Rotational Motion

Concept: MOI of remaining = MOI of full disc - MOI of removed part.

Step 1: Full disc


Step 2: Small disc mass
MOI about its centre:

Step 3: Shift to centre
Distance =

Step 4: Remaining MOI
Scaling gives: Conclusion:

Concept: Use parallel axis theorem.

Step 1: MOI of one rod about its centre


Step 2: Shift to square centre
Distance =

Step 3: Total for 4 rods
Conclusion:

Concept: Work done = change in rotational KE.

Step 1: Initial KE


Step 2: Final KE = 0
But opposing torque energy dissipated: Conclusion:

Concept: Use parallel axis theorem.

Step 1: Sphere MOI


Step 2: Disc MOI about edge


Step 3: Equate
Conclusion:

Step 1: Understanding the Concept:
For rolling without slipping, friction must be sufficient to provide the necessary torque for angular acceleration.

Step 2: Detailed Explanation:
Linear acceleration: . Friction force: . Normal: . Condition :

Step 3: Final Answer:
Condition for no slipping: .

Step 1: Understanding the Concept:
For rolling: .
Step 2: Detailed Explanation:
For solid sphere,

Step 3: Final Answer:
Acceleration is .