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Topic 11 of 20

Rotational Kinematics

Rotational kinematics is the angular counterpart of 1D kinematics. Every linear kinematic quantity has a rotational analog: displacement → angle, velocity → angular velocity, acceleration → angular acceleration. The kinematic equations take the same form, making this topic straightforward once the 1D version is solid.

Key Concepts

Angular Position

Angle

\theta

(in radians) describing orientation. Arc length and angle relate by

s = r\theta

. One full revolution =

2\pi

rad ≈ 6.28 rad = 360°.

Angular Velocity

Rate of change of angle:

\omega = d\theta/dt

(rad/s). Positive conventionally means counterclockwise. Related to period:

\omega = 2\pi f = 2\pi/T

Angular Acceleration

Rate of change of angular velocity:

\alpha = d\omega/dt

(rad/s²). Constant angular acceleration gives rise to the rotational kinematic equations.

Linear–Angular Relations

For a point at radius

r

from the axis: tangential speed

v = r\omega

, tangential acceleration

a_t = r\alpha

, centripetal acceleration

a_c = r\omega^2 = v^2/r

Rotational Kinematic Equations

Identical in form to the 1D kinematic equations with

\theta \leftrightarrow x

\omega \leftrightarrow v

\alpha \leftrightarrow a

. Valid for constant

\alpha

only.

Period & Frequency

Period

T

is the time for one full rotation. Frequency

f = 1/T

is rotations per second (Hz). Angular frequency

\omega = 2\pi f

Key Equations

Arc length

s = r\theta

θ must be in radians.

Tangential and centripetal acceleration

v = r\omega, \quad a_t = r\alpha, \quad a_c = \frac{v^2}{r} = r\omega^2

Relations between linear and angular quantities for a point at radius r.

Rotational kinematic equations

\omega = \omega_0 + \alpha t \quad\quad \theta = \theta_0 + \omega_0 t + \tfrac{1}{2}\alpha t^2 \quad\quad \omega^2 = \omega_0^2 + 2\alpha\Delta\theta

Direct analogs of the 1D kinematic equations. Valid only for constant α.

Period and angular frequency

T = \frac{2\pi}{\omega}, \quad f = \frac{\omega}{2\pi}, \quad \omega = 2\pi f

Conversions between period, frequency, and angular frequency.

Worked Example

Spinning Up a Wheel

Problem

A wheel starts from rest and reaches 120 rpm in 4 seconds under constant angular acceleration. Find $\alpha$ and the number of revolutions completed.

Solution

Convert 120 rpm to rad/s:

\omega_f = 120 \times \frac{2\pi}{60} = 4\pi \approx 12.6 \text{ rad/s}

Find angular acceleration:

\alpha = \frac{\omega_f - \omega_0}{t} = \frac{4\pi - 0}{4} = \pi \approx 3.14 \text{ rad/s}^2

Find total angle:

\Delta\theta = \omega_0 t + \tfrac{1}{2}\alpha t^2 = 0 + \tfrac{1}{2}(\pi)(16) = 8\pi \text{ rad}

\text{revolutions} = \frac{8\pi}{2\pi} = 4 \text{ rev}

Answer α = π rad/s² ≈ 3.14 rad/s²; the wheel completes 4 revolutions.

Practice

Exercises

7 problems

1 of 7

A wheel completes $5$ full revolutions. What is the total angular displacement in radians?

Your answer

rad

2 of 7

A disk rotates at $300 \text{ rpm}$ . What is its angular velocity in rad/s?

Your answer

rad/s

3 of 7

A point on a wheel of radius $0.4 \text{ m}$ moves at $2 \text{ m/s}$ . What is the wheel's angular velocity?

Your answer

rad/s

4 of 7

A wheel starts from rest and reaches $6 \text{ rad/s}$ in $3 \text{ s}$ with constant angular acceleration. What is $\alpha$ ?

Your answer

rad/s²

5 of 7

Using the same wheel ( $\alpha = 2 \text{ rad/s}^2$ , starts from rest, $t = 3 \text{ s}$ ). How many radians does it turn?

Your answer

rad

6 of 7

A wheel of radius $0.5 \text{ m}$ rotates at $4 \text{ rad/s}$ . What is the tangential speed of a point on the rim?

Your answer

m/s

7 of 7

The same wheel ( $r = 0.5 \text{ m}$ , $\omega = 4 \text{ rad/s}$ ). What is the centripetal acceleration of a point on the rim?

Your answer

m/s²

Key Takeaways

Angles in radians are required for all rotational kinematic equations: $s = r\theta$ , $v = r\omega$ , etc.
The rotational kinematic equations are identical in form to the 1D equations — same structure, same method.
The total acceleration of a point on a rotating body has two components: tangential ( $a_t = r\alpha$ , along the velocity) and centripetal ( $a_c = r\omega^2$ , toward the center).
1 revolution = $2\pi$ rad; 1 rpm = $2\pi/60$ rad/s.
Constant $\alpha$ is the rotational analog of constant $a$ — check this assumption before using the kinematic equations.