Why a Spinning Dancer Speeds Up by Pulling the Arms In

Push a door near its hinge and it barely moves; push at the handle and it swings open easily. The same force produces very different turning effects, so rotation needs its own quantities.

In rotation, torque plays the part of force and angular momentum plays the part of momentum — and both are built from the cross product.

This part covers the cross product, angular velocity, torque and angular momentum, and conservation of angular momentum.

**$\vec{A}\times \vec{B}$ has magnitude $AB\sin \theta$ and points perpendicular to both vectors, in the direction your right thumb points when your fingers curl from $\vec{A}$ towards $\vec{B}$; in components it is found with a determinant.**

$$\vec{A}\times \vec{B}=\hat{i}(A_y{B}_z-A_z{B}_y)-\hat{j}(A_x{B}_z-A_z{B}_x)+\hat{k}(A_x{B}_y-A_y{B}_x)$$

Useful results: $\hat{i}\times \hat{j}=\hat{k}$, $\hat{j}\times \hat{k}=\hat{i}$, $\hat{k}\times \hat{i}=\hat{j}$, and $\vec{B}\times \vec{A}=-\vec{A}\times \vec{B}$.

Worked example. $\vec{A}=2\hat{i}+3\hat{j}+\hat{k}$ and $\vec{B}=\hat{i}-\hat{j}+2\hat{k}$.

$$\vec{A}\times \vec{B}=\hat{i}(6+1)-\hat{j}(4-1)+\hat{k}(-2-3)=7\hat{i}-3\hat{j}-5\hat{k}$$

Check that it is perpendicular to both: $\vec{A}\cdot (\vec{A}\times \vec{B})=14-9-5=0$ and $\vec{B}\cdot (\vec{A}\times \vec{B})=7+3-10=0$. Its magnitude is $\sqrt{49+9+25}=\sqrt{83}\approx 9.11$.

An everyday example. Turning a screwdriver clockwise drives a screw into the wall — the direction a right-handed screw advances matches the cross product.

The substance. The dot-product check is a quick way to catch a sign error in a cross product.

**Every particle of a rigid body rotating about a fixed axis has the same angular velocity $\vec{\omega }$, directed along the axis, and its linear velocity is $\vec{v}=\vec{\omega }\times \vec{r}$, with magnitude $v=\omega r_{\perp }$.**

Here $r_{\perp }$ is the particle's perpendicular distance from the axis.

Worked example 1 — a wheel. A wheel of radius $0.3$ m turns at $120$ revolutions per minute.

$$\omega =\frac{120\times 2\pi }{60}=4\pi \approx 12.57\text{ rad/s}{v}_{rim}=12.57\times 0.3\approx 3.77\text{ m/s}$$

Worked example 2 — vector form. $\vec{\omega }=2\hat{k}$ rad/s and $\vec{r}=3\hat{i}+4\hat{j}$ m:

$$\vec{v}=2\hat{k}\times (3\hat{i}+4\hat{j})=6\hat{j}-8\hat{i}|\vec{v}|=10\text{ m/s}=\omega r$$

An everyday example. On a potter's wheel, clay near the rim rushes past the hands while clay near the centre barely moves, though the whole wheel turns together.

The substance. Angular velocity is the same for the whole rigid body; linear velocity is not.

**Torque is the moment of a force, $\vec{\tau }=\vec{r}\times \vec{F}$ with magnitude $rF\sin \theta$; angular momentum is the moment of momentum, $\vec{l}=\vec{r}\times \vec{p}$; and torque equals the rate of change of angular momentum, $\vec{\tau }=\frac{d\vec{l}}{dt}$.**

For a rigid body about a fixed axis, $L=I\omega$, where $I$ is the moment of inertia.

Worked example 1 — a spanner. A $40$ N force on a $0.25$ m spanner:

$${\tau }_{90°}=0.25\times 40=10\text{ N m}{\tau }_{30°}=0.25\times 40\times \sin 30°=5\text{ N m}$$

Worked example 2 — vector torque. $\vec{r}=\hat{i}+2\hat{j}$ m and $\vec{F}=3\hat{i}+\hat{j}$ N:

$$\vec{\tau }=(1\times 1-2\times 3)\hat{k}=-5\hat{k}\text{ N m}$$

Worked example 3 — a particle in a straight line. A $0.5$ kg particle moves at $4$ m/s along the line $y=3$ m, parallel to the x-axis. About the origin:

$$l=mv{r}_{\perp }=0.5\times 4\times 3=6{\text{ kg m}}^2\text{/s}$$

An everyday example. Pushing a door at the handle rather than near the hinges gives a larger $r_{\perp }$ and so a larger torque.

The substance. A force whose line passes through the axis gives zero torque, however large it is.

**If the net external torque on a system is zero, its total angular momentum stays constant, so for a body about a fixed axis $I_1{\omega }_1=I_2{\omega }_2$.

Worked example 1 — a skater.** Suppose a skater's moment of inertia drops from $4.0$ to $1.6$ kg m${}^2$ as the arms are pulled in, starting at $2$ revolutions per second.

$${\omega }_2=\frac{4.0\times 2}{1.6}=5\text{ rev/s}$$

In radians per second, $\omega$ rises from $4\pi$ to $10\pi$, and the rotational kinetic energy $\frac{1}{2}I{\omega }^2$ rises from about $316$ J to about $790$ J — the extra comes from the work done pulling the arms in.

Worked example 2 — a planet. Gravity from the Sun points along the line to the Sun, so it exerts no torque about the Sun, and $L=mvr$ at the nearest and farthest points stays the same:

$$v_{near}{r}_{near}=v_{far}{r}_{far}$$

If the farthest distance is $1.5$ times the nearest, the planet moves **$1.5$ times faster at its nearest point. Equal angular momentum also means the line to the Sun sweeps equal areas in equal times.

An everyday example. A kathak dancer spinning in chakkars turns faster by drawing the arms close to the body.

The substance. Angular momentum is conserved here, but rotational kinetic energy is not.**