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Class 11  »  Physics  »  System of Particles and Rotational Motion

Moment of Inertia and Radius of Gyration - UNSOLVED PRACTICE SET

Class 11

Chapter: System of Particles and Rotational Motion | Topic: Moment of Inertia and Radius of Gyration

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Class 11

MOMENT OF INERTIA AND RADIUS OF GYRATION - UNSOLVED PRACTICE SET

Topic: Moment of Inertia and Radius of Gyration

Time: 40 mins | Marks: 30 | Difficulty: Medium

Multiple Choice Questions

Q1. Moment of inertia of a particle of mass m at perpendicular distance r from the axis of rotation is:

  1. mr
  2. mr²
  3. m/r
  4. m/r²

Q2. The SI unit of moment of inertia is:

  1. kg
  2. kg·m
  3. kg·m²
  4. kg·m/s

Q3. The radius of gyration k is defined by:

  1. I = Mk
  2. I = Mk²
  3. I = M/k
  4. I = M/k²

Q4. The moment of inertia of a solid sphere about its diameter is:

  1. (2/5)MR²
  2. (2/3)MR²
  3. (1/2)MR²
  4. MR²

Q5. Two discs of the same mass and thickness but made of different materials (iron and aluminium) have moments of inertia in the ratio:

  1. 1:1
  2. Equal to the ratio of their densities
  3. Equal to the inverse ratio of their densities
  4. Equal to the ratio of their radii

Q6. When a figure skater in your school ice-skating demonstration pulls in her arms, her moment of inertia:

  1. Increases
  2. Decreases
  3. Remains the same
  4. Becomes zero

Short Answer Questions

Q7. Define moment of inertia. Why is it called "moment" of inertia? How does it differ from mass?

Q8. Define radius of gyration. What is its physical significance? Write its SI unit.

Q9. The moment of inertia of a body about a given axis is 20 kg·m² and its mass is 5 kg. Calculate its radius of gyration

Q10. Explain why the moment of inertia depends on:

(i) The mass of the body

(ii) The distribution of mass about the axis

(iii) The position and orientation of the axis

Q11. Compare the moments of inertia of a ring and a disc of the same mass and radius about their central perpendicular axes. Explain the difference.

Q12. A solid cylinder and a hollow cylinder of the same mass and outer radius roll down an inclined plane. Which one reaches the bottom first? Explain in terms of moment of inertia.

Long Answer Questions

Q13. Define moment of inertia and radius of gyration. Derive the expression for the moment of inertia of:

(i) A thin uniform rod about an axis through its centre and perpendicular to its length

(ii) A thin uniform ring about its central axis perpendicular to its plane

(iii) A solid disc about its central axis perpendicular to its plane

For each derivation, clearly state the assumptions and show all integration steps.

Q14. Discuss how moment of inertia depends on the axis of rotation. For a uniform rod of mass M and length L, calculate the moment of inertia about:

(i) An axis through the centre, perpendicular to the rod

(ii) An axis through one end, perpendicular to the rod

(iii) An axis through the centre, parallel to the rod

(iv) An axis through one end, at 30° to the rod

Use the parallel and perpendicular axis theorems where applicable.

Q15. A student is given a composite body consisting of a solid disc of radius R and mass M, with a smaller disc of radius R/2 and mass M/4 cut out from its edge, with the centre of the hole at R/2 from the centre of the large disc.

(i) Calculate the moment of inertia of the complete disc about its central axis.

(ii) Calculate the moment of inertia of the removed disc about the central axis of the large disc.

(iii) Use the subtraction method to find the moment of inertia of the remaining body.

(iv) Calculate the radius of gyration of the remaining body.

Numerical / Application-Based Problems

Q16. Four point masses are placed at the corners of a square of side 0.5 m. The masses are 1 kg, 2 kg, 3 kg, and 4 kg at corners A, B, C, and D respectively.

(i) Calculate the moment of inertia about an axis through A and perpendicular to the plane.

(ii) Calculate the moment of inertia about an axis through the centre and perpendicular to the plane.

(iii) Calculate the moment of inertia about a diagonal AC.

(iv) Calculate the radius of gyration for each case.

(v) Verify the perpendicular axis theorem for this system.

Q17. A uniform solid cylinder has mass 10 kg and radius 0.2 m.

(i) Calculate its moment of inertia about its central axis.

(ii) Calculate its moment of inertia about a parallel axis tangent to its surface.

(iii) Calculate its moment of inertia about a diameter of its circular face.

(iv) A thin-walled hollow cylinder of the same mass and radius is placed coaxially around the solid cylinder. Calculate the total moment of inertia.

(v) If the system rotates at 10 rad/s, calculate the rotational kinetic energy for each configuration.

Q18. In a school physics project, a student builds a "rotational inertia demonstrator" using a metre scale (mass 0.15 kg) with two identical metal discs (each mass 0.5 kg, radius 0.05 m) that can slide along the scale.

(i) Calculate the moment of inertia when the discs are at the 25 cm and 75 cm marks.

(ii) Calculate the moment of inertia when the discs are at the 10 cm and 90 cm marks.

(iii) The scale is rotated about its centre at 2 rad/s in both cases. Calculate the rotational kinetic energy in each case.

(iv) The student pulls the discs from the outer position to the inner position while rotating. Calculate the change in angular velocity if no external torque acts.

(v) Explain where the extra kinetic energy comes from when the discs are pulled inward.


Total: 30 Marks | Time: 40 mins

Explore more topics in System of Particles and Rotational Motion

Centre of Mass Definition and Position Motion of Centre of Mass Angular Velocity and Angular Acceleration Torque and Angular Momentum Conservation of Angular Momentum Equilibrium of Rigid Bodies Theorems Parallel and Perpendicular Axis Rolling Motion Without Slipping