Theorems Parallel and Perpendicular Axis - UNSOLVED PRACTICE SET
Chapter: System of Particles and Rotational Motion | Topic: Theorems Parallel and Perpendicular Axis
THEOREMS PARALLEL AND PERPENDICULAR AXIS - UNSOLVED PRACTICE SET
Topic: Theorems Parallel and Perpendicular Axis
Multiple Choice Questions
Q1. The parallel axis theorem states that the moment of inertia about any axis is equal to the moment of inertia about a parallel axis through the centre of mass plus:
- Ma
- Ma²
- M/a
- M/a²
Q2. The perpendicular axis theorem applies to:
- Any three-dimensional body
- Plane laminar bodies only
- Spherical bodies only
- Cylindrical bodies only
Q3. For a plane lamina, if I_x and I_y are moments of inertia about two perpendicular axes in the plane, then the moment of inertia about the axis perpendicular to the plane is:
- I_x + I_y
- I_x – I_y
- I_x × I_y
- √(I_x² + I_y²)
Q4. The moment of inertia of a disc about a tangent in its plane is:
- (1/2)MR²
- (3/2)MR²
- (5/4)MR²
- (7/5)MR²
Q5. For a ring of mass M and radius R, the moment of inertia about a tangent perpendicular to its plane is:
- MR²
- 2MR²
- 3MR²
- (3/2)MR²
Q6. A thin rectangular plate lies in the xy-plane. The perpendicular axis theorem tells us that:
- I_z = I_x + I_y
- I_z = I_x – I_y
- I_x = I_y + I_z
- I_z = I_x × I_y
Short Answer Questions
Q7. State the parallel axis theorem. Write its mathematical expression and explain each term.
Q8. State the perpendicular axis theorem. What are the conditions for its applicability?
Q9. The moment of inertia of a solid sphere about its diameter is (2/5)MR². Use the parallel axis theorem to find the moment of inertia about a tangent.
Q10. The moment of inertia of a uniform rod of mass M and length L about its centre is (1/12)ML². Find the moment of inertia about one end using the parallel axis theorem.
Q11. For a circular disc in the xy-plane with centre at origin, I_x = I_y = (1/4)MR². Use the perpendicular axis theorem to find I_z.
Q12. Explain why the perpendicular axis theorem cannot be applied to a solid cylinder.
Long Answer Questions
Q13. State and prove the parallel axis theorem. Consider a body of mass M with moment of inertia I_cm about an axis through its centre of mass. Show that the moment of inertia about a parallel axis at distance d is I = I_cm + Md². Draw a clear diagram and show all steps of the proof.
Q14. State and prove the perpendicular axis theorem. Consider a plane lamina in the xy-plane. Show that I_z = I_x + I_y, where z-axis is perpendicular to the plane. Discuss the limitations of this theorem and why it cannot be applied to three-dimensional bodies.
Q15. Using the parallel and perpendicular axis theorems, derive the moments of inertia for:
(i) A solid disc about a diameter
(ii) A solid disc about a tangent in its plane
(iii) A solid disc about a tangent perpendicular to its plane
(iv) A solid sphere about a tangent
For each case, start from the known moment of inertia about the central axis and apply the appropriate theorem(s).
Numerical / Application-Based Problems
Q16. A uniform rod of mass 2 kg and length 1 m has the following moments of inertia:
About its centre, perpendicular to its length: I_c = (1/12)ML²
Calculate:
(i) The moment of inertia about one end, perpendicular to the rod
(ii) The moment of inertia about a point 0.25 m from one end, perpendicular to the rod
(iii) The moment of inertia about its centre, parallel to the rod
(iv) The moment of inertia about an axis through one end, making 30° with the rod
(v) Verify your answers for parts (i) and (ii) by direct integration
Q17. A uniform circular disc of mass 5 kg and radius 0.4 m lies in the xy-plane with its centre at the origin.
(i) Calculate I_x and I_y (moments of inertia about diameters along x and y axes).
(ii) Use the perpendicular axis theorem to calculate I_z.
(iii) Calculate the moment of inertia about a tangent in the plane of the disc (parallel to x-axis).
(iv) Calculate the moment of inertia about a tangent perpendicular to the plane of the disc.
(v) A point mass 2 kg is attached to the rim at (R, 0). Calculate the new I_z.
Q18. In a school physics lab, a student has a composite body made of a solid disc (mass 3 kg, radius 0.3 m) and a concentric ring (mass 2 kg, radius 0.3 m) welded together. The body rotates about an axis.
(i) Calculate the moment of inertia about the central axis perpendicular to the plane.
(ii) Calculate the moment of inertia about a diameter.
(iii) The body is now shifted so that the axis passes through a point on the rim, perpendicular to the plane. Calculate the new moment of inertia.
(iv) Calculate the radius of gyration for each case.
(v) The student claims that the perpendicular axis theorem can be used to find the moment of inertia about any axis through the centre. Is this correct? Explain with reasoning.