE requirements.

Category 2

• Can be used with instructor permission — permission must be given at the time of use.
• Does not require instructor supervision during operation.
• Requires peer supervision present at all times during use.
• Both the operator and the peer supervisor must follow all safety procedures and PPE requirements.

Find the equipment category

These are the equipment categories for the Woods / Construction lab.

If an item is not on the list, ask the instructor to place it before you use it.

Category 1

• Hand tools (excluding saws)
• Paint / stain
• Sandpaper
• Glue
• Clamps

See Category 1 permission and supervision requirements before use.

Category 2

• Hand tools (saws)
• Power tools
• Belt/disk sander
• Drill press
• CNC router

See Category 2 permission and supervision requirements before use.

Category 3

• Table saw
• Band saw
• Miter saw
• Planer
• Jointer

See Category 3 permission and supervision requirements before use.

Find the equipment category

These are the equipment categories for the Rapid Prototyping Center.

If an item is not on the list, ask the instructor to place it before you use it.

Category 1

• 3D printers
• Hand tools (excluding knives)

See Category 1 permission and supervision requirements before use.

Category 2

• Laser cutter
• Hand tools (knives)

See Category 2 permission and supervision requirements before use.

Category 3

None.

See Category 3 permission and supervision requirements.

Find the equipment category

These are the equipment categories for the Metals lab.

If an item is not on the list, ask the instructor to place it before you use it.

Category 1

• Hand tools
• Fixture table
• Clamps
• Sanding / polishing
• Deburring kit
• Part setup

See Category 1 permission and supervision requirements before use.

Category 2

• CNC mill
• Drill press
• Belt/disk sander
• Grinder
• Power tools
• Waterjet

See Category 2 permission and supervision requirements before use.

Category 3

• Horizontal band saw
• Vertical band saw
• Manual mill
• CNC lathe
• Manual lathe
• Desktop injection molder

See Category 3 permission and supervision requirements before use.

During equipment operation

• Stay alert — maintain focus, and never leave equipment unattended while it is running.
• Follow proper procedures — use equipment only as trained and as intended.

Machine-specific hazards — pinch points, measurement, housekeeping — are covered in Machine Hazards and Housekeeping.

Personal Protective Equipment

Five topics. One of them — eye protection — applies the moment you walk in. The rest are set by what you are doing.

Eyes

Always, in the lab spaces. No exceptions.

Hearing

Prolonged exposure to loud equipment.

Feet

Closed-toe shoes, always.

Respiratory

Any grinding or sanding; hazardous dust or fumes.

Clothing & hair

Always, near anything that rotates.

Foot protection

• Minimum requirement: closed-toe shoes are MANDATORY — no sandals, no flip-flops, no open-toe footwear.

Recommended upgrades

• Slip-resistant soles, for traction on surfaces that may be slippery.
• Puncture-resistant soles, against sharp metal debris.
• Steel-toe or composite-toe protection — recommended, not required.

Avoid

• Canvas shoes or thin-soled footwear.
• High heels or unstable footwear.
• Anything that does not fully enclose your feet.

You will be standing for most of the lab period — choose something supportive and comfortable.

Hair, and what to wear instead

Hair

• Long hair must be secured back and kept away from equipment.
• Use hair ties, clips or caps to prevent entanglement.

Recommended

• Cotton or fire-retardant materials — preferred, not required.
• Close-fitting, short-sleeved shirts.
• Pants without excessive loose material.

Electrical Sources

What hurts you at 5 V is not what hurts you at 240 V; one is a burn hazard while the other is a shock hazard.

5 V / 12 V

The power supply on the bench, batteries, and the battery pack out of a cordless drill. Lower shock risk, not a magic cutoff. Not below the burn threshold.

120 V

Wall outlets. Dangerous — can cause serious injury or death.

240 V / 480 V

Industrial equipment. Extremely dangerous — often fatal.

What is a Safety Data Sheet (SDS)?

A Safety Data Sheet is the manufacturer's written answer to what their product can do to you, what to wear, and what to do when it goes wrong.

There are sixteen sections, always in the same order. You do not need to read a SDS front to back but you should read the following three sections before you use a product.

Section 2

Hazard(s) identification. What the chemical can do to you, in the manufacturer's own words — a signal word, the hazard statements, the red-bordered pictograms.

Section 4

First-aid measures. On skin, in eyes, breathed in, swallowed.

Section 8

Exposure controls / personal protection. Gloves, eye protection, ventilation — the PPE answer for that specific chemical, rather than the general one.

None of this is a local rule. OSHA requires a sheet for every hazardous chemical, and requires the sixteen sections and the order they come in — 29 CFR 1910.1200(g).

Supporting detail: Addendum 3 — All sixteen sections, and the four OSHA does not enforce.

Supporting detail: Addendum 4 — Signal words, pictograms and H-numbers — how Section 2 is written.

Where is SDS binder located?

• All Safety Data Sheets are organized in a designated binder for easy reference.
• Review before use — check the sheet for any unfamiliar chemical or liquid.
• Ask the instructor if you cannot locate the sheet for a material.

The binder is one way of satisfying the rule underneath it. 1910.1200(g)(8) requires an employer to keep the sheets "readily accessible during each work shift to employees when they are in their work area(s)", and allows electronic copies in place of paper "as long as no barriers to immediate employee access in each workplace are created by such options." Binder or screen, the test is the same: the sheet is something you can reach now, not something you have to go and ask for.

Supporting detail: Addendum 5 — Telling a current sheet from an old one.

Material Storage and incompatible materials

• Flammable liquids must be stored in an approved flammables cabinet when not in use.
• NEVER store incompatible chemicals together.
• All containers must be properly labeled.

“Incompatible” is not a judgement call, and answering it needs no chemistry. It is written on the sheet, by name, in two places:

Section 10

Stability and reactivity. One of its required entries is "incompatible materials" — what this chemical must not contact. Beside it sit "conditions to avoid" and "hazardous decomposition products", which is what the two of them would make.

Section 7

Handling and storage. Required to give "conditions for safe storage, including any incompatibilities" — the same question answered for the shelf rather than for the reaction.

So the test before two things share a cabinet: open both sheets and read Section 10 on each. The entry may name a specific substance or a whole class of them — oxidizers, strong acids. If either sheet names the other material, they do not go together. Read both, because the pair may only be called out on one of them.

Fumes and dust have a PPE requirement of their own — see respiratory protection.

Supporting detail: Addendum 6 — What makes a flammables cabinet “approved”, and how to spot one.

The environment around the machine

Slip hazards

Clean dust, metal chips, water and coolant off the floor immediately.

Lighting

Ensure lighting is adequate before operating any machine.

Walkways

Keep aisles and emergency exits unobstructed.

Operating habits

• The chuck key never leaves your hand. It is not released until it is out of the chuck. There is no moment where the key is in the chuck and not in your grip — not to reach for something, not to answer someone, not for a second.
• Stop machines completely before taking measurements or making adjustments.
• Secure all workpieces before operation — use clamps, vises or fixtures.
• Never leave tools on or near operating machinery.

Supporting detail: Addendum 12 — Why the chuck key rule is worded as a grip and not as a reminder.

The shutdown and cleaning steps that close every session are in After operating any equipment.

Using an extinguisher — PASS

Remember the acronym: P–A–S–S.

When it looks out, back away watching the fire, exit behind you. Do not assume it is over — a knocked-down fire can reignite.

Fire classes, and what puts them out

Fires are classed by what is burning. An extinguisher is rated for the classes it can handle, and using the wrong one ranges from useless to dangerous.

Most extinguishers on a wall are ABC dry chemical — one unit covering the first three classes. Check the label; the classes it is rated for are printed on it.

Hot work and the things that start fires

• Grinding throws sparks a surprising distance. Clear rag, sawdust, solvent and packaging out of the area before you start, not after it smolders.
• Oily rags — it depends entirely on which oil. Drying oils — linseed, tung, oil-based stains, varnishes and finishes — cure by reacting with oxygen, and that reaction gives off heat. Bunched in a pile the rags hold the heat in, and the pile can reach ignition on its own with nothing to light it. Those rags NEVER go in a pile: spread them flat to dry, drop them in water, or use the designated container.
• Petroleum oils do not do that. Way oil, cutting fluid, hydraulic oil and ordinary shop oil do not cure by oxidation, so a rag wiped with them will not self-heat and will not spontaneously combust. It is still combustible — a spark or a hot chip will light it like any oily cloth — so keep it away from hot work and out of a bin of paper or sawdust. If you do not know which oil is on the rag, treat it as a drying oil.
• Flammable liquids live in a flammables cabinet when they are not in your hand — see Chemicals and Safety Data Sheets.
• A smoldering fire can restart long after you leave. If something got hot enough to smoke, tell the instructor before you go.

Four reasons a chip is dangerous

A chip is not scrap until it has cooled and stopped moving. Straight off the tool it is every one of these at once.

Hot

Where an aluminum chip forms, the cut runs roughly 250–650 °C (500–1200 °F), depending on the alloy and the speed. Chips do not look it.

Sharp

The leading edge of a chip is always razor sharp. It is a freshly cut edge and it will open a hand.

Strong

A long continuous chip is a steel ribbon. If it wraps a sleeve or a limb it can pull you into the machine.

Slippery

Chips on the floor roll underfoot. This is one of the most common ways people get hurt without the machine touching them.

Staying alert

• Stay focused — maintain concentration while operating equipment.
• Take breaks when tired or distracted.
• Ask questions if you are unsure about any procedure.
• Report unsafe conditions immediately to the instructor.

Workspace responsibility

• Clean as you go — do not wait until the end.
• Report equipment problems before they become safety hazards.

Why 5S sits in a safety module

5S is a lean manufacturing method for organizing a workplace. It earns its place here because most of what it asks for is also what keeps people from getting hurt.

A cluttered bench hides a hazard. An unlabeled container is an unknown chemical. A floor with chips on it is a slip. Every one of those is a 5S failure before it is an incident — which is why the shutdown steps in After operating any equipment are 5S in everything but name.

This page is the basics. 5S will get its own expanded treatment later.

Sort — Seiri

• Separate necessary items from unnecessary ones.
• Remove or relocate items that do not belong in the work area.
• Keep only what you need for the task in front of you.

