Rigid body mechanics practice questions.

A force of 12 N is applied to the end of a rod 0.50 m from the pivot, at an angle of 30° to the rod. Calculate the magnitude of the torque about the pivot.

• 3.0 N m
• 6.0 N m
• 5.2 N m
• 1.5 N m

Answer: 3.0 N m

τ = Fr sin θ = 12 × 0.50 × sin 30° = 3.0 N m; using cos 30° instead of sin 30° gives the distractor 5.2 N m.

Two point masses, each of mass m, are attached to a light rod and rotate about an axis through one end. One mass is at distance r from the axis and the other at distance 2r. What is the moment of inertia of the system?

• 3mr²
• 4mr²
• 5mr²
• 9mr²

Answer: 5mr²

I = Σmr² = m r² + m(2r)² = mr² + 4mr² = 5mr²; the distance must be squared for each mass separately.

A flywheel has moment of inertia 2.0 kg m². A constant net torque of 8.0 N m acts on it. Calculate its angular acceleration.

• 16 rad s⁻²
• 4.0 rad s⁻²
• 0.25 rad s⁻²
• 8.0 rad s⁻²

Answer: 4.0 rad s⁻²

The rotational form of Newton's second law gives α = τ/I = 8.0/2.0 = 4.0 rad s⁻².

An ice skater spinning freely pulls her arms in so that her moment of inertia halves. Air resistance and friction are negligible. What happens to her rotational kinetic energy?

• It halves
• It stays the same
• It quadruples
• It doubles

Answer: It doubles

Angular momentum L = Iω is conserved, so ω doubles; since E_k = L²/(2I) and I halves, the kinetic energy doubles — the extra energy comes from the work done pulling the arms in.

A solid sphere and a hollow sphere of equal mass and radius are released from rest at the top of the same incline and roll without slipping. Which reaches the bottom first, and why?

• The solid sphere, because its moment of inertia is smaller so less energy goes into rotation
• The hollow sphere, because its moment of inertia is larger so it stores more rotational energy
• They arrive together, because their masses and radii are equal
• The solid sphere, because it experiences a larger gravitational force

Answer: The solid sphere, because its moment of inertia is smaller so less energy goes into rotation

The body with the smaller I/mr² converts a larger share of the lost potential energy into translational kinetic energy, so the solid sphere (I = 2/5 mr²) is faster than the hollow one (I = 2/3 mr²).

A uniform solid cylinder (I = ½mr²) rolls without slipping along level ground. What fraction of its total kinetic energy is rotational?

• 1/2
• 1/3
• 1/4
• 2/3

Answer: 1/3

With v = ωr, E_rot = ½(½mr²)ω² = ¼mv² and E_trans = ½mv², so E_rot/E_total = (1/4)/(3/4) = 1/3.