1. State the theorems of parallel and perpendicular axes with reference to moment of inertia.

2. Write the expressions for torsional rigidities of a solid cylinder and a hollow cylinder.

3. Show that the profile of a liquid flowing through a capillary tube is parabolic.

4. What are Newton's laws of viscous flow? Derive an expression for rate of flow of a liquid through capillaries in series.

5. Prove that for an elastic medium

where the symbols have their usual meanings.

6. What must be the relation between length l and radius R of a cylinder if the moment of inertia about the axis is to be same as its moment of inertia about a the equatorial axis?

7. Calculate the velocity of water along the axis of a capillary tube through which it is flowing under a pressure difference of 10 cm of water. The length of the capillary tube is 1 m and its diameter is 1 mm. The coefficient of viscosity of water is 0.001 kg/m s.

8. Show that torque is given by the time rate of change of angular momentum.

9. Define elastic constants.

10. Write the physical significance of moment of inertia.

11. State the theorems of parallel and perpendicular axis for moment of inertia

12. Show that torque is given by the time rate of change of angular momentum.

13. State and prove the theorem of perpendicular axis for moment of inertia.

14. Show that

where symbols have their usual meanings.

15. Define angular momentum. How is it related with torque?

16. State and prove the law of conservation of angular momentum.

17. Explain why hollow shafts are preferred over solid ones for transmitting large torques in a rotating machinery.

18. Define moment of inertia and radius. of gyration. Find the moment of inertia of a solid cylinder about an axis passing perpendicularly through the middle of its length.

19. Define Young's modulus, rigidity modulus and Poisson's ratio. Obtain the relation Y = 2η(1 + σ), where Y= Young's modulus, η = rigidity modulus and σ = Poisson's ratio

20. Show that torque is given by the time rate of change of angular momentum.

21. Explain elasticity and reason of elasticity.

22. What do you mean by restoring torque?

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

24. Define moment of inertia. What is its physical significance? Calculate the moment of inertia of a uniform circular disc about a diameter of the disc.

25. Connecting the three elastic constants, derive the following relation :

9 Y = 3 η + 1 K

Here the symbols have their usual meanings.

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26. A cylinder of length l and of radius a is clamped at one end and a torque is applied at the other end. Establish the restoring torque that comes to play during the twisting of the cylinder is given by

τ = πηQ⁴ 2l φ

where η is the modulus of rigidity and φ is the angle of twist.

27. Define moment of inertia and radius of gyration of a body about an axis.

28. The trajectory of a particle of unit mass is given by the radius vector

r = ia cos ωt + jb sin ωt

where a and b are constants. Calculate the angular momentum of the particle about the origin.

28. State Hooke's law. What do you mean by modulus of elasticity?

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29. State the two theorems of moment of inertia.

30. Calculate the moment of inertia of a solid cylinder about
(i) its own axis
(ii) an axis passing through its centre of mass and perpendicular to its own axis.

31. Define Young's modulus (Y), bulk modulus (K), modulus of rigidity (η) and Poisson's ratio (σ).

32. Show that —

(i)  Y = 3K(1 − 2σ)

(ii)  Y = 2η(1 + σ)