🔋College Physics I – Introduction Unit 10 Review

Rotational kinematics gives you the tools to describe how objects spin, speed up, and slow down around an axis. It mirrors linear kinematics almost exactly, with angular versions of displacement, velocity, and acceleration replacing their straight-line counterparts. If you already understand linear kinematics equations, you're most of the way there.

Rotational Motion Kinematics

Key variables in rotational kinematics

These angular quantities each have a direct linear counterpart. The key difference is that everything is measured relative to rotation around an axis rather than movement along a line.

Angular displacement ( $\Delta\theta$ ) is the change in angular position of a rotating object, measured in radians (rad) or degrees (°). It's analogous to linear displacement ( $\Delta x$ ). A door swinging from closed (0°) to a quarter-open position has an angular displacement of 90°, or $\frac{\pi}{2}$ rad.
Angular velocity ( $\omega$ ) is how fast the angular position changes, measured in rad/s. It's analogous to linear velocity ( $v$ ). A ceiling fan spinning at a constant 120 rpm has an angular velocity of $4\pi$ rad/s. To convert rpm to rad/s, multiply by $\frac{2\pi}{60}$ .
Angular acceleration ( $\alpha$ ) is how fast the angular velocity changes, measured in rad/s². It's analogous to linear acceleration ( $a$ ). A washing machine that goes from rest to 1200 rpm in 30 seconds has an angular acceleration of about $\frac{40\pi}{30} \approx 4.19$ rad/s².
Radius of rotation ( $r$ ) is the distance from the axis of rotation to the point you're looking at. This quantity links the angular world to the linear world. A point on the rim of a bicycle wheel has a larger $r$ than a point near the hub, so it covers more linear distance per revolution.
Tangential velocity ( $v_t$ ) is the linear speed of a point on a rotating object: $v_t = r\omega$ . It points tangent to the circular path. A point on Earth's equator ( $r \approx 6{,}371$ km) has a tangential velocity of about 1,670 km/h due to Earth's rotation.
Tangential acceleration ( $a_t$ ) is the linear acceleration due to changing angular velocity: $a_t = r\alpha$ . It also points tangent to the circular path. Any time a spinning object speeds up or slows down, points on it experience tangential acceleration.
Centripetal acceleration ( $a_c$ ) points toward the center of rotation and exists whenever an object moves in a circle, even at constant angular velocity: $a_c = r\omega^2$ . This is what keeps the object on its circular path.

Key variables in rotational kinematics, Rotational Variables – University Physics Volume 1

Problem-solving with rotational equations

These four equations are the rotational versions of the constant-acceleration kinematic equations you already know. They only apply when angular acceleration ( $\alpha$ ) is constant.

Rotational Equation	Linear Equivalent
$\omega_f = \omega_0 + \alpha t$	$v_f = v_0 + at$
$\Delta\theta = \omega_0 t + \frac{1}{2}\alpha t^2$	$\Delta x = v_0 t + \frac{1}{2}at^2$
$\omega_f^2 = \omega_0^2 + 2\alpha\Delta\theta$	$v_f^2 = v_0^2 + 2a\Delta x$
$\Delta\theta = \frac{1}{2}(\omega_0 + \omega_f)t$	$\Delta x = \frac{1}{2}(v_0 + v_f)t$

Pick the equation that contains the three quantities you know and the one you're solving for. Here's how to approach a typical problem:

Identify what's given (any three of $\omega_0$ , $\omega_f$ , $\alpha$ , $\Delta\theta$ , $t$ ).
Identify what you need to find.
Choose the equation that connects those four quantities.
Make sure all your units are consistent (use rad/s and rad, not rpm and degrees, in the equations).
Solve algebraically, then plug in numbers.

Example: A flywheel starts at $\omega_0 = 10$ rad/s with $\alpha = 2$ rad/s². What is $\Delta\theta$ after $t = 6$ s?

$\Delta\theta = \omega_0 t + \frac{1}{2}\alpha t^2 = (10)(6) + \frac{1}{2}(2)(6^2) = 60 + 36 = 96 \text{ rad}$

Example: A wind turbine starts at $\omega_0 = 2$ rad/s, accelerates at $\alpha = 0.5$ rad/s² over $\Delta\theta = 10$ rad. Find $\omega_f$ .

$\omega_f^2 = \omega_0^2 + 2\alpha\Delta\theta = 4 + 2(0.5)(10) = 14$

$\omega_f = \sqrt{14} \approx 3.74 \text{ rad/s}$

Common mistake: Mixing units. If a problem gives you rpm, convert to rad/s before plugging into equations. If it gives degrees, convert to radians. The equations assume radian measure.

Key variables in rotational kinematics, Quantities of Rotational Kinematics | Boundless Physics

Linear vs. rotational motion comparisons

The structure of rotational kinematics mirrors linear kinematics almost perfectly. The table below summarizes the correspondence:

Linear Quantity	Rotational Quantity	Relationship
Displacement $\Delta x$	Angular displacement $\Delta\theta$	$\Delta x = r\Delta\theta$
Velocity $v$	Angular velocity $\omega$	$v = r\omega$
Acceleration $a$	Angular acceleration $\alpha$	$a_t = r\alpha$

The main conceptual difference: linear motion describes movement along a straight line (a car on a highway), while rotational motion describes movement around an axis (a Ferris wheel turning). Linear quantities use meters and m/s; rotational quantities use radians and rad/s.

The factor of $r$ connecting them means that points farther from the axis move faster and accelerate more in linear terms, even though every point on a rigid body shares the same $\omega$ and $\alpha$ .

Rotational dynamics

These concepts connect rotational kinematics to the forces that cause rotation. You'll explore them in more depth later in this unit, but here's the overview:

Moment of inertia ( $I$ ) measures an object's resistance to angular acceleration, just as mass ( $m$ ) measures resistance to linear acceleration. It depends on both the mass and how that mass is distributed relative to the axis.
Torque ( $\tau$ ) is the rotational equivalent of force. Just as $F = ma$ , the rotational version is $\tau = I\alpha$ .
Angular momentum ( $L$ ) is the rotational analog of linear momentum. A spinning object tends to keep spinning at the same rate unless a net torque acts on it.

🔋College Physics I – Introduction Unit 10 Review