Conservation of angular momentum

A figure skater spins on the ice with her arms extended. At a certain moment, she pulls her arms in close to her body and, without pushing off again, starts spinning much faster.

1. a) What external forces act on the skater, and why is their moment about the axis of rotation practically zero?
2. b) State the principle of conservation of angular momentum and apply it to this case.
3. c) When she pulls in her arms, which quantity decreases: the moment of inertia $I$ or the angular velocity $\omega$? Deduce why the other increases.

A skater spins with a moment of inertia $I_1 = 5\ \text{kg} \cdot \text{m}^2$ and an angular velocity $\omega_1 = 4\ \text{rad/s}$. By pulling in her arms, her moment of inertia is reduced to $I_2 = 2\ \text{kg} \cdot \text{m}^2$. The figure summarises the change:

1. a) Calculate her initial angular momentum, $L = I_1 \cdot \omega_1$.
2. b) Apply the conservation of angular momentum, $I_1 \cdot \omega_1 = I_2 \cdot \omega_2$, and calculate the final angular velocity $\omega_2$.
3. c) Express the result as "times faster": how many revolutions does she complete now for each revolution she completed before?

The angular momentum of a body is not conserved in all situations. Answer with reasoning:

1. a) What is the condition on the torque of the external forces for angular momentum to be conserved?
2. b) A spinning top rotates supported on the ground and eventually stops after a while: has its angular momentum been conserved? What torque is responsible?
3. c) A planet has been orbiting its star for millions of years without appreciable changes: relate this fact to the condition in part a).

A diver begins a somersault spinning at one revolution per second with her body extended, with a moment of inertia of $12\ \text{kg} \cdot \text{m}^2$. Tucking in mid-air, her moment of inertia decreases to $4\ \text{kg} \cdot \text{m}^2$.

1. a) Express the initial angular velocity in $\text{rad/s}$ (one revolution is equal to $2 \cdot \pi\ \text{rad}$).
2. b) Apply the conservation of angular momentum and calculate the final angular velocity, in $\text{rad/s}$.
3. c) Express it in revolutions per second and explain why gymnasts tuck to perform multiple somersaults and extend before entering the water.

When a star exhausts its fuel, its core can collapse while virtually conserving its angular momentum. Model the core as a sphere whose moment of inertia is proportional to the square of its radius, $I \propto R^2$. The figure summarises the collapse:

1. a) If the radius of the core is halved, reason by what factor its moment of inertia changes.
2. b) Apply the conservation of angular momentum and deduce by what factor its angular velocity changes.
3. c) If initially the core rotated with a period of $10\ \text{h}$, calculate its final rotation period.

A skater spins with a moment of inertia $I_1 = 5\ \text{kg} \cdot \text{m}^2$ and an angular velocity $\omega_1 = 6\ \text{rad/s}$. By pulling in her arms, her moment of inertia becomes $I_2 = 2\ \text{kg} \cdot \text{m}^2$. The rotational kinetic energy is given by $E_c = \frac{1}{2} \cdot I \cdot \omega^2$.

1. a) Calculate the final angular velocity by applying the conservation of angular momentum.
2. b) Calculate the initial and final rotational kinetic energy.
3. c) Verify that the final energy is greater than the initial energy: where has this extra energy come from if no one pushes the skater from the outside? Reason your answer considering the work done by her muscles when bringing her arms closer to the axis.

A child of $30\ \text{kg}$ runs at $3\ \text{m/s}$ tangentially to the edge of a merry-go-round of radius $2\ \text{m}$ and jumps onto it, holding onto the edge. The merry-go-round was at rest and its moment of inertia is $I = 400\ \text{kg} \cdot \text{m}^2$. Model the child as a point mass. The diagram shows the situation viewed from above:

1. a) Calculate the angular momentum of the child about the axis of the merry-go-round just before jumping, $L = m \cdot v \cdot r$.
2. b) Calculate the total moment of inertia after the jump, $I_{total} = I + m \cdot r^2$.
3. c) Apply the conservation of angular momentum and calculate the angular velocity at which the combined system begins to spin.

