Rolling Motion Without Slipping - UNSOLVED PRACTICE SET

Q7. Define rolling motion without slipping. Write the condition for pure rolling.

Q8. A cylinder of mass M and radius R rolls without slipping with speed v. Write expressions for:

(i) Translational kinetic energy

(ii) Rotational kinetic energy

(iii) Total kinetic energy

Q9. Show that for a body rolling without slipping down an inclined plane, the acceleration is a = g sin θ / (1 + I/MR²).

Q10. Why does a rolling body slow down on a horizontal surface even though the point of contact is instantaneously at rest?

Q11. A solid sphere and a hollow sphere of the same mass and radius roll down the same inclined plane. Which one has greater acceleration? Explain.

Q12. What is the velocity of the highest point of a rolling wheel? Explain your reasoning.

Q13. Derive the expression for the acceleration of a body rolling down an inclined plane without slipping. Consider a body of mass M, radius R, and moment of inertia I about its central axis, rolling down a plane inclined at angle θ. Show that:

(i) a = g sin θ / (1 + I/MR²)

(ii) The frictional force is f = Mg sin θ · (I/MR²) / (1 + I/MR²)

(iii) The body rolls without slipping if μ ≥ tan θ / (1 + MR²/I)

Discuss why the acceleration is independent of mass and radius but depends on shape.

Q14. A solid cylinder rolls down an inclined plane of height h and angle θ.

(i) Using energy conservation, find the speed of the cylinder at the bottom.

(ii) Compare this with the speed of a block sliding down a frictionless incline of the same height.

(iii) Calculate the ratio of translational to rotational kinetic energy at the bottom.

(iv) Show that the cylinder takes longer to reach the bottom than a sliding block.

Q15. A student rolls various objects down a ramp in the school lab: a ring, a disc, a solid sphere, and a hollow sphere, all of the same mass and radius.

(i) Predict the order in which they reach the bottom. Explain using the concept of moment of inertia.

(ii) Calculate the acceleration for each object.

(iii) Calculate the speed of each at the bottom if the ramp height is 0.5 m.

(iv) If the ramp is covered with oil (reducing friction), what happens to the motion? Explain.

Q16. A solid cylinder of mass 5 kg and radius 0.2 m rolls without slipping down an inclined plane of angle 30° and length 5 m.

(i) Calculate the moment of inertia of the cylinder.

(ii) Calculate the acceleration of the cylinder.

(iii) Calculate the time taken to reach the bottom.

(iv) Calculate the speed at the bottom using kinematics and verify using energy conservation.

(v) Calculate the minimum coefficient of friction required for rolling without slipping.

(Given: g = 9.8 m/s²)

Q17. A sphere of mass 2 kg and radius 0.1 m rolls without slipping on a horizontal surface with initial velocity 4 m/s. It encounters a rough inclined plane of angle 20°.

(i) Calculate the initial total kinetic energy.

(ii) Calculate how far up the incline the sphere rolls before stopping.

(iii) Calculate the time taken to reach the highest point.

(iv) Will the sphere roll back down? If yes, calculate the speed at the bottom of the incline.

(v) If the coefficient of friction is insufficient for rolling without slipping, describe qualitatively what happens.

Q18. In a school physics demonstration, a student releases a solid disc (mass 0.5 kg, radius 0.15 m) and a ring (mass 0.5 kg, radius 0.15 m) from rest at the top of a ramp of height 0.8 m and length 3 m.

(i) Calculate the acceleration of each object.

(ii) Calculate the time taken by each to reach the bottom.

(iii) Calculate the speed of each at the bottom.

(iv) The student places the disc and ring side by side at the bottom and lets them roll up a second identical ramp. Which one goes higher? Calculate the height reached by each.

(v) A third student claims that if both objects had the same mass but the disc had twice the radius of the ring, the result would be different. Is this correct? Explain.

(Given: g = 9.8 m/s²)