Rotational Kinematics
A bicycle wheel, a ceiling fan, a vinyl record: every point on them moves in a circle around the same axis. Instead of tracking each point separately, we can describe the whole object with just three quantities. How can one number tell you how fast an entire wheel is spinning, and how does it connect to how fast the rim moves along the road?
Angle, angular displacement, and units
For straight-line motion we used position $x$. For a spinning object we use an angle $\theta$ measured from a reference line. The change in angle is the angular displacement, $\Delta\theta = \theta - \theta_0$.
Angles can be measured in three ways. One revolution is a full turn:
$$1\ \text{rev} = 360^\circ = 2\pi\ \text{rad}$$Why radians? A radian is defined by the arc length: $\theta = s/r$, where $s$ is the arc length along the circle of radius $r$. One radian is the angle where the arc equals the radius. A full circle has arc $2\pi r$, so it has $2\pi$ radians. Radians are the unit the equations below need.
By convention, counterclockwise is the positive direction for rotation, and clockwise is negative.
A bicycle wheel turns through $12$ complete revolutions as Emma rides down her street. Express the wheel's angular displacement in radians. Give your answer in terms of…
A bicycle wheel turns through $12$ complete revolutions as Emma rides down her street. Express the wheel's angular displacement in radians. Give your answer in terms of $\pi$ and also as a decimal number.
Angular velocity and angular acceleration
Angular velocity tells how fast the angle changes, just like velocity tells how fast position changes:
$$\omega_{avg} = \frac{\Delta\theta}{\Delta t}, \qquad \alpha_{avg} = \frac{\Delta\omega}{\Delta t}$$The units are rad/s for $\omega$ and rad/s² for $\alpha$. Spin rates are often given in revolutions per minute (rpm), so convert:
$$1\ \text{rpm} = \frac{2\pi\ \text{rad}}{60\ \text{s}} \approx 0.105\ \text{rad/s}$$The period and angular velocity of steady rotation are linked: one revolution takes $T$ seconds, so $\omega = 2\pi/T$.
As with linear motion, signs matter. If $\omega$ and $\alpha$ have the same sign, the object spins faster. If they have opposite signs, it slows down.
A vinyl record on a turntable rotates at a constant $33\tfrac{1}{3}$ revolutions per minute. Convert this rate to an angular velocity in rad/s, and then find the time…
A vinyl record on a turntable rotates at a constant $33\tfrac{1}{3}$ revolutions per minute. Convert this rate to an angular velocity in rad/s, and then find the time the record takes to complete one revolution.
Constant angular acceleration
When $\alpha$ is constant, the rotational equations have exactly the same form as the linear ones. Just swap the symbols:
| Linear | Rotational |
|---|---|
| $x$ | $\theta$ |
| $v$ | $\omega$ |
| $a$ | $\alpha$ |
| $v = v_0 + at$ | $\omega = \omega_0 + \alpha t$ |
| $x = x_0 + v_0 t + \tfrac12 a t^2$ | $\theta = \theta_0 + \omega_0 t + \tfrac12 \alpha t^2$ |
| $v^2 = v_0^2 + 2a\Delta x$ | $\omega^2 = \omega_0^2 + 2\alpha\Delta\theta$ |
The reason is that the definitions are identical in form: $\omega$ is the rate of change of $\theta$, and $\alpha$ is the rate of change of $\omega$. As in linear motion, the area under the $\omega$–$t$ graph equals $\Delta\theta$, and the slope equals $\alpha$.
A ceiling fan starts from rest and speeds up with a constant angular acceleration of $0.80$ rad/s$^2$ for $10$ s. (a) Find the fan's angular velocity at $t = 10$ s. (b)…
A ceiling fan starts from rest and speeds up with a constant angular acceleration of $0.80$ rad/s$^2$ for $10$ s. (a) Find the fan's angular velocity at $t = 10$ s. (b) Find the angular displacement during these $10$ s. (c) Find how many revolutions the fan makes in that time.
