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Angular momentum

Forces and motion · level 8, lesson 4 of 6

A question to start

A skater pulls her arms in and spins faster. A crew on Mars wants to start a big spinning room turning. The planet Mars has been spinning for four billion years. What is the quantity that all three have, and when does it stay the same?

The idea

The angular momentum of a spinning object is

\[L = I\omega.\]

The rotational second law can be written \(\tau_\text{net} = \Delta L/\Delta t\). So with no net outside torque, angular momentum is conserved: \(I\omega\) stays the same.

  • The skater. Pulling her arms in cuts \(I\), so \(\omega\) rises to keep \(I\omega\) fixed.
  • Starting a spinning room. A motor that turns the room pushes back on its base just as hard. If the base were free, it would turn the other way. On Mars the base is fixed to the ground, so the planet takes the opposite angular momentum; it is far too big to notice. In space, a station has to spin up with thrusters, or a counter-spinning wheel.
  • Spacecraft pointing. Many spacecraft turn by spinning internal reaction wheels: spin a wheel one way and the craft turns the other way.
arms out: big I, slowarms in: small I, fast

The spin of a planet. For a planet, \(I\) is written as \(kMR^2\), with \(k = 0.4\) for a ball of even density and less if heavy matter is concentrated in the middle. NASA's fact sheets give the measured values:

Mars Earth
Moment of inertia factor \(k = I/MR^2\) 0.366 0.331
Mass \(6.42 \times 10^{23}\) kg \(5.97 \times 10^{24}\) kg
Mean radius 3,390 km 6,371 km
\(\omega\) \(7.09 \times 10^{-5}\) rad/s \(7.29 \times 10^{-5}\) rad/s
Angular momentum \(L = kMR^2\omega\) \(1.9 \times 10^{32}\) kg m²/s \(5.8 \times 10^{33}\) kg m²/s

Mars's \(k\) is closer to 0.4 than Earth's: its iron-rich core holds a smaller share of its mass, so its mass is more evenly spread.

Worked examples

1. The skater. \(I = 3.0\) kg m², \(\omega = 4\) rad/s, arms out. Arms in, \(I = 1.2\) kg m². New \(\omega = 3.0 \times 4 / 1.2 = 10\) rad/s.

2. Mars's angular momentum. \(0.366 \times 6.42\times10^{23} \times (3.39\times10^6)^2 \times 7.09\times10^{-5} = 1.9\times10^{32}\) kg m²/s.

3. A reaction wheel. A 5 kg wheel with \(I = 0.05\) kg m² spins up to 600 rad/s, \(L = 30\) kg m²/s. A spacecraft with \(I = 1{,}500\) kg m² turns the other way at \(30/1500 = 0.02\) rad/s, about 1.1° a second.

Try it

  1. When is angular momentum conserved?
  2. A diver with \(I = 12\) kg m² spins at 2 rad/s and tucks to \(I = 4\) kg m². Find the new \(\omega\).
  3. What does a \(k\) of 0.366 tell you about the inside of Mars, compared with a uniform ball?
  4. A motor in a free-floating space station spins a room one way. What happens to the rest of the station?
Answers
  1. When there is no net outside torque.
  2. \(12 \times 2 / 4 = 6\) rad/s.
  3. It is a little less than 0.4, so the mass is somewhat concentrated towards the centre: there is a dense core.
  4. It starts turning the other way, so the total angular momentum stays zero.

Common mistake

"Spinning faster means energy was conserved." For the skater, angular momentum is conserved but kinetic energy is not: \(\tfrac12 I\omega^2 = \tfrac12 L\omega\) goes up as \(\omega\) does. The extra energy comes from the work of pulling the arms in.

Where it leads