Exercises: 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 torque 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 $I_1 = 5\ \text{kg} \times \text{m}^2$ and an angular velocity $\omega_1 = 4\ \text{rad/s}$. By pulling in her arms, her moment of inertia is reduced to $I_2 = 2\ \text{kg} \times \text{m}^2$. The figure summarizes the change:

1. a) Calculate her initial angular momentum, $L = I_1 \times \omega_1$.
2. b) Apply the conservation of angular momentum, $I_1 \times \omega_1 = I_2 \times \omega_2$, 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?

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 farthest 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: $m \times v_a \times r_a = m \times v_p \times r_p$.
3. c) Reason, from this equation, at which point the planet moves fastest and at which it moves slowest.

A planet orbits around a star and at aphelion is at $4 \times 10^{11}\ \text{m}$ from it, moving at $10\ \text{km/s}$. At perihelion, the distance decreases to $2 \times 10^{11}\ \text{m}$. The figure shows the orbit (not to scale):

1. a) Write the equation for conservation of angular momentum between aphelion and perihelion.
2. b) Calculate the speed of the planet at perihelion, in $\text{km/s}$.
3. c) Explain physically why its speed has increased.

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).

The main rotor of a helicopter spins in one direction and, in doing so, the fuselage tends to spin in the opposite direction; the small tail rotor is responsible for preventing this. The diagram shows the helicopter seen from above:

1. a) Before takeoff, the total angular momentum of the helicopter is zero. Applying the conservation of angular momentum, explain why the fuselage tends to rotate when the main rotor starts up.
2. b) What does the tail rotor do to prevent this rotation? Explain your reasoning.
3. c) Helicopters with two main rotors have them spin in opposite directions: explain, using the conservation of angular momentum, why they do not need a tail rotor.

A carnival merry-go-round rotates with angular velocity $\omega_1$ with all the children seated near the center. When they move to the seats at the edge, the total moment of inertia of the system doubles.

1. a) Write the equation for the conservation of angular momentum for this situation.
2. b) Deduce, without further data, the final angular velocity in terms of $\omega_1$.
3. c) In reality, the merry-go-round hardly slows down because the motor maintains the speed: is angular momentum conserved in the real merry-go-round? Reason your answer.

Comet 67P, visited by the Rosetta probe, passes through perihelion at $1.86 \times 10^{11}\ \text{m}$ from the Sun and through aphelion at $8.5 \times 10^{11}\ \text{m}$.

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 farthest point from the Sun, at $1.521 \times 10^{11}\ \text{m}$ with a speed of $29.3\ \text{km/s}$; it passes through perihelion at $1.471 \times 10^{11}\ \text{m}$.

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.

Mercury is the planet with the most elongated orbit in the Solar System: it passes through perihelion at $4.6 \times 10^{10}\ \text{m}$ from the Sun with a speed of $59\ \text{km/s}$ and through aphelion at $7.0 \times 10^{10}\ \text{m}$.

1. a) Apply the conservation of angular momentum and calculate Mercury's speed at aphelion, in $\text{km/s}$.
2. b) Express both speeds in $\text{m/s}$, in scientific notation.
3. c) Calculate the ratio $\frac{v_p}{v_a}$ and the distance ratio $\frac{r_a}{r_p}$: check that they match and explain why they must do so.

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

1. a) Express the initial angular velocity in $\text{rad/s}$ (one revolution is equal to $2 \times \pi\ \text{rad}$).
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 divers tuck to perform multiple somersaults and extend before entering the water.

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 $I = 400\ \text{kg} \times \text{m}^2$. 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, $L = m \times v \times r$.
2. b) Calculate the total moment of inertia after the jump, $I_{total} = I + m \times r^2$.
3. c) Apply the conservation of angular momentum and calculate the angular velocity at which the combined system begins to spin.

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 summarizes 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.

Comet Halley crosses perihelion at $8.8 \times 10^{10}\ \text{m}$ from the Sun with a speed of $54.5\ \text{km/s}$; at aphelion it moves away to $5.2 \times 10^{12}\ \text{m}$.

1. a) Calculate the speed of the comet at aphelion by applying the conservation of angular momentum, in $\text{km/s}$.
2. b) Express that speed in $\text{m/s}$, in scientific notation.
3. c) The mass of the comet does not appear in the calculation: justify why it can be canceled in the conservation equation.

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 $I = 0.2\ \text{kg} \times \text{m}^2$. 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 gyroscope is a wheel that rotates at high angular velocity about its axis. Satellites and the International Space Station carry reaction wheels based on this principle to orient themselves without expending fuel.

1. a) The vector $\vv{L}$ of a wheel rotating freely in space is conserved in magnitude, direction, and sense: explain this using the torque of external forces.
2. b) Deduce why a gyroscope maintains a fixed orientation of its axis in space and what usefulness this has as a navigation reference.
3. c) If a reaction wheel on the satellite increases its speed of rotation in one direction, what happens to the rest of the satellite? Explain your answer using the conservation of total angular momentum.

The exoplanet HD 80606 b has a highly eccentric orbit: at its closest point to its star, it passes at $4.5 \times 10^{9}\ \text{m}$ from it with a speed of $1.2 \times 10^{5}\ \text{m/s}$, and at its farthest point it reaches a distance of $1.35 \times 10^{11}\ \text{m}$.

1. a) Justify why the planet's angular momentum is conserved throughout its orbit.
2. b) Calculate its speed at the farthest point, in scientific notation.
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.

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

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.

