Chapter Review

Circular Motion

Angular Variables and Linear Relations · Centripetal Force and Circular Motion

Angular Position and Arc Length

Angular displacement measured in radians links rotational geometry to linear distance along a curved path. The radian is defined so that arc length equals radius times angle.

Key Points

•
One radian is the angle where arc length equals radius; a full circle is $2\pi$ rad = 360°
•
$s = r\theta$ works only when $\theta$ is in radians — not degrees or revolutions
•
Points closer to the center travel shorter arcs for the same angular displacement
•
Angular displacement is positive for counterclockwise rotation, negative for clockwise

Formula

$$s = r\theta$$

Angular and Linear Velocity

Angular velocity ($\omega$) measures the rate of rotation in rad/s. Every point on a rigid body shares the same $\omega$, but linear velocity increases with distance from the axis.

Key Points

•
$\omega = \Delta\theta / \Delta t$; related to frequency and period by $\omega = 2\pi f = 2\pi / T$
•
$v = r\omega$ — tangential velocity is always perpendicular to the radius vector
•
To convert RPM to rad/s: multiply by $2\pi / 60$
•
Angular velocity direction is given by the right-hand rule (thumb along axis)
•
At the center ($r = 0$), linear velocity is zero regardless of $\omega$

Formula

$$v = r\omega$$

Angular Acceleration and Its Effects

Angular acceleration ($\alpha$) changes the rotation rate. It produces tangential acceleration that changes speed, while centripetal acceleration always exists to change direction.

Key Points

•
$\alpha = \Delta\omega / \Delta t$; positive $\alpha$ speeds up counterclockwise rotation
•
Tangential acceleration: $a_t = r\alpha$ (along the tangent, changes speed)
•
Centripetal acceleration: $a_c = \omega^2 r$ (toward center, changes direction)
•
Total acceleration magnitude: $a = \sqrt{a_t^2 + a_c^2}$ since $\vec{a}_t \perp \vec{a}_c$
•
In uniform circular motion ($\alpha = 0$): only centripetal acceleration exists

Formula

$$a_t = r\alpha, \quad a_c = \omega^2 r$$

Rotational Kinematic Equations

When angular acceleration is constant, four kinematic equations predict rotational motion — exact analogues of the linear equations with $x \to \theta$, $v \to \omega$, $a \to \alpha$.

Key Points

•
$\omega_f = \omega_i + \alpha t$ — relates velocities, acceleration, and time
•
$\Delta\theta = \omega_i t + \frac{1}{2}\alpha t^2$ — relates displacement, time, and acceleration
•
$\Delta\theta = (\omega_i + \omega_f) t / 2$ — average-velocity form, no $\alpha$ needed
•
$\omega_f^2 = \omega_i^2 + 2\alpha \Delta\theta$ — the timeless equation
•
Strategy: list the three knowns, identify the unknown, pick the equation containing all four

Formula

$$\omega_f^2 = \omega_i^2 + 2\alpha \Delta\theta$$

Centripetal Acceleration and Force

Circular motion requires a center-directed acceleration. Centripetal force is not a separate force — it is the net radial component of real forces (tension, friction, gravity, or normal) that provides this acceleration.

Key Points

•
$a_c = v^2 / r = r\omega^2$ — two equivalent forms with opposite $r$-dependence
•
$a_c \propto v^2$: doubling speed quadruples centripetal acceleration
•
In vertical circles: at top $T + mg = mv^2/r$; at bottom $T - mg = mv^2/r$
•
Minimum speed at top of vertical circle: $v_{min} = \sqrt{gr}$ (when $T = 0$)
•
If the constraint breaks (string snaps), the object flies off tangentially — not radially

Formula

$$F_c = \frac{mv^2}{r} = mr\omega^2$$

Moment of Inertia

Moment of inertia ($I$) quantifies rotational resistance based on mass distribution. It depends on both total mass and how far that mass sits from the rotation axis — via the $r^2$ term.

Key Points

•
$I = \sum m_i r_i^2$ — mass farther from the axis contributes quadratically more
•
Thin rod (center): $I = \frac{1}{12}mL^2$; hoop: $I = mr^2$; solid disk: $I = \frac{1}{2}mr^2$; solid sphere: $I = \frac{2}{5}mr^2$
•
For same $m$ and $r$: $I_{hoop} > I_{disk} > I_{sphere}$ — sphere wins rolling races
•
$I$ changes with axis choice, unlike mass which is a fixed scalar
•
All formulas have the form $I = k \cdot mr^2$; memorize the coefficient $k$ for each shape

Formula

$$I = \sum m_i r_i^2$$

Torque and Rotational Newton's Law

Torque is the rotational analogue of force — it causes angular acceleration. It depends on force magnitude, distance from the pivot, and the angle between them.

Key Points

•
$\tau = rF\sin\theta$ — maximum when $\theta = 90°$ (force perpendicular to lever arm)
•
$\tau = I\alpha$ is the rotational form of Newton's 2nd law
•
Lever arm is the perpendicular distance from pivot to the line of action of force
•
Counter-clockwise torque is conventionally positive
•
Rotational equilibrium: $\sum \tau_{CW} = \sum \tau_{CCW}$ (net torque = 0)

Formula

$$\tau = rF\sin\theta = I\alpha$$

Rotational Kinetic Energy

Spinning objects store energy in their rotation. Rolling objects split total KE between translational and rotational forms, with the partition determined by the shape's moment of inertia coefficient.

Key Points

•
$K_{rot} = \frac{1}{2}I\omega^2$ for pure spinning; total for rolling: $K = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2$
•
Rolling without slipping constraint: $v = r\omega$ links translational and rotational motion
•
Energy fraction that is rotational: $k/(1+k)$ — hoop 1/2, disk 1/3, sphere 2/7
•
From height $h$: $v = \sqrt{2gh/(1+k)}$ — lower $k$ means higher final speed
•
A rolling object is always slower than a sliding block from the same height

Formula

$$K_{total} = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2$$

Angular Momentum and Its Conservation

Angular momentum is conserved when no external torque acts. Redistributing mass inward decreases $I$ and increases $\omega$, while total $L = I\omega$ stays constant.

Key Points

•
For rigid body about fixed axis: $L = I\omega$; for particle: $L = mvr\sin\theta$
•
$I_i\omega_i = I_f\omega_f$ when net external torque is zero
•
Halving $I$ doubles $\omega$; KE changes ($K = L^2/2I$) — internal work is done
•
Direction of $\vec{L}$ is conserved too (gyroscopic stability)
•
Ice skater and diver are classic examples: tucking reduces $I$, increases spin rate

Formula

$$I_i\omega_i = I_f\omega_f$$

Formulas

Arc Length

Arc length equals radius times angle in radians.

Formula

$$s = r\theta$$

Linear-Angular Velocity

Tangential speed depends on radius and angular velocity.

Formula

$$v = r\omega$$

Centripetal Acceleration

Center-directed acceleration, two equivalent forms.

Formula

$$a_c = \frac{v^2}{r} = r\omega^2$$

Centripetal Force

Net inward force maintaining circular motion.

Formula

$$F_c = \frac{mv^2}{r} = mr\omega^2$$

Moment of Inertia (Point Mass)

Rotational resistance from mass at distance squared.

Formula

$$I = mr^2$$

Torque

Twisting force causes angular acceleration.

Formula

$$\tau = I\alpha$$

Rotational KE

Energy stored in spinning motion.

Formula

$$K_{rot} = \frac{1}{2}I\omega^2$$

Angular Momentum Conservation

When no external torque acts, Iω remains constant.

Formula

$$I_i\omega_i = I_f\omega_f$$