Torque and Angular Momentum - UNSOLVED PRACTICE SET

Q7. Define torque. Write its vector form and explain the significance of each quantity.

Q8. Define angular momentum. Show that for a particle in circular motion, L = mvr = mr²ω.

Q9. A force of 10 N is applied perpendicular to a door at a distance of 0.8 m from the hinge. Calculate the torque.

Q10. Explain why torque is a vector quantity. What is the direction of torque?

Q11. A particle of mass 0.5 kg moves in a circle of radius 2 m with angular velocity 3 rad/s. Calculate its angular momentum about the centre.

Q12. Show that the rate of change of angular momentum of a particle is equal to the torque acting on it.

Q13. Define torque and angular momentum for a single particle. Derive the relation τ = dL/dt. Starting from Newton's second law for linear motion, show how the rotational analogue emerges. Extend this to a system of particles and discuss the conditions under which angular momentum is conserved.

Q14. A particle of mass m moves with velocity v along a straight line at perpendicular distance b from a fixed point O.

(i) Calculate the angular momentum of the particle about O.

(ii) Show that this angular momentum remains constant as the particle moves.

(iii) If a force acts on the particle directed towards O, what happens to the angular momentum?

(iv) Explain how this is analogous to planetary motion.

Q15. A student tries to open a heavy classroom door by pushing at different points.

(i) She pushes with 20 N at the edge farthest from the hinge (0.9 m away). Calculate the torque.

(ii) She pushes with the same force but at the midpoint (0.45 m from hinge). Calculate the torque and compare.

(iii) She pushes with 40 N at the midpoint. Is the torque the same as in part (i)?

(iv) She pushes at the hinge with 100 N. What is the torque? Explain why the door doesn't move.

(v) What is the minimum force needed to produce the same torque as in part (i), if applied at 0.3 m from the hinge?

Q16. A particle of mass 2 kg moves in the xy-plane with position vector r = (3t i + 4t² j) metres.

(i) Calculate the velocity vector at t = 2 s.

(ii) Calculate the linear momentum at t = 2 s.

(iii) Calculate the angular momentum about the origin at t = 2 s.

(iv) Calculate the torque about the origin at t = 2 s.

(v) Verify that τ = dL/dt.

Q17. A uniform rod of length 2 m and mass 5 kg is hinged at one end and held horizontally. It is released from rest.

(i) Calculate the torque about the hinge due to gravity when the rod is horizontal.

(ii) Calculate the initial angular acceleration.

(iii) Calculate the angular velocity when the rod becomes vertical. (Use energy method)

(iv) Calculate the angular momentum about the hinge when the rod is vertical.

(v) Calculate the torque about the hinge when the rod makes 30° with the horizontal.

Q18. In a school physics demonstration, a student sits on a rotating stool holding two 2 kg dumbbells with arms outstretched. The student-stool system has moment of inertia 5 kg·m² with arms outstretched and 2 kg·m² with arms pulled in. Initially, the student rotates at 2 rad/s with arms outstretched.

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

(ii) The student pulls the dumbbells close to the body. Calculate the new angular velocity.

(iii) Calculate the ratio of final to initial kinetic energy. Where does the extra energy come from?

(iv) The student now extends the arms again. What happens to the angular velocity?

(v) A second student claims that angular momentum is not conserved because the student does work in pulling the dumbbells. Is this correct? Explain.