Set in Order — Seiton

• Organize and label the remaining items clearly, for easy identification.
• A place for everything, and everything in its place.
• Create logical, efficient storage systems.

Shine — Seiso

• Clean work areas to maintain high hygiene standards.
• Reduce accident risk through cleanliness and organization.
• Inspect while cleaning — that is how problems get found early.

This is the step that overlaps most directly with safety — see the environment around the machine.

Standardize — Seiketsu

• Establish consistent procedures for maintaining the first three S’s.
• Create habits through regular practice and routine.
• Ensure everyone follows the same organizational systems.

Sustain — Shitsuke

• Continuously review and maintain all 5S practices.
• Build the discipline to follow established procedures long-term.
• Regular audits, and improvements to the system itself.

A measurement is a number and a unit

A dimension on a drawing, a speed in a robot program, and a torque setting are all quantities: a number paired with a unit. The number tells how many; the unit tells what kind of thing and how large each part is.

25 mm

A length: twenty-five millimeters. Writing only 25 leaves the measurement incomplete.

4.25 min

A CNC cycle time. It is not interchangeable with 4.25 seconds or 4.25 hours.

3.0 m/s

A speed or velocity magnitude. The slash means meters per second.

Use the unit symbol exactly as specified: m for meter, s for second, kg for kilogram. Symbols are not pluralized and do not take a period: 12 mm, not 12 mms.

Two systems you will encounter

The International System of Units (SI) is the standard system for science and much technical work. U.S. drawings, hardware, materials, and equipment also use U.S. customary units. Neither system lets us skip the unit; the job is to state which one is being used and convert deliberately when they meet.

SI base units used here

meter (m) for length, second (s) for time, and kilogram (kg) for mass.

U.S. customary units used here

inch (in), foot (ft), mile (mi), second (s), pound mass (lbm), and U.S. liquid gallon (U.S. gal).

Coherent BG engineering system

The source slides call the coherent engineering form BG (British Gravitational): foot, slug, and second. In that system, slug · ft/s2 = lbf. A pound force and a pound mass are not the same unit.

A gallon must be named. This unit set uses the U.S. liquid gallon (U.S. gal), not an Imperial gallon.

Supporting detail: Addendum 14 — Why technical writing cares about the space.

SI prefixes change scale, not quantity

A prefix is a power of ten attached to a unit. It makes a practical-sized number without changing the physical quantity: 0.025 mm and 25 µm describe the same length.

Case matters. M is mega (10⁶); m is milli (10⁻³). A wrong capital letter changes a result by a billion.

Prefix explorer

Choose two prefixes for the same base unit. The calculation compares their powers of ten; it does not change what is being measured.

Powers apply to the whole conversion

Area and volume are not lengths with decorative superscripts. The conversion factor itself must be squared or cubed.

For example, 4 m2 × (1 ft / 0.3048 m)2 = 43.0556 ft2, which is sensibly reported as 43.1 ft².

For liquid capacity, 1 m3 = 1,000 L and 1 U.S. gal = 3.785411784 L. The converter labels this as a U.S. liquid gallon so it cannot be mistaken for an Imperial gallon.

Supporting detail: Addendum 16 — Dimensional analysis is also a design check.

Conversion bench

This tool uses exact internal factors where standards define them. It only converts within one quantity type; a length cannot become a speed without additional information.

CNC production calculation

A CNC program estimates 4 min 15 s per part. The operator uses 30 min of an 8 h shift for setup. How many complete parts fit in the shift?

1. Convert the cycle: 4 min 15 s = 4.25 min.
2. Find machine-running time: 8 h × (60 min / 1 h) − 30 min = 450 min.
3. Divide running time by cycle time: 450 min / 4.25 min per part = 105.882....

Supporting detail: Addendum 17 — Two more conversion cases from the unit.

Path length and displacement answer different questions

Distance is how much path was traveled. Displacement is the directed change from the starting position to the ending position. A return trip can add distance while leaving displacement at zero.

Distance

A nonnegative scalar path length. A conveyor that moves a tote 12 m out and 12 m back travels 24 m of distance.

Displacement

A change in position with direction. If the tote returns to its start, its displacement is 0 m even though it moved.

Coordinate choice

Choose a positive direction before calculating. Along a straight robot rail, right might be positive and left negative.

Speed and velocity are not synonyms

Average speed is total distance divided by elapsed time. Average velocity is displacement divided by elapsed time. Velocity includes direction; speed does not.

average speed = total distance / elapsed time
average velocity = displacement / elapsed time

For the 12 m out-and-back path completed in 16 s, average speed is 24 m / 16 s = 1.5 m/s. Average velocity is 0 m / 16 s = 0 m/s. Both statements can be true at the same time.

A production-line example

A mechanism must carry a part 12 m in 8 s. Its required average speed is 12 m / 8 s = 1.5 m/s. Converting with exact factors gives 1.5 m/s × (39.37007874 in / 1 m) = 59.055... in/s, or 59.1 in/s. If the programmed direction is stated, its average velocity is +59.1 in/s in that direction.

Average and instantaneous readings

An average uses a time interval. An instantaneous reading describes one moment: a car speedometer reports instantaneous speed; a robot controller’s live velocity value is an instantaneous velocity in its coordinate frame.

• Use average speed for throughput, travel-time, and trip planning.
• Use average velocity for a net movement over a stated interval.
• Use an instantaneous reading when clearance, collision avoidance, or a machine limit depends on what is happening now.

Supporting detail: Addendum 19 — The calculus idea, after the practical one.

Out-and-back path lab

Set positions along a straight path and the outbound and return times. The text results make the distance and displacement distinction visible even when the track is not used.

Acceleration is change in velocity over time

Acceleration is not simply “speeding up.” It is a change in velocity, so it can describe speeding up, slowing down, or turning.

aavg = Δv / Δt = (vf − vi) / (tf − ti)

Δv

Change in velocity. With one chosen positive direction, Δv = vf − vi.

m/s²

Meters per second per second. Each second, velocity changes by the stated number of meters per second.

ft/s²

Feet per second per second. The coherent foot–slug–second system uses this acceleration unit with slugs for mass.

Average is an interval; instantaneous is a moment

Average acceleration is calculated over stated start and end conditions. An accelerometer or control system can report instantaneous acceleration at one moment. They match at every moment only when the acceleration stays constant through the interval.

Machine arm example

A machine arm starts from rest and reaches 3 m/s in 4 s:

aavg = (3 m/s − 0 m/s) / 4 s = 0.75 m/s2.

This is the average acceleration for the four-second move. If the drive ramps gently at the start and end, its instantaneous acceleration changes during the move even though the average remains 0.75 m/s².

Robot gripper example

A SolidWorks motion study programs a gripper from 0 to 5 ft/s in 0.5 s to pick an electronic component. Its average acceleration is 5 ft/s / 0.5 s = 10 ft/s2. In SI, 10 ft/s2 × (0.3048 m / 1 ft) = 3.048 m/s2.

Acceleration lab

Change the initial and final velocities and the elapsed time. The last result converts the calculated foot-per-second-squared value to SI.

Volume is capacity only after you name the unit

CAD mass properties often report volume in a model’s document units. Before treating that number as a tank capacity or a material quantity, state the volume unit and convert it to the form the drawing or specification requires.

Static solved reservoir example

A SolidWorks model gives a hydraulic reservoir internal volume of 2.5 ft³. Its capacity is 2.5 ft3 × 7.48051948 U.S. gal/ft3 = 18.7013 U.S. gal, or 18.7 U.S. gal. This assumes the reported CAD volume is the usable internal volume and no design headspace has yet been subtracted.

Density connects volume to mass

Density is mass per volume, and its symbol is ρ (rho): ρ = m/V. Turned around, when material density and CAD volume use compatible units, m = ρV produces mass.

(kg/m3) × (m3) = kg

For example, a volume of 0.13824 m³ made of material with a density of 2,700 kg/m³ has mass m = 2,700 kg/m3 × 0.13824 m3 = 373.248 kg.

Static solved steel example

A steel I-beam has a volume of 0.2 m³ and a density of 7,850 kg/m³. Its mass is m = 7,850 kg/m3 × 0.2 m3 = 1,570 kg. That is the same mass the weight example below starts from — a CAD volume and a material card are usually where a weight calculation actually begins.

Supporting detail: Addendum 22 — Check the CAD assumptions before trusting mass properties.

Force and weight are separate quantities

Force is a push or pull. For constant mass, Newton’s second law is Fnet = ma. Weight is one particular force: the force due to gravity, written W = mg.

F_net

Net force, in newtons (N) or pound force (lbf). It is the combined force that produces acceleration.

W

Weight, a gravitational force. Near Earth’s standard gravity, g0 = 9.80665 m/s2; local gravity can differ slightly.

m

Mass, in kilograms, pound mass, or slugs. It is not measured in newtons or pound force.

In coherent SI, 1 N = 1 kg · m/s2. In the coherent foot–slug–second system, 1 lbf = 1 slug · ft/s2. Do not silently replace lbm with lbf.

Static solved weight example

For a 1,570 kg mass at standard gravity, W = mg = 1,570 kg × 9.80665 m/s2 = 15,396.44 N, or 15.4 kN. This is a weight calculation at standard gravity, not a claim that the mass is 15.4 kN.

CAD scale, mass, and weight lab

This model begins with an area and volume in SI units, scales them, then uses density and gravitational acceleration to show the separate mass and weight results.

Rotation is measured three ways

Angular displacement is how far something has turned. Its symbol is θ (theta), and there are three units in everyday use for it. They measure the same rotation; they differ only in how much turning counts as one unit.

Revolution (r)

One complete turn. This is what a machine nameplate counts.

Degree (°)

One revolution cut into 360 parts. This is what a drawing and a protractor use.

Radian (rad)

The angle whose arc is exactly as long as the radius. This is the SI unit, and it is the one formulas such as v = ωr expect.

The two factors worth keeping to hand: 1° = 0.01745329 rad and 1 rad = 180°/π = 57.29578°.

The degree sign sits tight against the number — 90°, not 90 °. Every other unit symbol takes a space: 1.57 rad, 45 r/min. The degree, minute, and second of plane angle are the only exceptions to that rule.

