Rotational Motion

Given:

• A force is applied horizontally at the center of mass of a rolling cylindrical object of mass and radius
• Coefficient of friction = , and the object rolls without slipping
• Acceleration due to gravity =
• We need to find the linear acceleration of the center of mass

Important condition: No slipping means the point of contact has zero relative velocity, hence:

Translational motion:
Net force on the center of mass:

Rotational motion about center:

For a thin-walled hollow cylinder:
Moment of inertia about the center is: Substituting in equation (2):

Now substitute (3) into (1):

✓ Correct Answer: Option (D): For a thin-walled hollow cylinder,

Given:

• Mass of particle:
• Initial position:
• Constant acceleration along +x:
• Initial velocity = 0

We are to calculate: Angular momentum and torque w.r.t. origin, especially at the point

At t = 0:

• Initial position:
• Initial velocity:
• Initial angular momentum:
• Torque:
• ✓ Angular momentum and torque at t = 0 are both zero

At the point (x = +10, y = -1):

This is a displacement of 20 m along x-axis from the starting point, so using:

Velocity at this point:

Position vector:

Linear momentum:

Angular momentum:

✓ Correct: Option C — Angular momentum is at

Correct Answer: Option C

The resultant velocity of point P is represented as , making an angle of 60° with the horizontal, where

Answer: 0.52

(1) For 𝜇 ≠ 0 and 𝑎 → 0, . This statement suggests that when the coefficient of friction is nonzero and the inner radius (a) tends to 0, the height (ℎ_𝑚) at which the impulse is applied is equal to half the outer radius( ). This statement is correct. When the inner radius approaches 0, the entire disk participates in the rolling motion, and the height at which the impulse is applied becomes equal to half the outer radius. Therefore, this option is correct.
(2) For 𝜇 ≠ 0 and 𝑎 → 𝑏, ℎ_𝑚 = 𝑏. This statement suggests that when the coefficient of friction is nonzero and the inner radius (a) tends to the outer radius (b), the height (ℎ𝑚) at which the impulse is applied is equal to the outer radius (b). This statement is correct. When the inner radius approaches the outer radius, the entire disk participates in the rolling motion, and the height at which the impulse is applied becomes equal to the outer radius. Therefore, this option is correct.
(3) For ℎ = ℎ_𝑚, the initial angular velocity does not depend on the inner radius a. This statement is correct. The initial angular velocity of the disk does not depend on the inner radius a. The initial angular velocity is determined by the impulse applied and the moment of inertia of the disk. The inner radius does not affect these factors. Therefore, this option is correct.
(4) For 𝜇 = 0 and ℎ = 0, the wheel always slides without rolling. This statement is correct. When the coefficient of friction is zero and the impulse is applied at the same height as the center of mass (ℎ = 0), the wheel will slide without rolling. In this case, there is no friction force to initiate rolling motion, so the wheel will simply slide. Therefore, this option is correct.
Therefore correct statements are (A), (B), (C) and (D)

By impulse – momentum theorem : J=MVCM

Total time of journey=
By angular impulse-momentum theorem,

rad/s

Number of rotations =

Angular impulse about


Linear impulse =



As disc is pure rolling,



is independent of radius a…… (Ans.1)
Now, If , then the annular disc is a ring of radius b




…….(Ans.2)

if h=0 and =0,
as h=0 = 0
but ………(Ans.3)
If a 0then annular disc is disc of radius b

……..(Ans.4)