Conservation of Angular Momentum - UNSOLVED PRACTICE SET

Q7. State the law of conservation of angular momentum. Write the mathematical condition for its validity.

Q8. A spinning disc has moment of inertia I and angular velocity ω. If its moment of inertia is halved by pulling mass inward, what happens to its angular velocity?

Q9. Explain why a planet sweeps equal areas in equal times (Kepler's second law) using conservation of angular momentum.

Q10. A student on a rotating platform holds two heavy books with outstretched arms. What happens to the platform's rotation when she drops the books? Explain.

Q11. A comet has a highly elliptical orbit around the Sun. Explain why its speed is maximum at perihelion and minimum at aphelion.

Q12. Can angular momentum be conserved even when kinetic energy is not conserved? Give an example.

Q13. State and prove the law of conservation of angular momentum. Starting from τ = dL/dt, show that when τ_external = 0, L = constant. Discuss at least three real-life applications:

(i) A spinning ice skater

(ii) A diver performing somersaults

(iii) Planetary motion (Kepler's second law)

For each, explain how conservation of angular momentum explains the observed phenomenon.

Q14. A uniform disc of mass M and radius R rotates about its central axis with angular velocity ω₀. A small mass m is gently placed on the disc at a distance R/2 from the centre and sticks to it.

(i) Calculate the initial angular momentum of the system.

(ii) Calculate the new moment of inertia of the system.

(iii) Calculate the final angular velocity.

(iv) Calculate the change in kinetic energy. Where does this energy go?

(v) What would happen if the mass were placed at the centre instead?

Q15. A student investigates conservation of angular momentum using a rotating platform.

(i) Describe an experiment to verify conservation of angular momentum.

(ii) She stands on the platform holding a spinning bicycle wheel. When she flips the wheel upside down, what happens to the platform? Explain.

(iii) She drops a ball vertically onto the spinning platform. Does angular momentum remain conserved? Explain carefully.

(iv) How would the experiment differ if performed on a frictionless surface versus a rough surface?

Q16. A uniform solid disc of mass 4 kg and radius 0.5 m rotates about its central axis at 10 rad/s. A thin ring of mass 2 kg and radius 0.5 m is gently placed coaxially on the disc and sticks to it.

(i) Calculate the initial angular momentum.

(ii) Calculate the final moment of inertia.

(iii) Calculate the final angular velocity.

(iv) Calculate the loss in kinetic energy.

(v) If the ring were placed on the disc when the disc was at rest and then a torque was applied, would the final angular velocity be different? Explain.

Q17. A planet of mass m orbits the Sun in an elliptical path. At perihelion (closest approach), its distance from the Sun is r₁ and speed is v₁. At aphelion (farthest point), the distance is r₂ = 2r₁.

(i) Apply conservation of angular momentum to find the speed at aphelion.

(ii) Calculate the ratio of kinetic energies at perihelion and aphelion.

(iii) Calculate the ratio of potential energies at perihelion and aphelion.

(iv) Show that the total mechanical energy is the same at both points.

(v) If the planet's orbit were circular with radius r₁, what would be its orbital speed?

Q18. In a school physics demonstration, a student stands on a rotating platform (moment of inertia 4 kg·m²) holding a 5 kg mass in each hand. Initially, the arms are outstretched (total moment of inertia 10 kg·m²) and the platform rotates at 1 rad/s. The masses are 0.8 m from the axis.

(i) Calculate the initial angular momentum.

(ii) The student pulls the masses to 0.2 m from the axis. Calculate the new moment of inertia and angular velocity.

(iii) Calculate the initial and final kinetic energies. Account for the difference.

(iv) The student now throws both masses tangentially outward at 2 m/s relative to the platform. Calculate the new angular velocity of the platform.

(v) A teacher asks: "If the student had simply dropped the masses, would the platform's angular velocity change?" Answer with reasoning.