Supporting detail: Addendum 24 — Why a radian is barely a unit at all.

Angular acceleration is how fast the turning changes

Angular acceleration uses the symbol α (alpha), and it is change in angular velocity divided by the time it took:

αavg = Δω / Δt

Its SI unit is rad/s2. Everything true of straight-line acceleration on Acceleration and Squared Time is true here as well: it is a change, so a shaft slowing down has angular acceleration opposite its angular velocity (negative if it is turning in the positive direction), and the squared second means a squared factor whenever the time unit changes.

Worked example: a motor speeding up

A motor shaft is turning at 17 rpm. Power to the motor is increased, and 1.1 s later it is turning 8 rpm faster. What is the angular acceleration in rad/s2?

1. Only the change matters, so convert the 8 rpm increase: 8 r/min × 0.1047198 rad/s per r/min = 0.837758 rad/s.
2. Divide by the time: 0.837758 rad/s ÷ 1.1 s = 0.7616 rad/s2.

Worked example: a spindle running up

A machining-centre spindle goes from rest to 8,000 rpm in 2.5 s.

1. 8,000 r/min × 0.1047198 rad/s per r/min = 837.758 rad/s.
2. (837.758 rad/s − 0 rad/s) ÷ 2.5 s = 335.10 rad/s2.

That is the average over the run-up. A drive that ramps gently at each end has a changing instantaneous angular acceleration inside the same interval, exactly as a linear drive does.

Worked example: a conveyor drive roller

A conveyor belt runs over a drive roller 200 mm in diameter, turning at 45 rpm. What is the belt speed, and how long does a package take to travel 15 m?

1. Halve the diameter: r = 100 mm = 0.100 m.
2. Convert the speed: 45 r/min × 0.1047198 rad/s per r/min = 4.712389 rad/s.
3. Multiply: v = 4.712389 rad/s × 0.100 m = 0.4712 m/s.
4. Divide the distance by the speed: 15 m ÷ 0.4712389 m/s = 31.83 s.

Worked example: a roll-forming feed rate

Forming rolls 150 mm in diameter rotate at 25 rpm. What is the feed rate of the sheet in m/min, and how long does 20 m of sheet take?

1. r = 75 mm = 0.075 m.
2. 25 r/min × 0.1047198 rad/s per r/min = 2.617994 rad/s.
3. v = 2.617994 rad/s × 0.075 m = 0.19635 m/s.
4. The question asks for metres per minute, so convert the time unit: 0.19635 m/s × 60 s/min = 11.78 m/min.
5. 20 m ÷ 11.78 m/min = 1.70 min, which is about 1 min 42 s.

Note what step 4 did and did not do. Changing seconds to minutes in a speed takes one factor of 60. If the quantity had been an acceleration, the same change would take 60² — see Acceleration and Squared Time.

Worked example: running the conversion backwards

Sometimes the linear quantity is what you know. The formulas simply rearrange: ω = v/r and α = a/r.

A wheel, from angular to linear

A tyre 28 in in diameter turns at 3.5 rad/s. How fast is the vehicle moving, in miles per hour?

1. r = 14 in × 0.0254 m/in = 0.3556 m.
2. v = 3.5 rad/s × 0.3556 m = 1.2446 m/s.
3. 1.2446 m/s ÷ 0.44704 m/s per mph = 2.78 mph.

A wheel, from linear to angular

A vehicle accelerates at 5 m/s² on tyres 78 cm in diameter. What is the angular acceleration of a wheel?

1. r = 39 cm = 0.39 m.
2. α = 5 m/s2 ÷ 0.39 m = 12.8 rad/s2.

Check the unit on the answer, not just the number. Dividing m/s2 by m leaves 1/s2, and because the radian is a ratio of two lengths, rad/s2 is the same thing named for what it describes.

The unit, and the word order that identifies it

Case matters, and this one bites. The SI torque unit is N·m with a lower-case m. Written N·M it is not a valid SI torque unit: M is the prefix for 10⁶, but a prefix cannot stand alone as a unit. The same trap catches Kg for kg and S for s. See Measurement Language.

Worked example: can the motor hold the arm up?

A robot-arm joint motor produces 15 N·m. The arm segment is 0.35 m long and carries a 25 N load at its end. Will the motor hold the arm horizontal?

1. The load acts straight down, at right angles to a horizontal arm, at the full 0.35 m: τ = 0.35 m × 25 N = 8.75 N·m.
2. Compare with what the motor produces: 8.75 N·m < 15 N·m.

This is the shape of most torque questions: work out what the load demands, then check it against what the drive can supply, and be explicit about what you have left out.

Worked example: torque from a tangential force

A wheel 250 mm in diameter needs 3.5 kN of tangential force to overcome resistance. What torque must the motor supply to the axle?

1. r = 125 mm = 0.125 m.
2. 3.5 kN = 3,500 N.
3. τ = 0.125 m × 3,500 N = 437.5 N·m.

The same formula, rearranged

When the torque is what you know and the force is what you want, F = τ/r. That single rearrangement explains a family of everyday machines: a longer wrench, a bigger steering wheel, and a larger pulley all trade radius against force. Double the radius and the same torque delivers half the force — and, at the same rotational speed, the rim travels twice as far, which is why Power and Efficiency comes out unchanged.

Count the teeth to find the speed

The driver supplies motion; the driven gear receives it. For one external gear mesh, each tooth passing on the driver advances the driven gear by one tooth space. A gear with twice as many teeth needs twice as many passing teeth to complete a turn.

This lesson uses reduction ratio i = input speed ÷ output speed, with speed magnitudes. A 2:1 reduction means two input turns for one output turn. Ratios above 1 reduce speed; ratios below 1 increase it. At zero speed, use tooth counts to find the ratio; do not divide 0 by 0.

Notation map: this page's original equations use N for teeth and n for speed. The course uses n for teeth, ω for angular velocity and τ for torque. Thus page N = course n; page speed n is written ω in the additions below. GR and i name the same reduction ratio.

In course notation, using magnitudes: GR = nout/nin = dout/din = ωin/ωout = τout/τin for an ideal single pair. Here d is pitch diameter; the torque equality assumes no losses. For a compound train, multiply the stage ratios; the first and last tooth counts alone do not give the overall ratio.

Supporting detail: Addendum 30 — Why tooth counts work, and what the dashed circles mean.

An idler and a compound shaft do different jobs

Same shaft means the same angular speed. A large B and a small C fixed to one shaft make the same number of turns in the same direction. Their rim speeds differ because their radii differ. They do not mesh with each other.

In the compound example, 20 → 40 teeth halves the speed. A second 20 → 40 stage halves it again: 120 → 60 → 30 rpm. The two external meshes make the final output turn in the same direction as the input.

An idler is not another reduction stage. Drop its tooth count when finding the end-to-end ratio, but count every external mesh for direction: odd means opposite; even means the same, viewed from the same end of the parallel shafts. Do not drop a gear that shares a shaft with a different gear in a compound train.

More torque costs speed; it does not create power

Torque is turning effort. In this ideal, lossless train at steady load, output torque magnitude = input torque magnitude × reduction ratio. A 2:1 reduction changes 120 rpm and 2 N·m into 60 rpm and 4 N·m. The 4:1 compound example gives 30 rpm and 8 N·m.

Rotational power is torque × angular speed: P = T × 2πn/60 when T is in N·m and n is in rpm. Both sides of either example carry 25.13 W ideally. Real friction and other losses leave less output power. Tooth strength, bearings, shafts, lubrication and the motor all limit usable torque.

At zero speed, mechanical power is zero. A loaded, stationary gear train can still carry torque. The lab shows ideal static torque balance in that case; it does not model motor stall behavior or promise a safe load.

Supporting detail: Addendum 31 — Primary references and model boundaries.

Connect the gears to Angular Motion, Rotation and Linear Speed and Torque.

Stage efficiencies multiply; losses accumulate

Each stage receives only the power that survived the stage before it. Use decimal efficiencies and multiply: η_total = η₁ × η₂ × η₃. Never add or average them. See Power and Efficiency for the general meaning of efficiency.

At 98% per stage, three stages give 0.98³ = 0.941192: about 5.88% total loss, not 6%. Fewer, higher-quality stages often beat many cheap stages for efficiency, provided they can meet the ratio and load requirements.

Supporting detail: Addendum 32 — Why adding losses is only an approximation.

Positive drives keep the speed ratio; losses reduce torque

A positive drive keeps input and output linked in a fixed ratio without slipping. Gears, chains and timing belts are positive drives. A slipping flat belt or V-belt is not. Follow Rotation and Linear Speed for the connection from rotation to belt travel.

The teeth still set the speed exactly in this model. Efficiency reduces output torque from its ideal value. It does not reduce output rpm as well. Use matching torque units on both sides: lbf·ft or N·m. The course conversion is 1 rpm ≈ 0.1047 rad/s when angular speed in rad/s is needed.

Supporting detail: Addendum 33 — Follow both stages of one loaded train.

Count stages for ratio, meshes for direction and stated losses

For a simple train A → I → B, the idler cancels from (NI/NA) × (NB/NI) = NB/NA. The two external meshes still reverse direction twice.

Follow the loss assumption in the problem. Real meshes lose some power, but a problem may explicitly ignore it. If every actual mesh is 96% efficient, A → I → B has ηtotal = 0.96². If a whole stage including its idler is assigned 96%, use that stage value once. If η = 1.00 is stated, ignore losses. Do not infer the loss count from the reduction-stage count when an idler is present.

Work is a force moving something through a distance

Holding a heavy box still is exhausting, and in this sense it is zero work. Work needs a component of force along the displacement; work is negative when that component opposes the movement.

W = Fs  —  work equals force times displacement for a constant force parallel to the motion.

Work is one of the ways energy moves from one place to another, so it is measured in energy units and is often written E when what matters is the energy delivered rather than the act of delivering it.

Worked example: lifting a load

How much work is required to lift a 2,000 lb load to a height of 10 ft?

1. The force needed to lift steadily is the load’s weight, 2,000 lbf, and it acts straight up along the 10 ft of travel.
2. W = 2,000 lbf × 10 ft = 20,000 ft·lbf.
3. In SI: 20,000 ft·lbf × 1.355818 J per ft·lbf = 27,116 J, or 27.1 kJ.

