Angular momentum: a skater and a gyroscope.

Step 1 of 6

Pulling in

A body moving on a circle needs a force pulling it towards the centre. A body spinning about itself raises a new question: what keeps the spin going, and what changes it? A skater on ice is a good example. The blade is sharp and the ice smooth, so almost nothing outside pushes or brakes her.

She starts with her arms out, Arm reach 0.8 m from the axis, turning about once a second. Predict what will happen when she pulls them in, then look.

Step 2 of 6

The moment of inertia

A spinning body has angular momentum, $L$⁠. It depends on two things: how fast the body turns, its angular velocity $\omega$⁠, and how its mass is spread around the axis, its moment of inertia $I$⁠. Mass far from the axis counts for much more than mass close in: by the square of the distance. So

$L = I\omega$⁠

Angular momentum is kept unless something outside twists the body. Such a twist is called a torque. With Show angular momentum set to On, the board above the skater shows $I$⁠, $\omega$⁠ and their product. Gyroscope spin is fixed.

Step 3 of 6

The energy of the spin

The angular momentum is kept, but the energy is not. The energy of a spin is

$E = \frac{1}{2}I\omega^2 = \frac{L^2}{2I}$$E =$$\frac{1}{2}I\omega^2 =$$\frac{L^2}{2I}$⁠

so with $L$⁠ fixed and $I$⁠ smaller, the energy grows. Her arms are tucked now, and Arm reach is fixed.

Step 4 of 6

The gyroscope

Beside the skater stands a gyroscope: a heavy brass wheel on an axle, with one end of the axle resting on a stand. When the wheel is not spinning, what you expect happens: gravity pulls the far end down, and the gyroscope falls against the post. Predict what will happen when the wheel spins fast.

Step 5 of 6

The rate of precession

The wheel’s weight and the stand’s push make a torque about the pivot, pointing level, at right angles to the axle. A torque does not set a direction of motion: it changes the angular momentum, and the change points the same way as the torque. So the blue arrow, the angular momentum, always moves the way the violet arrow, the torque, points, and the axle goes round the stand.

For a fast wheel the rate is the torque divided by the angular momentum:

$\Omega = \frac{Mg\ell}{I\omega}$$\Omega =$$\frac{Mg\ell}{I\omega}$⁠

Here $M$⁠ is the wheel’s mass, $\ell$⁠ its distance from the pivot, and $I\omega$⁠ its angular momentum.

Step 6 of 6

A slowing wheel

At 800 rpm, the gyroscope already dips and wobbles before it settles, and goes round the stand quickly. A real gyroscope loses spin to friction in its bearings. The Earth is a spinning top too: the pull of the Sun and the Moon on its bulge at the equator makes a torque, and its axis traces a circle in the sky about once every 26,000 years.