Rotational Motion – Stage 2 (Page 3)

Long Answer Questions & Derivations

This section covers important derivations and explanations frequently asked in Board examinations. Write answers neatly with definitions, equations, and conclusions.

Q1. Derive the expression for angular momentum and state its conservation.

Definition:

Angular momentum of a particle about a point is defined as the vector product of position vector and linear momentum.

L = r × p

Relation with torque:

τ = dL/dt

If net external torque acting on a system is zero, angular momentum remains constant.

Conclusion: Angular momentum is conserved in absence of external torque.

Q2. Derive the expression for rotational kinetic energy.

Consider a rigid body rotating with angular velocity ω about a fixed axis. Each particle has velocity v = rω.

K = Σ (½ m v²)

Substituting v = rω,

K = ½ ω² Σ m r² = ½ I ω²

Hence, rotational kinetic energy = ½ Iω².

Q3. State and explain the parallel axis theorem.

Parallel axis theorem states that the moment of inertia of a body about any axis is equal to the moment of inertia about a parallel axis through the center of mass plus Md².

I = I_cm + Md²

This theorem helps calculate moment of inertia about axes not passing through the center of mass.

Q4. Explain rolling motion and derive the condition for pure rolling.

Rolling motion is a combination of translational and rotational motion.

For pure rolling, the point of contact must be momentarily at rest.

Hence, v = ωR

This is called the no-slipping condition.

Q5. Derive the equation τ = Iα.

Consider a particle of mass m at distance r from axis.

Torque τ = rF

Using F = ma and a = rα,

τ = mr²α

For a rigid body:

τ = Iα

This is Newton’s second law for rotational motion.

Board Exam Writing Tips

✔ Always start with definition
✔ Write formulas clearly
✔ Draw neat diagram if required
✔ End with final result

Next: Stage 2 – Page 4
Numerical Problems (Board-level, step-wise)

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