Two things this calculation assumes, both worth saying out loud: the load is raised at a steady speed, so no extra force goes into accelerating it, and the 2,000 lb figure is a weight in pound-force, not a mass in pound-mass. On Geometry, Mass, and Force those are separate quantities, and here it is the force that belongs in the formula.

The energy units you will actually meet

Energy is billed, rated, and specified in a lot of different units. They all convert; the only work is knowing which one a number is in.

“Btu” on its own is ambiguous. NIST lists six different British thermal units, from 1,054.35 J to 1,059.67 J depending on which definition is meant. The spread is about half a percent, which is small until it is multiplied by a fuel bill. Use BtuIT unless a specification names another one.

Power is how fast work gets done

Two hoists that lift the same load to the same height do the same work. The one that does it in half the time needs twice the power.

P = W / t  —  average power equals work divided by elapsed time.

The symbol W does double duty: upright W after a number is the watt, while W standing in a formula for a quantity is usually work. Context separates them: P = W/t uses W for work, while 1 W = 1 J/s uses it for the watt.

Worked example: powering the lift

On Work and Energy, lifting 2,000 lbf through 10 ft took 20,000 ft·lbf, or 27.1 kJ. How much power does doing it in 4 s require?

1. In U.S. units: 20,000 ft·lbf ÷ 4 s = 5,000 ft·lbf/s, and 5,000 ÷ 550 = 9.09 hp.
2. In SI: 27,116 J ÷ 4 s = 6,779 W, or 6.78 kW.
3. Cross-check the two: 9.0909 hp × 745.6999 W/hp = 6,779 W. They agree, which is the point of doing both.

That is the power delivered to the load. A motor sized for this job needs more than that, because no drive is perfectly efficient.

Two shortcut formulas, and the conditions attached to them

These two save a lot of arithmetic in practice. Both are unit conversions with an assumption hidden inside.

From volts and amps

hp = (volts × amps) ÷ 746

The 746 is exact: NIST defines a separate unit, horsepower (electric), as 746 W. So the formula is really two steps — find the electrical power in watts, then convert watts to horsepower.

From torque and rpm

hp = (torque in lbf·in × speed in rpm) ÷ 63,025

This one is the P = τω row above, with the rpm-to-rad/s conversion and the inch-to-foot conversion already folded into the constant. It only works with the torque in pound-force inches and the speed in rpm; feed it lbf·ft and the answer is twelve times too small.

Supporting detail: Addendum 37 — Where 63,025 comes from.

Three ways to carry rotation from one shaft to another

A drive transfers motion and turning effort from a driver to a driven shaft. Belts and chains bridge a gap. Gears engage directly and fit into a compact housing. Start with the job the drive must do, then choose the parts.

Gears, chains and timing belts are positive drives: engagement maintains a tooth-count ratio without ordinary frictional slip. Flat and V-belts are friction drives: their grip depends on friction and tension, so slip can change the speed ratio.

Positive engagement does not mean perfect positioning. Tooth clearance, elastic deflection and chain speed ripple still matter. For ratios, compound trains and losses, use Gear Ratios. For turning effort and power, use Torque and Power and Efficiency.

Trace the whole torque path, including where each pulley, sprocket or gear meets its shaft. Keys and keyways are one way to connect a hub to a shaft; they are not a feature of every drive. Bearings have a different job: supporting and locating the shaft relative to its housing.

Compare timing, space and upkeep side by side

Belt contact running dry does not mean the shaft bearings need no lubrication. A sealed gearbox can keep lubricant away from a product; an exposed chain generally cannot. Special materials and lubrication-free products need their own ratings.

Read the parts in detail: Belt Drives, Chain Drives and Gear Drives and Tooth Geometry.

A key joins the shaft slot to the hub slot to transmit torque

A key is a separate piece fitted between a shaft and the bore of a hub, the part of a pulley, sprocket or gear that fits around the shaft. Contact between the key and slot faces transfers turning force. Here, keyseat means the recess in the shaft and keyway means the matching slot in the hub. Other references sometimes use keyway for both; always identify which part is being cut.

Trace the turning load through the connection: shaft slot face → key → hub slot face, or the reverse when the hub drives the shaft. At the rim, the pulley, sprocket or gear transfers the load through its belt, chain or mating gear teeth. Not every hub is keyed; check how the actual assembly connects the hub to the shaft.

Identify the key by its cross-section and seating shape

Supporting detail: Addendum 40 — Public reference for key forms.

Practice identifying keys

Check key fit and axial retention as separate requirements

For a parallel key, the side fit controls how the turning load is carried. Excessive side clearance permits movement; an overly tight fit can prevent assembly. Verify the specified top clearance too. A tapered key has a different seating action, so do not apply a parallel-key fit rule to it.

Use the drawing's key designation, dimensions and tolerances. Do not choose a key size from shaft diameter alone or assume that half the key height belongs in each part.

The key does not by itself define how the hub is held against movement along the shaft. Check the assembly drawing for axial retention as well as torque transmission. A hub that turns with the shaft is not necessarily secured against sliding along it.

Bearings support the shaft relative to the housing; the keyed hub transfers torque between the shaft and the attached rotating part. For the bearing arrangement and the shaft's axial location, see Shaft Support and Axial Adjustment.

To make and inspect the recesses, continue with Machining Keyseats and Keyways. That lesson connects shaft milling to broaching a through hub keyway.

Flat and V-belts grip; timing belts engage teeth

Flat belt

A broad, flat band transmits force through friction against a pulley. Useful for smooth, quiet motion and long spans when the drive is designed for them. Tracking and tension matter.

V-belt

A trapezoidal section runs in a matching V-groove. Contact at the sides gives a wedging action that increases grip. The belt should not rely on rubbing the bottom of the groove.

Timing / synchronous belt

Teeth on the belt engage matching grooves in the pulleys. Load is carried by this engagement, so the drive keeps its tooth-count ratio without frictional slip in normal operation.

Timing does not come from extra tension. It comes from matching teeth. An undertensioned or overloaded timing belt can jump teeth or strip them; that is a fault, not its normal way of transmitting power. A notched V-belt remains a friction belt: relief notches for bending do not make it a timing belt.

Belt contact transfers load at the pulley rim; the pulley still needs a connection to its shaft. For a keyed pulley, follow the torque path through the key and hub. Other pulley mountings use different connections.

Read length, tension and wrap before ordering

Pulley / sheave

The wheel that carries a belt. “Sheave” is commonly used for a grooved belt pulley. A timing pulley has matching tooth grooves.

Pitch and pitch length

Timing-belt pitch is the spacing between adjacent tooth centers along the pitch line. Pitch length is the loop length along that reference line, near the tensile cords. It is not the outside circumference. Match pitch and tooth profile to the pulley.

Tension

The pulling force along the belt. The difference between the loaded and return spans transmits torque. Installation tension keeps the belt seated; too little encourages slip or tooth jumping, and too much loads belts and bearings.

Wrap angle

The angle through which the belt contacts a pulley. Less wrap gives a friction belt less contact and a timing belt fewer teeth in engagement. Check the smaller pulley in particular.

Catalogs may specify inside, outside, datum, effective or pitch lengths. Use the length convention and measuring method for that belt family; do not substitute a tape measurement of the outside for the catalog value.

One-stage reminder: n_out = n_in × D_driver / D_driven using matching effective/pitch diameters and assuming no slip. For timing belts, use tooth counts N in place of diameters. Write the reduction as i:1, where i = input speed / output speed. Follow Gear Ratios for the ratio method and Rotation and Linear Speed for belt linear speed.

Here N means teeth and n means rotational speed. Some class slides use n for teeth and ω for angular speed; check the quantity, not just the letter.

Choose a belt for the load and environment

Belts can span separated shafts, soften vibration and avoid oil at the transmission contact. Flat and V-belts suit jobs where a small speed variation is acceptable. Choose a timing belt when the shafts must stay synchronized. Compare the alternatives in Belt, Chain and Gear Drives.

Check belt section or tooth profile, width, minimum pulley size, available length, wrap, power rating, temperature and chemical compatibility. There is no single maximum speed or temperature range for every belt. Reinforcement cords limit stretch, but the belt still needs correct alignment and installation tension.

Some specially designed flat-belt layouts connect nonparallel shafts. That does not make misalignment acceptable in an ordinary V-belt or timing-belt installation. Frictional slip may soften an overload, but it can also generate heat and destroy the belt; it is not a guaranteed overload-protection device.

Use the wear pattern to find the cause

Inspect the pulleys as well as the belt. Replacing the worn part without fixing its cause often repeats the failure. Follow the product's tension and inspection instructions; use Equipment Operating Procedure before touching the drive.

Practice a timing-pulley speed

Supporting detail: Addendum 41 — Manufacturer references for belt geometry and faults.

Sprocket teeth engage a chain of moving joints

A chain drive is a positive drive: sprocket teeth engage the chain instead of relying on frictional grip. It can bridge separated shafts and carry substantial load, but its joints and sprockets need attention.

Pitch, p

The center-to-center distance between adjacent chain pins. A chain's pitch must match its sprocket.

Roller

A sleeve that turns around a bush as it contacts a sprocket tooth. Some small chains are bush chains and have no separate roller.

Link

A connecting unit of the chain. Ordinary roller chain alternates inner links, carrying bushes and rollers, with outer links, carrying pins and side plates.

Sprocket

The toothed wheel that engages the chain. Tooth count sets the speed relationship; tooth shape and width must suit the chain series.

Master / connecting link

The link used to join the ends into a loop. Use the specified type and installation method; retention and strength depend on the product.

The chain engages the sprocket teeth, but torque must also pass between each driven or driving sprocket and its shaft. A keyed sprocket uses a key between the shaft and hub; tooth engagement alone does not explain that connection.

Read #25, #35 and #40 as pitch codes

For the common ANSI short-pitch chain numbers here, remove the final digit of the base number and divide the remaining number by 8 to get pitch in inches. Thus #40 uses 4/8 in, not 40/8 in. #100 uses 10/8 in.

