What is the moment of inertia of a solid sphere?

I = (2/5)MR² about any diameter. For a hollow sphere it is (2/3)MR².

State and explain parallel axis theorem.

I = I_cm + Md². The moment of inertia about any axis equals the MI about a parallel axis through the centre of mass plus the product of mass and square of the distance between the axes.

Why does a skater spin faster when arms are pulled in?

No external torque → angular momentum L = Iω is conserved. Pulling arms in reduces I, so ω must increase to keep L constant.

Moment of Inertia

System of Particles and Rotational Motion • Class 11 Physics • NCERT • CBSE

Moment of Inertia I = Σmr². Torque τ = Iα. Angular momentum L = Iω. Parallel axis theorem: I = I_cm + Md². Perpendicular axis theorem (lamina): I_z = I_x + I_y.

Key Formulas

I = Σmr² = ∫r² dm
τ = Iα (rotational Newton's second law)
L = Iω (angular momentum)
I = I_cm + Md² (parallel axis theorem)
I_z = I_x + I_y (perpendicular axis theorem)

Frequently Asked Questions

What is the moment of inertia of a solid sphere?: I = (2/5)MR² about any diameter. For a hollow sphere it is (2/3)MR².
State and explain parallel axis theorem.: I = I_cm + Md². The moment of inertia about any axis equals the MI about a parallel axis through the centre of mass plus the product of mass and square of the distance between the axes.
Why does a skater spin faster when arms are pulled in?: No external torque → angular momentum L = Iω is conserved. Pulling arms in reduces I, so ω must increase to keep L constant.

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