The core of a giant star, of radius $R_1 = 2 \times 10^{8}\ \text{m}$, rotates with a period of $20\ \text{h}$. When its fuel is exhausted, it collapses and forms a neutron star of radius $R_2 = 1 \times 10^{4}\ \text{m}$, conserving its angular momentum. Model the core as a sphere, $I \propto R^2$.

1. a) Write the conservation equation $I_1 \times \omega_1 = I_2 \times \omega_2$ in terms of the radii and deduce the factor $\frac{\omega_2}{\omega_1}$.
2. b) Calculate this factor using the given data.
3. c) Express the initial period in seconds and calculate the final period of the neutron star, in scientific notation.
4. d) Real pulsars rotate with periods of milliseconds: comment on whether your result is compatible and what it implies about the violence of stellar collapse.

A playground merry-go-round ($I = 100\ \text{kg} \times \text{m}^2$) 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 center of the merry-go-round.

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

The Earth-Moon system conserves its total angular momentum almost perfectly. Due to tidal effects, the Earth's rotation slows down: the day lengthens by about $2\ \text{ms}$ per century, and the Moon moves away from us by about $3.8\ \text{cm}$ per year.

1. a) The total angular momentum of the system is the sum of the Earth's rotational angular momentum and the Moon's orbital angular momentum: write this sum, identifying each term.
2. b) If the Earth's rotational angular momentum decreases, reason what must happen to the Moon's orbital angular momentum for the total to be conserved.
3. c) The orbital angular momentum of the Moon is $m \times v \times r$: explain why moving away (increasing $r$) increases this momentum even though the Moon moves somewhat slower (in a circular orbit $v$ decreases only as $\frac{1}{\sqrt{r}}$, whereas $r$ increases).
4. d) Extrapolate the situation into the very long term: what will happen to the length of the Earth's day and to the distance to the Moon if the process continues?

A satellite describes a circular orbit of radius $r_1 = 7 \times 10^{6}\ \text{m}$ with a speed of $7.5 \times 10^{3}\ \text{m/s}$. To move to a higher orbit, it fires its engine briefly in the direction of motion (tangential thrust) and increases its speed by 20%. The diagram shows the orbit and the engine burn:

1. a) Calculate the angular momentum per unit mass before the burn, $\frac{L}{m} = v \times r$.
2. b) Is the engine thrust a central force? Is angular momentum conserved during the burn? Reason both answers.
3. c) Calculate the new speed and the new angular momentum per unit mass immediately after the burn.
4. d) With the engine now switched off, the satellite follows an elliptical orbit moving away from the burn point: describe, using the conservation of angular momentum, how its speed and its distance from the center of the Earth evolve until it reaches the highest point.

The International Space Station (total moment of inertia $I_{EEI} = 1 \times 10^{8}\ \text{kg} \times \text{m}^2$) is oriented using reaction wheels: internal flywheels of $I_r = 100\ \text{kg} \times \text{m}^2$ each. Assume the station is initially at angular rest, with the wheels stationary, and neglect the moment of inertia of the wheels compared to that of the station.

1. a) What is the total angular momentum of the station plus wheels system in this initial state?
2. b) A wheel starts rotating at $\omega_r = 300\ \text{rad/s}$. Applying the conservation of total angular momentum, calculate the angular velocity at which the station begins to rotate in the opposite direction.
3. c) Express the result in revolutions per hour and comment on whether this method allows slow but controlled orientation maneuvers without expending fuel.

A skater spins on ice holding a dumbbell of $2\ \text{kg}$ in each hand with arms extended, at $0.8\ \text{m}$ from the axis of rotation, with $\omega_1 = 5\ \text{rad/s}$. The moment of inertia of his own body (with arms) is $I_p = 3\ \text{kg} \times \text{m}^2$. At a given moment, he lets go of the dumbbells without throwing them: he simply opens his hands.

1. a) Calculate the initial total moment of inertia, summing that of the skater and that of the two dumbbells, treated as point masses at $0.8\ \text{m}$ from the axis.
2. b) Calculate the initial total angular momentum.
3. c) Once released, the dumbbells move in a straight line, carrying their angular momentum about the axis with them: calculate what fraction of the initial angular momentum corresponded to the dumbbells and deduce the skater's final angular velocity.
4. d) Compare the result with what would have happened if he had pulled in his arms without dropping anything: is it the same effect? Justify your answer.

Unit challenge. Design a scientific-technical report linking angular momentum dynamics to relativistic effects in extreme orbits.

1. Home experiment: Tie a small object ($m = 25\text{ g} \pm 1\text{ g}$) to a string. Spin it horizontally under a central force with radii $r \in \{0.30; 0.45; 0.60\}\text{ m}$ ($\pm 0.01\text{ m}$). Time $10$ revolutions ($\pm 0.2\text{ s}$) to calculate the period $T$ and the speed $v$. Build a data table with uncertainties, plot $L = m \times r \times v$ versus $r$, and discuss whether angular momentum is conserved.

2. Relativistic leap: Look up the speed of a fast probe on the NASA website and compare it to a hypothetical case in a central field at $v = 0.80c$. Determine the factor $\gamma$, the contraction of a proper length $L_0$ parallel to the velocity ($L' = L_0/\gamma$), the daily time dilation ($\Delta t = \gamma \Delta t_0$), and the total relativistic energy ($E = \gamma m c^2$).

3. Final product: Submit a reasoned proposal that includes the experimental data table, the graph, and the comparative analysis between classical and relativistic mechanics.

Grading criteria:

• Rigor in home measurement, calculation of uncertainties, and graph quality.

• Conceptual correctness in applying angular momentum, central forces, and relativistic transformations.

• Critical coherence and expository clarity in the final report.