A suffix such as -1 or -2 counts parallel strands: simplex or duplex. #40-2 still has a 0.500 in pitch. Heavy-series suffixes and manufacturer prefixes carry other information; do not apply the pitch shortcut to every chain product code.

Supporting detail: Addendum 43 — Checked public wording behind the ANSI numbers.

“Metric chain” still needs a series and a pitch

ISO chain data are commonly tabulated in millimeters. That does not make every pitch a whole number of millimeters. ISO 606 covers both A-series chains derived from American practice and B-series chains derived from European practice.

The common 06B, 08B and 10B sizes correspond to 6/16, 8/16 and 10/16 in pitch. Use the table for the exact designation: small sizes such as 05B do not follow that shortcut. B identifies the series; -1 identifies one strand.

Same pitch does not mean interchangeable. #40 and 08B both have 12.7 mm pitch, but roller diameter and inner width differ. Check series, pitch, width, roller diameter and strand count when matching a sprocket or replacement link.

Supporting detail: Addendum 44 — ISO scope and the source of the millimeter dimensions.

Positive engagement still permits wear and speed ripple

Chain is worth considering when separated shafts need a fixed average ratio and lubricant and noise are acceptable. Select it for the actual power, speed, starts, shock loads and environment. Tensile breaking strength is not an allowable continuous working load.

One-stage reminder: n_out = n_in × N_driver / N_driven; reduction i = N_driven / N_driver, written i:1. N counts teeth and n is rotational speed. For the reasoning and further stages, use Gear Ratios.

The chain wraps a sprocket as a series of straight links, forming a polygon rather than a perfectly smooth circle. This chordal action produces speed ripple, especially with small sprockets. Backlash and elastic stretch also limit precise positioning, even without ordinary slip.

• Wear elongation: pin/bush wear increases joint clearance and measured chain length. The worn chain rides poorly on sprocket teeth and can jump. Take-up removes slack; it does not restore worn joints.
• Stiff or seized joints: check lubrication reaching the joints, contamination and corrosion.
• Fractured plates or pins: investigate fatigue, overload, misalignment and connecting-link installation.
• Hooked or uneven sprocket teeth: inspect the chain and sprockets together and follow replacement limits for that drive.

Use the specified lubricant, alignment and slack. A chain is not installed taut like a belt. Set inspection intervals and wear limits from the maker's guide; isolate the drive before measuring or servicing it. See Equipment Operating Procedure.

Practice reading chain pitch

Supporting detail: Addendum 45 — Chain selection and maintenance reference.

Spur teeth engage straight across; helical teeth engage progressively

Gears transmit rotation by direct tooth contact. They make a compact positive drive, but accurate mounting, tooth compatibility and lubrication matter. This page compares external spur and helical gears on parallel shafts.

Spur gear

Teeth run parallel to the shaft. It is comparatively simple to make and produces no axial thrust from the ideal tooth mesh. Engagement can be noisy at higher speed.

Helical gear

Teeth run at a helix angle around the gear. Contact enters progressively across the face and overlapping teeth share load. This generally gives smoother, quieter operation, with added manufacturing cost.

The inclined helical tooth force also pushes along the shaft. That axial thrust needs a suitable bearing arrangement. A spur gear still loads its bearings radially; “no axial thrust” does not mean “no bearing force.”

The driver supplies power. The pinion is the smaller gear in a pair; it can be the driver or the driven member. An idler passes motion between gears and still needs a shaft or other bearing support. Follow Gear Ratios for direction, tooth-count ratios and compound trains.

Tooth contact and shaft attachment are separate connections. Where a gear hub is keyed to its shaft, the key transfers torque at the hub. Check axial retention separately, especially when the tooth mesh produces thrust.

Measure teeth from the pitch circle

Pitch diameter, d

The diameter of the reference pitch circle. In a standard unshifted pair at its standard center distance, the pitch circles roll together without slipping.

Module, m

Metric tooth size: m = d/N, with d in millimeters and N teeth. Larger module means larger teeth.

Diametral pitch, P_d

Inch-based tooth size: P_d = N/d, with d in inches. More teeth per inch of pitch diameter means smaller teeth. Do not confuse it with circular pitch, the distance along the pitch circle between successive teeth.

Pressure angle, α

The angle between the line of action (the contact-force direction, neglecting friction) and the common tangent to the pitch circles at the pitch point. It describes tooth geometry, not the helix angle.

Addendum / dedendum

Radial tooth height from pitch circle to tip / radial depth from pitch circle to root. Neither is a diameter difference by itself.

Backlash

Clearance between mating tooth flanks. It allows clearance in the mesh but creates lost motion when the load reverses.

Supporting detail: Addendum 47 — Check the size definitions with one gear.

Matching pitch is necessary, but not sufficient

Ordinary spur gears intended to mesh must share module or diametral pitch and pressure angle. Matching outside diameter or tooth count does not establish compatibility. Check the tooth system, center distance, clearance and manufacturer's mating information too.

For external helical gears on parallel shafts, match tooth size, pressure angle and helix angle. Use one right-hand and one left-hand gear (teeth slanting in opposite directions), and confirm the manufacturer's mating specification. Equal tooth counts are not required.

Supporting detail: Addendum 48 — Normal and transverse measurements use different planes.

A larger tooth size can support more load when other conditions are comparable, but pitch alone cannot determine capacity, backlash or achievable reduction. Material, face width, heat treatment, alignment and speed remain part of the design.

Supporting detail: Addendum 49 — Primary references for the geometry and helical mesh.

Gear damage tells you where the load went

Choose gears for compact packaging, a positive ratio and a supported shaft arrangement. Choose helical teeth when smoothness and noise justify their cost and thrust bearings. Check tooth bending strength, surface durability, speed, lubricant, seals and bearing loads before approving a pair.

A symptom is a clue, not a unique diagnosis. Correct the cause before replacing parts. Most loaded metal gear sets need a controlled lubricant supply; some specially selected materials can run dry. Follow the product data and Equipment Operating Procedure for access.

Compare family trade-offs in Belt, Chain and Gear Drives, and follow Power and Efficiency for power losses. A positive drive is not automatically lossless.

A fastener holds parts together; a power screw moves a load

A thread is a ridge wrapped around a cylinder. A matching internal thread in a nut follows that ridge. Relative rotation can produce straight-line travel along the screw's axis.

A machine screw usually holds parts together. A lead screw uses the thread to position or drive a load: a slide, a vise jaw, or a linear actuator. Belts, chains and gears commonly transfer rotation between shafts; a lead screw converts rotation into linear motion. See Lead Screws for the calculations.

Keep four descriptions separate: form is the thread's cross-sectional shape; series selects diameter and pitch combinations; hand determines direction; starts count the separate helices. A helix is the spiral path around the cylinder.

Match the thread form to fastening, repeated travel, or one-way thrust

No thread form wins every comparison. Actual strength also depends on diameter, pitch, engaged length, material and the failure being checked. ACME is not automatically stronger than every V thread. Nor does its name guarantee a larger lead. ACME and metric trapezoidal are related forms, not interchangeable parts.

Supporting detail: Addendum 50 — Why the efficiency and strength comparisons need conditions.

No radial wedging (outward force on the nut) does not mean no wear

In the ideal square-thread geometry, axial load does not create the outward wedging associated with angled flanks. That reduces a source of radial loading on the nut. The loaded surfaces still slide against each other. Friction, lubrication, materials, load and speed still affect wear.

Wear can increase backlash: clearance that must be taken up after reversing direction before the load moves. An adjustable nut can reduce that clearance; adjustment does not restore lost material or guarantee longer life. Do not infer nut life from the thread profile alone.

Supporting detail: Addendum 51 — Where wear and operating limits enter the selection.

Inch callouts count threads; metric callouts state pitch

Pitch is the axial distance between adjacent thread crests. Inch fastener callouts commonly give threads per inch (TPI), so pitch = 1 / TPI in inches. Metric callouts give pitch directly in millimetres.

Fine means a smaller pitch at the same nominal diameter. It does not mean a better-quality thread. Metric threads also have coarse and fine pitch choices. The 6 in M6 is a diameter, not a pitch; the 20 in 1/4-20 is a count per inch, not a lead.

These are basic size-and-pitch callouts, not complete manufacturing specifications. Fit and tolerance information may follow them. A fastener-series label does not tell you the efficiency of a lead screw; for a multiple-start drive, obtain both pitch and starts, or the lead directly.

Supporting detail: Addendum 52 — The exact standard entries used here.

Practice reading an inch callout

Right-hand and left-hand threads advance in opposite directions

Fix the nut and look along the screw from the end you turn. Clockwise turns a right-hand screw away from you into the nut. The same clockwise rotation moves a left-hand screw toward you. Reverse the rotation and the travel reverses.

If the screw is held against axial motion and the nut travels instead, the nut moves opposite to the screw travel just described. Handedness changes the direction, not the magnitude of travel per revolution.

Supporting detail: Addendum 53 — Direction reference.

Next, distinguish pitch from lead in Lead Screws.

Pitch is thread spacing; lead is travel per revolution

Lead = pitch × number of starts. This is the distinction to get right before calculating either speed or force.

A start is one continuous helical ridge. A single-start screw has one ridge, so its lead equals its pitch. A triple-start screw has three interleaved ridges: follow one ridge through a full turn and it advances three pitch distances.

At the same pitch, adding starts increases travel per turn. At the same torque it reduces theoretical force. The thread's cross-sectional form and its number of starts are separate choices; see Screw Threads.

Supporting detail: Addendum 54 — Check the definition, not just the label on a catalog.

Multiply rpm by lead to find linear speed

For an engaged screw and nut, with the nut restrained from rotating, v = nL. Here n is screw speed in revolutions per minute and L is lead in distance per revolution. The result is distance per minute. Divide by 60 for distance per second.

Do not multiply that speed by efficiency. The teaching rule is: “RPM → velocity is 100% efficient unless the screw is malfunctioning; all efficiency losses are found in the torque → force conversion.” This describes a positive-drive motion relationship, not 100% energy efficiency. It is the same tooth-count principle used in Gear Ratios.

Use the actual screw rpm. If the motor slows under load, the lower rpm gives a lower travel speed. Backlash at reversal and elastic deflection can affect positioning without being an efficiency factor on speed. These calculations describe steady engaged travel, after clearance is taken up. See Rotation and Linear Speed for the general rotation-to-travel connection.