A grinding wheel spinning at $60$ rad/s is switched off and slows uniformly to rest in $15$ s. (a) Find the magnitude of its angular acceleration. (b) Find the angular…
A grinding wheel spinning at $60$ rad/s is switched off and slows uniformly to rest in $15$ s. (a) Find the magnitude of its angular acceleration. (b) Find the angular displacement while it slows. (c) A point on the wheel is $0.10$ m from the axis. Find the tangential acceleration of that point.
Two students study a spinning disk. The disk's angular velocity is given in the table below. | $t$ (s) | 0 | 2.0 | 4.0 | 6.0 | 8.0 | |---|---|---|---|---|---| | $\omega$…
Two students study a spinning disk. The disk's angular velocity is given in the table below.
| $t$ (s) | 0 | 2.0 | 4.0 | 6.0 | 8.0 |
|---|---|---|---|---|---|
| $\omega$ (rad/s) | 4.0 | 10.0 | 16.0 | 22.0 | 28.0 |
(a) Show that the angular acceleration is constant and find its value. (b) Find the angular displacement from $t = 0$ to $t = 8.0$ s. (c) Aaliyah says the disk makes about $22$ revolutions in this interval. Check whether she is correct, and explain your reasoning with a calculation.
Connecting rotation to linear motion
A point at distance $r$ from the axis travels along a circle. In one revolution it moves a distance $2\pi r$, so it is the same arc-length idea as $s = r\theta$. Dividing by time gives:
$$v = r\omega \qquad a_t = r\alpha \qquad a_c = \frac{v^2}{r} = \omega^2 r$$Here $v$ is the linear (tangential) speed, $a_t$ is the tangential acceleration (it changes the speed, and points along the circle), and $a_c$ is the centripetal acceleration (it changes the direction, and points toward the center). The two are perpendicular, so the magnitude of the total acceleration is
$$a = \sqrt{a_t^2 + a_c^2}$$If $\alpha$ is constant and $\omega_0 = 0$, then $\omega = \alpha t$, so $a_c = (\alpha t)^2 r$ grows with time while $a_t = r\alpha$ stays fixed.
All points on a rigid object share the same $\omega$ and $\alpha$, but points farther from the axis have larger $v$ and $a_t$.
A point on the rim of a merry-go-round at a park in Denver is $2.0$ m from the center. The merry-go-round rotates with a constant angular velocity of $1.5$ rad/s. Find…
A point on the rim of a merry-go-round at a park in Denver is $2.0$ m from the center. The merry-go-round rotates with a constant angular velocity of $1.5$ rad/s. Find the linear speed of this point.
Marcus's car tire has a radius of $0.32$ m. The car speeds up from rest to $20$ m/s in $8.0$ s on a straight road, and the tires roll without slipping. Assuming constant…
Marcus's car tire has a radius of $0.32$ m. The car speeds up from rest to $20$ m/s in $8.0$ s on a straight road, and the tires roll without slipping. Assuming constant acceleration, find (a) the tire's angular acceleration and (b) the tire's angular velocity at $t = 8.0$ s.
A clothes dryer drum of radius $0.30$ m starts from rest and has a constant angular acceleration of $2.0$ rad/s$^2$. A sock sits against the inside wall of the drum and…
A clothes dryer drum of radius $0.30$ m starts from rest and has a constant angular acceleration of $2.0$ rad/s$^2$. A sock sits against the inside wall of the drum and moves with it. At $t = 3.0$ s, find (a) the sock's tangential acceleration, (b) its centripetal acceleration, and (c) the magnitude of its total acceleration. (d) Explain why the angle between the total acceleration vector and the radial direction changes as time passes, and say whether it increases or decreases.
Back to the start: one number, $\omega$, describes the whole wheel, and multiplying by the radius gives the speed of any point, such as the rim moving along the road.