In a laboratory, a dart of $0,05\ \text{kg}$ is fired at $40\ \text{m/s}$ and embeds itself into the edge of a horizontal disc of radius $0,25\ \text{m}$, free to rotate about its vertical axis. The disc was at rest and its moment of inertia is $I = 0,2\ \text{kg} \cdot \text{m}^2$. The dart impacts tangentially to the edge; neglect the moment of inertia of the embedded dart.

1. a) Calculate the angular momentum of the dart about the axis of the disc just before impact.
2. b) Justify why angular momentum is conserved in the collision even though kinetic energy is not conserved, as it is an inelastic collision.
3. c) Calculate the angular velocity of the disc with the dart embedded in it.

A playground roundabout ($I = 100\ \text{kg} \cdot \text{m}^2$) rotates freely at $\omega_1 = 2\ \text{rad/s}$ with a child of $40\ \text{kg}$ sitting at the edge, at $1,5\ \text{m}$ from the axis (treat the child as a point mass). The child crawls slowly to the centre of the roundabout.

1. a) Calculate the total initial moment of inertia, $I_{total} = I + m \cdot r^2$.
2. b) Calculate the initial angular momentum of the system.
3. c) With the child at the centre, where his moment of inertia is negligible, calculate the final angular velocity of the roundabout.
4. d) If the child returned to the edge, would the roundabout regain its initial velocity? Justify this and comment on how you would answer someone who claims that "the roundabout remains spinning faster forever".

A planet describes an elliptical orbit around its star, which occupies one of the foci of the ellipse. Call $r_a$ the distance at the furthest point (aphelion) and $r_p$ that at the closest point (perihelion).

1. a) Justify, using the central nature of the gravitational force, that the angular momentum of the planet is the same throughout the orbit.
2. b) Write the conservation equation between aphelion and perihelion: $m \cdot v_a \cdot r_a = m \cdot v_p \cdot r_p$.
3. c) Reason, from this equation, at which point the planet moves fastest and at which it moves slowest.

Comet 67P, visited by the Rosetta probe, passes through perihelion at $1,86 \cdot 10^{11}\ \text{m}$ from the Sun and through aphelion at $8,5 \cdot 10^{11}\ \text{m}$.

1. a) Write the conservation of angular momentum between both points and rearrange for the ratio $\frac{v_p}{v_a}$.
2. b) Calculate this ratio and interpret it: how many times faster does the comet travel near the Sun compared to far from it?
3. c) The comet takes about $6$ and a half years to complete a full orbit: reason in which part of the orbit it spends most of that time.

The Earth passes through aphelion, its furthest point from the Sun, at $1,521 \cdot 10^{11}\ \text{m}$ with a speed of $29,3\ \text{km/s}$; it passes through perihelion at $1,471 \cdot 10^{11}\ \text{m}$.

1. a) Apply the conservation of angular momentum and calculate the speed of the Earth at perihelion, in $\text{km/s}$.
2. b) Find the difference between both speeds and comment on whether it is large or small in relative terms.
3. c) In January, the Earth is at perihelion and yet it is winter in the Northern Hemisphere: in light of the data, comment on why the seasons are not caused by the distance from the Sun.

The exoplanet HD 80606 b has a highly eccentric orbit: at its closest point to its star, it passes at $4,5 \cdot 10^{9}\ \text{m}$ from it with a speed of $1,2 \cdot 10^{5}\ \text{m/s}$, and at its furthest point it reaches a distance of $1,35 \cdot 10^{11}\ \text{m}$.

1. a) Justify why the planet's angular momentum is conserved throughout its orbit.
2. b) Calculate its speed at the furthest point, in standard form.
3. c) Calculate the distance ratio $\frac{r_{lejos}}{r_{cerca}}$ and compare it with the speed ratio: what relationship is there between them?
4. d) The planet takes about $111$ days to complete a full orbit, but spends most of its time far from its star: explain this fact using the conservation of angular momentum.