Divide torque times 2π by lead, then apply efficiency

For steady motion driving a resisting axial load, Ftheoretical = 2πT / L and Factual = η × Ftheoretical. Here T is input torque and η (eta) is efficiency written as a decimal: 40% is 0.40.

To find the torque required for a target actual force, rearrange: T = FL / (2πη). For the same load and lead, lower efficiency requires more torque. Keep the efficiency boundary consistent: a whole screw assembly efficiency includes losses in the bearings or collar (a supporting thrust-contact surface) as well as the threads.

Supporting detail: Addendum 55 — Derive the 2π from equal power, or one revolution of work.

Worked actuator: keep inch torque and inch lead together

A single-start ACME actuator has 0.20 in pitch, runs at 900 rpm and receives 12 lbf·in of torque. Its assembly efficiency is given as 60%. Find travel speed and useful axial force.

1. L = 0.20 × 1 = 0.20 in/rev.
2. v = 900 × 0.20 = 180 in/min = 3 in/s.
3. Ftheoretical = 2π × 12 / 0.20 = 376.99 lbf.
4. Factual = 376.99 × 0.60 ≈ 226 lbf.

The 60% belongs in step 4, not step 2. It is an example input, not an efficiency assigned to every ACME screw.

Practice actuator force

Worked vise: turn hand force into torque before finding clamp force

For an illustrative square-thread vise, a 120 N hand force acts perpendicular to the handle at a radius of 0.18 m from the screw axis. Pitch is 4 mm with one start. Assume a given overall tightening efficiency of 30%; this is not a measured or universal vise efficiency.

1. Use torque = radius × perpendicular force: T = 0.18 × 120 = 21.6 N·m. Radius is the distance from the axis to the hand, not necessarily the full handle length.
2. L = 4 mm/rev = 0.004 m/rev.
3. Factual = 0.30 × 2π × 21.6 / 0.004 ≈ 10,179 N = 10.18 kN.

This is a tightening estimate using the supplied efficiency. It is not a vise rating or a prediction of residual force after the hand stops turning. A separate free-travel calculation at 24 rpm gives 24 × 4 / 60 = 1.6 mm/s; the jaws do not continue at that speed through a rigid clamped part.

Practice handle torque and clamp force Practice a triple-start CNC screw

The dial counts screw rotation; the indicator finds the carriage

A handwheel dial does not measure the carriage. It counts how far the screw has turned and multiplies by the lead. That is the carriage's travel only while the screw keeps pushing the nut in one direction.

A screw and its nut need some clearance between their thread flanks (a flank is the sloping side of a thread). After the screw changes direction, it must turn through that clearance before its other flank meets the nut and the carriage moves again. That clearance is backlash, the same idea defined for threads in Screw Threads and for gear teeth in Gear Drives and Tooth Geometry. The turn spent crossing it is lost motion: input that produces no output.

• Dial reading: travel predicted from screw rotation, turns × lead. See Lead Screws.
• Carriage position: where the carriage really is, as an indicator touching it would show.
• The difference is zero after a forward approach (the direction the dial was zeroed in) and one full clearance after a reverse approach.

Supporting detail: Addendum 56 — Where the definition comes from.

Predict the dead band: clearance divided by lead, as a fraction of a turn

Once the screw reverses, the carriage stays still until the screw's helix has moved one full clearance B. One turn moves the helix one lead L, so the dead band is B ÷ L of a turn.

The carriage error after a reverse approach is the same 0.20 mm in both rows. A finer lead only spreads that same clearance over more of the handwheel turn.

Land on a dimension: finish every approach in the direction the dial was set

If you overshoot a target, turning straight back to the number leaves the carriage one clearance away from it, because the screw is now pushing on the other flank. Instead:

1. Back the handwheel past the target by more than the backlash.
2. Turn in the original direction again and stop on the number.

The same reasoning applies when setting a dial. MIT's 2.670 milling notes say to “be certain that the backlash in the mechanism driving the table is taken up prior to setting the dial indicator”. A dial set with the clearance taken up in one direction reads true for approaches in that direction. For edge finding, Mill Indicators and Part Zero repeats the touch from the same direction for the same reason.

Supporting detail: Addendum 57 — A readout that measures the slide, and when the clearance still matters.

What the lab model includes and leaves out

• Included: a rigid screw that turns but never slides along its axis, a nut fixed to the carriage, and a total axial clearance B. The carriage position c stays within B/2 of the screw's helical phase s = −Lφ/360° (right-hand thread, φ the screw rotation); inside that band it does not move.
• Assumed: friction holds the carriage still inside the clearance, and no outside force pushes it along the axis.
• Left out: forces, friction values, stiffness and elastic wind-up, preload, stick-slip and wear. Drives designed to remove the clearance, such as preloaded ball nuts or adjustable split nuts, are beyond this model.
• Illustrative: the leads, clearances and the 3D screw (5 mm lead, 1.0 mm clearance, 30° flanks) are chosen to be seen, not taken from a standard thread or a measured machine.

For drive design in depth, Alexander Slocum's Precision Machine Design (1992) covers actuators and transmissions in Chapter 10, including section 10.8, Linear Power Transmission Elements.

A bearing allows useful motion and restrains unwanted motion

A loose rigid part can move in six independent ways: translate along three axes and rotate about those three axes. These are its degrees of freedom. A bearing transfers load to a support while allowing the motion the machine needs, with low friction.

For a rotating shaft, the goal is usually one remaining freedom: rotation about its own axis. Two spaced radial supports resist sideways movement and tilting; axial location stops the shaft walking lengthwise. A single short bearing does not automatically provide that whole constraint system.

Bearings guide motion; motors and drives supply it. Belt, chain and gear drives load their supporting shafts. See Torque for turning effort and Shaft Support and Axial Adjustment for the complete support arrangement.

Radial load crosses the shaft; thrust acts along it

Radial load acts perpendicular to the shaft axis. Axial load, or thrust, acts parallel to it. Combined load has both components. A bearing's orientation and internal geometry determine which it can carry.

Do not turn “roller” into “radial only,” or “thrust” into a rule for every product bearing that name. Tapered rollers carry combined loads; some spherical roller thrust designs also accept radial load under specified conditions.

Plain and rolling bearings trade friction, upkeep and cost

A plain bearing supports sliding surfaces, often with a replaceable sleeve called a bushing. A rolling-element bearing places balls or rollers between rings. “Anti-friction” is a traditional name, not a promise of zero friction.

Rolling contact generally reduces resistance compared with sliding contact. It does not guarantee higher efficiency than every fluid-film bearing, nor does a cheap bushing guarantee lower lifetime cost. Include lubrication equipment, replacement, downtime and power loss. Bearing losses contribute to Power and Efficiency and the stage efficiencies discussed in Gear Ratios.

Plain bearings differ in both lubrication and load direction

Dry or self-lubricating

A suitable polymer, composite or lubricant-containing material supplies a low-friction sliding surface. Follow that material's instructions; adding grease is not automatically an improvement.

Hydrodynamic

Relative motion draws lubricant into a converging gap, forming a pressure-supported film. Starts and stops require attention because the running film depends on motion.

Hydrostatic

An external pressure supply supports the load on a fluid film, including at rest while the supply operates. The pump, flow control and loss-of-pressure behavior are part of the system.

Those are lubrication descriptions. Shape answers a different question: a journal surrounds a shaft to carry radial load; a footstep supports axial load at a shaft end; a thrust collar presents an annular face to carry axial load. An assembly can combine a journal and a collar to handle both components.

Rings carry the load; the cage spaces the rolling elements

The inner ring fits around the shaft; the outer ring fits in the housing. The rolling elements transfer load between their raceways. The cage maintains spacing and guides the elements.

In a common arrangement the inner ring turns with the shaft while the outer ring stays still. The balls spin about their own axes and orbit around the shaft. The rings do not have to rotate in opposite directions.

Balls have idealized point contact; rollers have idealized line contact. Under load these become finite contact patches. The longer roller contact generally supports more load in a similar overall size. Contact angle, not just element shape, determines thrust capability. Thrust ball and roller bearings place their contact geometry to support axial force.

Supporting detail: Addendum 59 — Manufacturer reading for bearing selection.

Practice identifying the load

A pillow block is a housing arrangement, not another rolling-element shape

A pillow or plummer block supports a shaft from a base. A flange-mounted housing attaches around the shaft to a facing support. Different housings can contain different bearing families. For example, NSK Self-Lube units combine a sealed ball-bearing insert with a pillow-block or flanged housing.

These housings support rotation. For bearings that guide a carriage along a straight path, see Linear Bearing Guides, and see Shaft Support and Axial Adjustment for shaft location and support arrangements.

Read the basic designation before the extra letters

A bearing number is a compact description, not a universal sizing formula. Start with the maker and the bearing family. A basic designation identifies a series and bore; prefixes and suffixes add details. The same letters can mean different things in different families or manufacturers' catalogs.

For the ordinary metric deep-groove example 6205, read 6 | (0)2 | 05: type 6 is deep-groove ball; 02 is the dimension series, with width-series 0 omitted in the written number; bore code 05 means 25 mm. The diameter-series digit 2 is not an outside diameter in millimetres.

Width-series digits can be omitted differently: NSK lists 6005 as 6 | (1)0 | 05, and 6305 as 6 | (0)3 | 05. Parentheses here explain the omitted digit; they are not characters to add to an order. See NSK's bearing-series table.

Recognize the geometry first in Bearing Types and Loads. A matching bore alone does not make two bearings interchangeable.

00, 01, 02 and 03 are the bore-code exceptions

Thus 6202 has a 15 mm bore, not 10 mm; 6205 has a 25 mm bore. This rule supplies only the bore. It does not calculate D or B.

Do not apply it blindly to miniature, inch, letter-coded or special bearings. Slash forms can state the bore directly: NSK gives 63/22 as 22 mm and 230/560 as 560 mm. Its miniature 688 example has an 8 mm bore. NSK excludes double-direction thrust ball bearings from its general 04–96 rule.

Supporting detail: Addendum 60 — Primary references for the bore-code scope.

