Murray State University School of Engineering CourseWiki
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CourseWiki/Engineering Fundamentals/Motion and Rotation

Angular Motion

Measure rotation in degrees, radians, and revolutions, then use angular velocity and angular acceleration to describe a turning shaft.

Rotation is measured three ways

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.
One radian A sector drawn from the centre of a circle. Two radii of equal length open out from the centre, and the curved arc between their tips is drawn the same length as one radius. The angle enclosed at the centre is labelled one radian. radius r radius r arc length = r 1 rad
A radian is defined by the circle itself, not by a count somebody picked. Swing the radius around the rim until the arc it traces is as long as the radius, and the angle you have opened is one radian.
The three units, describing one and the same rotation.
In revolutionsIn degreesIn radians
1 r360°2π rad = 6.283185 rad
0.5 r180°π rad = 3.141593 rad
0.25 r90°π/2 rad = 1.570796 rad
≈ 0.159 r≈ 57.29578°1 rad

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.

Why a radian is barely a unit at all

A radian is an arc length divided by a radius — a length over a length. The two cancel, so NIST gives the base-unit expression for the radian as m/m and its value as simply 1: the radian and steradian are “special names for the number one that may be used to convey information about the quantity concerned.”

That is not trivia. It is why a torque in N·m multiplied by an angle in rad comes out as energy in joules with no conversion factor anywhere, and why a formula that mixes rotation with straight-line motion works only when the angle is in radians. Degrees and revolutions are counts somebody picked; the radian is the ratio the geometry already contains.

Angular velocity is how fast it turns; Watch out

Angular velocity is how fast it turns

Angular velocity uses the symbol ω (omega), and it is directed angle change per time interval for rotation about a fixed axis:

ωavg = Δθ / Δt. Instantaneous angular velocity is the rate at one moment; it equals this average throughout the interval only for constant angular velocity.

rad/s
Radian per second. The SI unit, and the form every formula on Rotation and Linear Speed needs.
r/min, or rpm
Revolutions per minute. What a motor nameplate, a spindle display, and a speed chart are marked in.

Nameplates speak rpm; formulas speak rad/s. Convert before you calculate, and convert back before you set a machine.

The conversion pair, rounded, either way round:

1 r/min = 0.1047198 rad/s  ·  1 rad/s = 9.549297 r/min

Worked example

Convert 12.5 rpm to rad/s. Put revolutions per minute on the bottom so they cancel:

12.5 r/min × = 1.309 rad/s.

Where 0.1047198 comes from

One revolution is 2π radians and one minute is 60 seconds, so one revolution per minute is 2π/60 = 0.104719755… radians per second. It is not a measured constant; it falls straight out of the two definitions. Turning it over gives 60/(2π) = 9.5492966… revolutions per minute in one radian per second.

Angular acceleration is how fast the turning changes

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.

0.76 rad/s2, as an average over those 1.1 s. The starting 17 rpm never enters the calculation — acceleration is built out of the change, not out of the speed.

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.

Sources

Sources

Where these values were checked

NIST SP 811, Appendix B gives revolution per minute to radian per second as 1.047 198 E−01, revolution to radian as 6.283 185 E+00, and degree to radian as 1.745 329 E−02. The 9.549297 r/min in one rad/s is the reciprocal of the first of those.

NIST SP 811, Chapter 4, Table 3, lists the radian with symbol rad, value 1, and base-unit expression m/m. The rule that every unit symbol takes a space except the degree, minute, and second of plane angle is NIST SP 811, section 7.2.

Appears in
DET 403 DET Rotation and Torque