Rotational dynamics

Angular Kinematics

Describe spin with radians: angular position, velocity, and acceleration, the v = rω link to linear motion, and the rotational kinematic equations.

Torque & Moment of Inertia

Build the rotational analogs of force and mass — lever arms and τ = rF sin θ, moment of inertia from mass distribution, and Newton’s second law for rotation.

Rolling & Rotational Energy

Combine translation and spin: rolling without slipping, kinetic energy split between ½mv² and ½Iω², and why shape decides the race down an incline.

Angular Momentum

Meet L = Iω and its conservation: spinning skaters, dropped disks, and why kinetic energy can change while angular momentum cannot.

Rotating Frames

Finish

Tell centripetal force apart from the apparent centrifugal and Coriolis effects using a rotating spaceship and a bead on a spinning rod.

Angular Kinematics

A spinning wheel is a strange object to describe with the kinematics of straight-line motion. Every piece of it is moving, but no two pieces share the same velocity — the rim races along while material near the axle barely crawls. The trick is to stop tracking positions and start tracking a single angle. One number, $\theta$ , pins down the orientation of the whole wheel, and everything else follows from it.

Radians: the Natural Angle

Wrap a distance around a circle and you get the natural definition of angle: the radian. An angle in radians is the arc length it spans divided by the radius,

\theta = \frac{s}{r} \quad\Longleftrightarrow\quad s = r\theta.

One full turn wraps the full circumference $2\pi r$ , so a revolution is $2\pi$ radians and a radian is about $57.3°$ . Degrees and revolutions are fine for bookkeeping, but $s = r\theta$ — and every formula descended from it — only works when $\theta$ is in radians. It is worth being paranoid about this; it is the single most common error in rotation problems.

The Angular Vocabulary

Each linear quantity has a rotational twin, built the same way:

Linear	Angular	Connection
position $x$	angle $\theta$ (rad)	$s = r\theta$
velocity $v = \dfrac{dx}{dt}$	angular velocity $\omega = \dfrac{d\theta}{dt}$ (rad/s)	$v = r\omega$
acceleration $a = \dfrac{dv}{dt}$	angular acceleration $\alpha = \dfrac{d\omega}{dt}$ (rad/s²)	$a_t = r\alpha$

The connections in the last column are what make the angular description powerful. A rigid body has just one $\omega$ — every point sweeps the same angle in the same time — but each point’s speed grows with its distance from the axis:

v = r\omega.

That is why the rim of a merry-go-round feels fast while the center feels calm: same $\omega$ , different $r$ .

Two markers ride the same wheel, so they share the same ω — but the outer one moves faster, since v = rω. Switch to rolling and watch the rim marker stop dead each time it touches the ground.

Angular speed ω (rad/s)3.0Roll along the ground

Signs work like they do on a number line, except the line is bent into a circle: pick a positive rotation direction (counterclockwise, by convention), and $\omega$ and $\alpha$ carry signs relative to it. A wheel spinning counterclockwise ( $\omega > 0$ ) while slowing down has a clockwise angular acceleration ( $\alpha < 0$ ) — just as a car moving in the $+x$ direction while braking has a negative $a$ .

One caution: $a_t = r\alpha$ is only the tangential part of a point’s acceleration — the part that changes its speed. Any point moving in a circle also has the inward centripetal part $a_c = r\omega^2$ , even at constant $\omega$ . The rotating frames page takes that inward piece apart in detail.

The Same Old Equations, Rotated

Because $\theta$ , $\omega$ , and $\alpha$ are defined exactly like $x$ , $v$ , and $a$ , constant- $\alpha$ rotation obeys the same kinematic equations with the letters swapped:

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

No new problem-solving skills are required — identify what is given, spot the missing variable, and pick the equation that avoids it, exactly as in linear kinematics.

A wheel spins up with constant angular acceleration. Scrub or play time and compare the curves: ω(t) is a straight line, θ(t) a parabola — the same shapes as v(t) and x(t) in linear kinematics.

t = 3.00sθ = 6.8radθ/2π = 1.07revω = 4.50rad/s

Angular acceleration α (rad/s²)+1.5Initial ω₀ (rad/s)0.0Time t (s)3.0

Problem Solving

From RPM to Radians per Second

Problem

An old hard drive spins at $300\ \text{rpm}$ . What is its angular speed in rad/s, and how fast does a point at radius $4.0\ \text{cm}$ move?

Use

One revolution is $2\pi$ radians, so $\omega = 2\pi \cdot \text{rpm}/60$ ; then $v = r\omega$ .

Solve

\omega = \frac{2\pi (300)}{60} = 10\pi \approx 31.4\ \text{rad/s},

v = r\omega = (0.040)(31.4) \approx 1.3\ \text{m/s}.

Check

Five revolutions per second times roughly $6.28$ radians per revolution is indeed about $31$ rad/s. The units only work because radians are dimensionless — a speed in “degrees·meters per second” would mean nothing.

Counting Revolutions During Spin-Up

Problem

A turbine starts from rest and accelerates at a constant $\alpha = 2.0\ \text{rad/s}^2$ for $10\ \text{s}$ . Find its final angular speed and the number of revolutions it completes.

Use

Constant- $\alpha$ kinematics from rest: $\omega = \alpha t$ and $\theta = \tfrac{1}{2}\alpha t^2$ .

Solve

\omega = (2.0)(10) = 20\ \text{rad/s},

\theta = \tfrac{1}{2}(2.0)(10)^2 = 100\ \text{rad} \quad\Rightarrow\quad \frac{100}{2\pi} \approx 15.9\ \text{revolutions}.

Check

The average angular speed is $10\ \text{rad/s}$ (half the final value), and $10\ \text{rad/s} \times 10\ \text{s} = 100\ \text{rad}$ — the two routes agree. You can replay these exact numbers in the grapher above.

Angular Kinematics Checkpoint

Question 1 of 4

Two children ride a spinning merry-go-round: one near the center, one at the rim. Which quantities do they share?

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