Series identify a family; the catalog supplies the envelope

A dimension series selects a standardized size family; it is a table key, not a digit-to-millimetres formula. Series examples follow NSK's designation guide. Different types also use different dimension letters. A tapered bearing distinguishes inner-ring width B, outer-ring width C and assembled width T. Thrust catalogs may label axial height T or H: use that manufacturer's diagram rather than substituting B automatically.

The 15 mm inner-ring width of HR30205J is not its 16.25 mm assembled width. These examples are not added to the three-row deep-groove viewer above. NSK's boundary-dimension diagrams also distinguish radial width from thrust height.

C3 means internal clearance, not accuracy or a loose shaft fit

Internal clearance is movement between a bearing's rings under the specified measuring conditions. A fit is the relationship between a ring and its shaft or housing seat. A precision class specifies dimensional and running accuracy. These are different quantities; see NSK's explanation of fits and clearance.

Prefixes matter too: NSK uses HR for its high-capacity tapered roller series. Read the complete maker-specific designation, including cage and lubrication details where required. Seals and shields affect exclusion of contaminants, friction and speed limits; they do not replace the catalog's load ratings. Sources: NSK TI/E/0111, auxiliary-symbol tables; NSK Bearing Designation Systems, sections 2 and 3.

For load direction and the existing basic rating-life calculation, return to Bearing Types and Loads. For how the bearing is located on a machine, see Shaft Support and Axial Adjustment.

Locate the shaft at one end and allow growth at the other

In a locating/non-locating arrangement, also called fixed/floating here, both supports carry radial load. The locating support also carries the applied axial load and establishes the shaft's axial position. The non-locating support allows axial movement as the shaft expands relative to the housing.

Location needs a load path from shaft to bearing to housing: shoulders, suitable spacers, nuts, retaining rings or covers can retain the rings. Which end is fixed follows the required position of the driven parts; it is not automatically the left end or the motor end.

The floating end can accommodate movement inside a suitable bearing, such as an NU cylindrical roller bearing, or by controlled sliding of a ring in its seat. Do not loosen an arbitrary fit: see Tolerances and Fits and the bearing maker's mounting guidance.

Supporting detail: Addendum 61 — Why two locating supports can create unwanted load.

Keep shaft support separate from torque transmission. Bearings support the shaft relative to the housing; a keyed shaft-hub connection transfers torque to or from a pulley, sprocket or gear. Securing a hub against sliding along the shaft is a separate assembly requirement.

End play is free movement; preload is an initial internal load

End play is the axial free movement available before the bearing arrangement takes up its clearance. Preload is a deliberately applied internal load before the external working load arrives. Reducing end play to a specified value and applying preload are different adjustments.

Do not tighten a bearing until it merely “feels solid.” Excess preload increases friction and temperature. Some arrangements require positive operating clearance; others use specified preload for stiffness or running accuracy. Temperature and fits change the operating condition.

A shim is a thin, accurately sized spacer. Changing its thickness sets a position in a rigid stack. Whether adding a shim increases or decreases end play depends on where the shim sits and which faces it separates.

Avoid burrs and rubbing contact that could release debris. Use the material and placement specified for the assembly.

A linear guide leaves travel along one axis

A linear bearing guides sliding motion just as a rotary bearing guides turning. Let x point along the travel. The guide supports transverse loads and, where its geometry allows, resists moments that would tilt or twist the carriage. A screw, belt or other actuator supplies travel force; the guide does not drive itself.

In the ideal model, translation along x is free. The desired constraints depend on whether rotation about x is acceptable. Real guide selection also checks load, moment capacity, stiffness, clearance, lubrication and alignment.

A round shaft does not provide an anti-rotation key

A plain round sleeve or linear ball bushing guides motion along a cylindrical shaft. In the ideal geometry it leaves two freedoms: T_x and R_x. The four other motions are restrained by the support's length or spaced bearings. A second parallel guide can stop the carriage rotating.

Geometric freedom is not a rotary service rating. A recirculating ball bushing may resist or wear under rotation even though its round shaft provides no positive anti-rotation feature. Use a product explicitly rated for combined linear and rotary motion when both are required.

LMU is a common catalog family label associated with round-shaft linear ball bushings. Treat it as a catalog name, not a universal standard or proof of rotary capability. Check the supplier's construction, dimensions and limits.

A profile rail constrains twist as well as sideways motion

A shaped rail and matching carriage have separated contact paths that resist twisting and tipping. Within their moment ratings, they leave one degree of freedom: T_x. Mounting surfaces must keep the rail aligned; adding constraints can also make an assembly sensitive to misalignment.

MGN commonly identifies a miniature, narrow-block profile-guide family. For example, HIWIN uses MGN within its miniature MG series, with recirculating balls. The label does not make every supplier's product interchangeable, and does not describe every plain or roller guide.

Opposed V-wheels guide a carriage; one wheel does not

A V-groove wheel runs on a matching edge or rail. A stable carriage commonly uses three or more wheels with contacts opposed across the rail and spaced along it. A triangular three-wheel arrangement can use an adjustable wheel to take up clearance.

The assembled carriage should leave only travel along the rail. Three wheels placed arbitrarily do not guarantee that result: orientation, spacing, contact retention and adjustment determine which loads and moments it can resist. A carriage that can lift away from its track has not constrained that motion.

Supporting detail: Addendum 64 — Manufacturer reading and the rotary-motion qualification.

Compare rotary support on Bearing Types and Loads. Practice counting freedoms

Three quantities, three units

Almost every electrical number you will read off a nameplate, a meter, or a schematic is one of three quantities. Each has a name, a quantity symbol used in formulas, and a unit symbol written after a value.

Current, I

The rate at which electric charge flows past a point. Its unit is the ampere, symbol A — one of the seven SI base units. One ampere is one coulomb of charge per second.

Voltage, V

Also called electric potential difference: the difference in electric potential between two points. Its unit is the volt, symbol V, defined as power per unit of current.

Resistance, R

How strongly something opposes the flow of current. Its unit is the ohm, symbol Ω, defined as voltage per unit of current.

Say “amp” if you like; write A. The unit name is the ampere, and the symbol is a capital A because the unit is named after a person. So is the volt (V) and the watt (W). Lower case changes the meaning of a prefix, not just the look of it: mA is a milliampere and MA is a megaampere, a billion times larger. See Measurement Language.

Supporting detail: Addendum 65 — Why voltage and resistance are called derived units.

Two equations, and everything else follows

For an ohmic resistor (constant resistance under the stated conditions), the set is generated by Ohm’s law and the power relationship:

V = IR  —  P = VI

Rearranging and substituting gives three expressions for each of the four quantities. There is nothing to memorise beyond the two above; the table is a lookup, not a list of new facts.

P = VI gives real power in watts on direct current or a purely resistive load. For single-phase AC, RMS volts times RMS amps is apparent power in volt-amperes, and the real power is that multiplied by the power factor — see Power and Efficiency.

Worked example: what a lamp draws

How much current does a 100 W resistive lamp draw on a 120 V supply (RMS for AC)?

1. Pick the form with the two quantities you have: I = P/V.
2. I = 100 W ÷ 120 V = 0.833 A.

Check the units rather than trusting the arithmetic. A volt is a watt per ampere, so watts divided by volts leaves amperes — which is what was asked for.

Worked example: voltage drop along a cord

A 100 ft extension cord has a total out-and-back conductor resistance of 0.1588 Ω. What is the resistive voltage drop across that path when 10 A flows through it?

1. V = IR = 10 A × 0.1588 Ω = 1.588 V.

Supporting detail: Addendum 66 — The dropped voltage does not disappear.

The same method keeps working

These are not the last units you will meet. Each new quantity arrives the same way: it is defined as a combination of quantities you already have, and its unit is that same combination.

Read the middle column and the right-hand column together and the pattern is the whole point: the unit is the definition, written in symbols. Once that clicks, a unit you have never seen tells you what the quantity is.

DC flows one way; AC reverses direction

Direct current (DC) flows in one direction. Its magnitude can change without making it AC. Alternating current (AC) reverses direction repeatedly. A battery supplies DC; a wall receptacle supplies AC.

Electronics generally run from a DC supply. That does not mean a transistor cannot handle an AC signal: transistors can amplify changing signals and act as switches. The supply and the signal are different parts of the story.

Start with Current, Voltage, and Resistance for the quantities and equations used here. For measurement, see Measuring Electric Circuits.

A 120 V sine wave peaks near positive and negative 170 V

The nominal 120 V of a US receptacle is an RMS value: root mean square. For a sine wave, Vpeak = VRMS × √2, and VRMS = Vpeak / √2. Thus 120 × √2 = 169.7 V. The instantaneous voltage swings between about +170 V and −170 V, or about 340 V peak to peak.

Frequency counts complete cycles per second, in hertz (Hz). A 60 Hz waveform completes 60 whole cycles per second; a 50 Hz waveform completes 50. A cycle returns to the same phase, rather than ending at the opposite peak. US household mains is 60 Hz; many other countries use 50 Hz (DOE, Electricity 101).

Supporting detail: Addendum 68 — Why use RMS instead of the average voltage?.

Higher transmission voltage reduces current and line loss

For the same transmitted power, a higher voltage allows a lower current. The line's resistive heating is Ploss = I2Rline. Halve the current and the loss becomes one quarter, if line resistance stays the same. Here Rline includes the complete current path.

Transformers made it practical to step AC voltage up for transmission and down for use. That advantage helped the AC systems associated with Westinghouse and Tesla prevail over Edison's early DC distribution system in the “war of the currents.” It was not one inventor winning every part of electrical engineering.

Supporting detail: Addendum 70 — A comparison with the same power and the same line.

A rectifier is one stage of an AC-to-DC power supply

A rectifier makes current flow in one direction from an AC input. Its output can still pulsate. Filtering smooths the output, and regulation controls the output voltage. A complete power supply may include all these functions as well as isolation.

• Linear supply: commonly uses a line-frequency transformer, rectifier, filter, and linear regulator. The regulator dissipates excess voltage as heat.
• Switch-mode supply: switches current rapidly and uses energy-storage components to convert voltage efficiently. An AC-input version also rectifies its input. Not every switch-mode converter provides isolation.

