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

Here

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.

Rolling Motion and Rotational Energy

A rolling wheel does two things at once: its center travels forward, and the wheel spins about that center. Rolling without slipping is the special case where the two motions are locked together — the tire grips the road instead of skidding across it — and the lock is the same relation that connected angular and linear speed on the angular kinematics page:

v_{\text{center}} = r\omega.

Each turn of the wheel carries the center forward by exactly one circumference. One startling consequence: the point of the wheel touching the ground is momentarily at rest. The forward motion of the center ( $+r\omega$ ) and the backward sweep of the rim at the bottom ( $-r\omega$ ) cancel exactly. That is why the rim marker’s velocity arrow in the wheel tracker shrank to zero at every ground contact — and why a rolling tire can be gripped by static friction even at highway speed.

The Kinetic Energy of Rolling

Because a rolling body both translates and spins, its kinetic energy has two parts:

K = \underbrace{\tfrac{1}{2}mv^2}_{\text{translation}} + \underbrace{\tfrac{1}{2}I\omega^2}_{\text{rotation}}.

For round shapes, the moment of inertia is some coefficient $c$ times $mr^2$ — one half for a disk, two fifths for a solid sphere, one for a hoop. Substituting $I = cmr^2$ and $\omega = v/r$ makes the radius disappear:

K = \tfrac{1}{2}mv^2 + \tfrac{1}{2}(cmr^2)\frac{v^2}{r^2} = \tfrac{1}{2}(1 + c)\,mv^2.

A rolling object is carrying a hidden energy surcharge: at the same speed, a hoop ( $c = 1$ ) holds twice the kinetic energy of a sliding block, with a full half of it tucked into the spin.

The Race Down the Incline

That surcharge has a famous consequence. Release a hoop, a disk, and a sphere from rest at the top of a ramp and they do not arrive together — and neither mass nor radius has anything to do with the finish order.

Why a = g sin θ / (1 + c)

Roll from rest down a slope of angle $\theta$ through a distance $s$ along the surface (a height drop of $s\sin\theta$ ). Rolling friction does no work at the contact point (it is static, and the contact point is at rest), so mechanical energy is conserved:

mgs\sin\theta = \tfrac{1}{2}(1 + c)\,mv^2 \quad\Rightarrow\quad v^2 = \frac{2g\sin\theta}{1 + c}\,s.

Comparing with the kinematic relation $v^2 = 2as$ identifies a constant acceleration

a = \frac{g\sin\theta}{1 + c}.

Both $m$ and $r$ have cancelled. Only the shape coefficient $c$ survives — the fraction of energy each shape is forced to divert into spinning.

The smaller the coefficient $c$ , the less energy the shape wastes on rotation and the faster it accelerates: solid sphere ( $\tfrac{2}{5}$ ) beats disk ( $\tfrac{1}{2}$ ) beats hollow sphere ( $\tfrac{2}{3}$ ) beats hoop ( $1$ ) — every time, on any planet, at any size. Watch the energy bars in the lab below: every racer converts the same potential energy, but the hoop locks half of it into rotation while the sphere keeps most of it moving forward.

Rolling Race Lab

Race a hoop, disk, solid sphere, and hollow sphere down an adjustable incline. The energy bars split each racer's budget into translational and rotational kinetic energy.

A sliding, frictionless block would beat them all ( $c = 0$ ): with no spin to feed, every joule of potential energy goes into forward motion.

Problem Solving

Solid Sphere on a Ramp

Problem

A solid sphere rolls without slipping from rest down a $30°$ incline. Find its acceleration and its speed after $6.0\ \text{m}$ along the slope.

Use

$a = g\sin\theta/(1 + c)$ with $c = \tfrac{2}{5}$ for a solid sphere, then $v^2 = 2as$ .

Solve

a = \frac{(9.8)\sin 30°}{1 + \tfrac{2}{5}} = \frac{4.9}{1.4} = 3.5\ \text{m/s}^2,

v = \sqrt{2(3.5)(6.0)} \approx 6.5\ \text{m/s}.

Check

A frictionless sliding block would have $a = g\sin 30° = 4.9\ \text{m/s}^2$ — the rolling sphere is slower because $\tfrac{2}{7}$ of its energy budget ( $c/(1+c) = \tfrac{2/5}{7/5}$ ) is spent on spin. Note that neither the sphere’s mass nor its radius was needed.

The Falling Toilet-Paper Roll

Problem

A fresh cylindrical roll is held by the loose end of its paper and released. Treating it as a uniform solid cylinder unrolling without slipping, what is its downward acceleration?

Use

This is a “rolling” problem turned vertical: the paper acts like an incline of $\theta = 90°$ , so $a = g/(1 + c)$ with $c = \tfrac{1}{2}$ .

Solve

a = \frac{g}{1 + \tfrac{1}{2}} = \frac{2g}{3} \approx 6.5\ \text{m/s}^2.

Check

The roll falls, but at only two thirds of $g$ — the remaining third of the gravitational energy is spinning the roll up. The paper tension that provides the slowing force is $T = mg/3$ , which you can confirm from $ma = mg - T$ .

Rolling & Energy Checkpoint

Question 1 of 3

A small steel sphere and a huge wooden sphere (both solid) roll from rest down the same ramp. Who wins?

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