Power-supply manufacturer ARCH gives illustrative typical efficiency categories: linear supplies about 50–60%, conventional switching supplies 75–85%, and modern high-efficiency designs 90–95%.

Both lose some input energy as heat. Efficiency depends on the design, input voltage and load; compare published efficiency curves at the intended operating point rather than assuming one percentage for a whole category. See Power and Efficiency for the input/output calculation.

An inverter makes AC from DC; the waveform matters

A pure-sine inverter produces an output close to a sine wave. A modified-sine inverter produces a stepped approximation. Equipment may run differently, make more noise or heat, or fail to operate on a stepped waveform. Check the equipment manufacturer's compatibility guidance; equal RMS voltages do not make the waveforms interchangeable.

For grid-tied PV inverters, efficiency is typically above 95% under most operating conditions, according to Sandia PVPMC’s CEC Inverter Test Protocol. Efficiency depends on AC output power, DC voltage and sometimes temperature.

Every real conversion loses some energy. Several conversions in a row multiply their efficiencies rather than adding them. Much of what we use today runs on DC internally, even when it plugs into AC. Phones, computers, and microcontroller electronics bring the instructor's closing question into focus: who gets the last laugh?

Motors, generation, and energy storage make use of both forms of electrical power.

The conductor carries current; insulation separates conductors

The metal inside a wire is the conductor. Its resistance depends on material, length, cross-sectional area and temperature. The covering is insulation: it keeps the conductor from contacting other conductors or surrounding objects. The outside diameter includes that covering and does not by itself tell you how much metal is inside.

Solid wire has one solid conductor; stranded wire has several smaller strands forming a conductor. Insulation type and condition matter separately from the metal size. See Electrical Safety for use and inspection guidance.

A smaller AWG number means a thicker conductor

American Wire Gauge (AWG) uses smaller numbers for larger conductor cross sections. For the same material, length and temperature, a thicker conductor has less resistance. See Adafruit’s wire-gauge explanation. Under comparable installation and temperature conditions, it can carry more current without exceeding its temperature limit.

Ampacity means the allowable current under specified conditions. Gauge alone is not an ampacity rating: material, insulation temperature rating, ambient temperature, bundling and installation all matter. Use the applicable manufacturer data and installation requirements for a real selection; this page is not a wire-sizing table. IEWC explains these ampacity factors. Choosing a wire involves voltage drop as well as allowable current.

Supporting detail: Addendum 71 — Why a long wire can matter even below its current limit.

Series has one path; parallel has separate branches

In a series circuit, the same current passes through every element. The voltage drops around the loop add up to the source voltage. One break stops current everywhere in that loop.

In a parallel circuit, each branch connects to the same two supply nodes (a node is an electrically connected junction), so each complete branch has the same voltage. The branch currents add up to the supply current. A break in one branch leaves the other branches working, provided the source and shared wiring remain intact.

Equal resistance gives equal drops in series and equal currents in parallel. Different resistances make the drops or currents different. The formulas for each individual resistor are on Current, Voltage, and Resistance.

Add resistances in series; add their reciprocals in parallel

Equivalent resistance is the single resistance that would draw the same total current at the supply terminals.

• Series: Req = R1 + R2 + R3.
• Parallel: 1/Req = 1/R1 + 1/R2 + 1/R3. A reciprocal is 1 ÷ R: one divided by the resistance. Include only connected branches.
• Two parallel resistors: Req = R1R2 / (R1 + R2).

Check the result: series resistance is greater than any one positive resistor; parallel resistance is smaller than the smallest connected resistor when two or more branches are connected. An open circuit has no conducting path and is represented by infinite equivalent resistance.

Supporting detail: Addendum 73 — Why the parallel formula uses reciprocals.

Power explains the bulb comparison, within the model

For fixed resistors in series, the larger resistance dissipates more power because both carry the same current. In parallel, the smaller resistance dissipates more power because both have the same voltage. Check the power column rather than assuming the same bulb wins in both arrangements.

Greater electrical power is a useful brightness comparison for otherwise comparable incandescent bulbs. Real filaments change resistance as they heat, and LED lamps contain additional circuitry. The fixed-resistor model is not a universal lamp-brightness predictor.

Build another circuit in PhET's Circuit Construction Kit: DC – Virtual Lab. Try an unequal pair of resistors, predict the effect of a break, and then test your prediction.

Add a series-current problem to practice

Voltage is measured across; current is measured through

A voltmeter measures the potential difference between two points. Connect it across the component, in parallel. A series ammeter becomes part of the current path: open the path and insert it in series with the element whose current you want.

A meter set for current has a low-resistance input; connecting that input across a source makes a short circuit. Read Electrical Safety before making physical measurements. The diagrams here explain connections, not a procedure for working on mains wiring.

Supporting detail: Addendum 74 — Real meters change the circuit a little.

A shunt or clamp lets you measure current indirectly

A shunt is a known small resistance in the current path. Measure the voltage across it and use the relationship in Current, Voltage, and Resistance to find current. The shunt is in series; the voltage measurement across it is in parallel. Many ammeters contain a shunt internally. An installed shunt avoids opening the circuit each time you read it; adding one still requires changing the path.

A clamp meter senses the magnetic field around a conductor without breaking the circuit. Clamp around one conductor, not the outgoing and returning conductors together: their fields can cancel. The instrument must support the type of current being measured; an AC-only clamp does not measure steady DC.

Resistance and continuity are measured with the circuit unpowered

An ohmmeter applies its own small test signal. Use it on an unpowered circuit, with stored energy discharged as described in Electrical Safety. Other connected paths can change the result; isolating the component lets you measure its resistance instead of the whole network.

Continuity is a resistance-threshold test. The beep means the measured resistance is below that meter's threshold. It does not prove exactly zero resistance, identify the intended path, or establish that a connection can carry its working current. The threshold comes from the meter's manual.

A multimeter combines voltage, current and resistance functions, often with continuity too. Select the function, AC or DC mode, range and lead sockets for the measurement. In a schematic, follow the actual connection points rather than the physical positions of the symbols.

Choose the drawing for the job

One electrical system can need several drawings. Start with the question: What does the part look like? Where does this wire go? How does the circuit work? Where does the equipment fit?

Read the drawing title, legend, revision, and notes before treating a line or mark as an instruction. The same habit applies to mechanical drawings; see Choosing Drawing Views and The Alphabet of Lines.

A layout, floor plan, and riser answer different location questions

A PCB footprint is the land pattern where a component attaches to a board. Traces are the board's conductive paths. An IC is a chip; its internal layout is not the same drawing as the board that holds it.

A floor plan places devices in rooms. Its legend tells you what each mark means. A riser makes the relationship between building levels easier to follow. Neither should be treated as a complete installation instruction by itself.

Read control logic without assuming physical placement

A ladder diagram arranges control conditions and outputs in horizontal rungs between two vertical rails. A relay drawing and a PLC program may use a similar ladder shape, but a PLC instruction tests a stored state; it is not a picture of a physical contact.

A logic diagram emphasizes operations such as AND, OR, and NOT. It helps answer which combinations of inputs produce an output. The diagram may omit supply connections and package pin numbers, so it does not replace a wiring drawing.

Practice selecting a useful drawing

Practice diagram selection

Continue with Reading Electrical Connections to recognize common component symbols, distinguish a joined wire from a crossing, and count electrical nodes.

Read the connection convention before tracing a wire

A schematic organizes electrical relationships, not the positions of parts on a bench. Stretching or bending a drawn wire does not change its connections. Moving a line onto a different terminal does.

For the examples on this page, a filled dot at a crossing means the wires join. A crossing with no dot means they do not join. Other drawings may use a bridge or different conventions. Check the legend; an unclear or damaged junction mark is a reason to consult the drawing source.

Count connected conductors, not corners or dots

A node is a set of points joined by ideal conductors. Every point along an uninterrupted wire belongs to the same node, including its bends and branches. A dot marks a connection; it is not a separate component or an extra node.

In the figure, the dotted cross is one node. The undotted cross is two nodes: the horizontal conductor and the vertical conductor. This count concerns the wires drawn in the figure; no connection elsewhere is implied.

Do not trace through a resistor, diode, or open switch as if it were wire. If an ideal switch closes, its two sides become connected; state the switch position before counting nodes. Matching net labels can also indicate a connection without a continuous line across the page. Check the label scope and drawing notes before joining similarly named nets.

Read component identity separately from its value

A reference such as R1 identifies a particular part; a value such as 1 kΩ describes its resistance. D1 can identify a diode or LED, so its symbol and parts information still matter. A ground label identifies a reference or connection according to the drawing; it does not by itself establish a connection to protective earth.

US practice (IEEE 315 / ANSI Y32.2) and international practice (IEC 60617) draw some symbols differently. Use the drawing legend; the forms below are teaching examples, not a claim of compliance with a standard.

Recognize common symbols before tracing their connections

Use the symbol to identify the kind of component, then read its reference and value. The sheet includes both resistor forms and an unpolarized capacitor. A battery is one kind of DC source; other DC supplies may use a circle marked with polarity.

Practice symbol identification

Count four nodes with the switch open, three with it closed

Trace this small component schematic with S1 open. Stop at each component terminal; do not count through the battery, resistor, or LED.

Supporting detail: Addendum 75 — Check the node count terminal by terminal.

Practice counting schematic nodes

The stator stays still; the rotor turns

An electric motor turns electrical input into mechanical motion. In a rotary motor, the stator is the stationary part and the rotor is the rotating part. An air gap separates them. Bearings support the rotating assembly; the shaft carries motion to the load.

The stator is often outside the rotor, but it does not have to be. The names describe motion, not inside versus outside. See Electric Motor Types for both arrangements.

Keep the magnetic target moving: carrot on a stick

Picture the rotor turning to align its magnetic field with the stator's field. If the fields simply settled into alignment, that turning effect would disappear. To keep turning, the motor switches a field or keeps the stator field rotating. Think carrot on a stick: keep moving the target ahead.

Commutation means switching current between windings at the right time. Brushes and a commutator can do it mechanically, or a drive can do it electronically. A winding is a coil of wire that creates a magnetic field when current flows through it.

Supporting detail: Addendum 76 — Where the alignment picture needs care.