A solid disk of radius 0.5 m rotates at 10 rad/s about its center. What are the linear speeds of a point at the rim (r = 0.5 m) and a point halfway to the rim (r = 0.25 m)?
ABoth move at 10 m/s — all points share the same angular velocity, so they have the same linear speed
BRim: 5 m/s; inner point: 2.5 m/s — linear speed is proportional to distance from the axis
CRim: 0.5 m/s; inner point: 0.25 m/s — linear speed equals the radius at that angular velocity
DThe inner point moves faster because it has a shorter arc to complete per revolution
Linear speed v = rω. For the rim: v = 0.5 × 10 = 5 m/s. For the inner point: v = 0.25 × 10 = 2.5 m/s. All points on a rigidly rotating body share the same angular velocity ω — this is the defining property of fixed-axis rigid rotation. But their linear speeds differ because they travel different arc lengths per radian: a rim point covers a longer path per revolution. Option A is the classic error: shared ω does not imply shared v. Only the angular quantities are uniform across the body; the linear quantities vary with r.
Question 2 Multiple Choice
Two cylinders have identical total mass M and radius R. Cylinder A is solid; Cylinder B is a hollow thin ring with all mass at the rim. The same torque is applied to each. Which angularly accelerates more quickly?
ACylinder B — hollow objects are lighter at the center and therefore easier to spin
BBoth equally — they have the same total mass, so they have the same rotational inertia
CCylinder A — its mass is distributed closer to the axis on average, giving a smaller moment of inertia (I = ½MR² vs I = MR²)
DCylinder B — the hollow ring concentrates mass at large radius, increasing the torque arm
From τ = Iα, for the same torque, smaller I means larger α. The solid cylinder has I = ½MR², while the hollow ring has I = MR² — twice as large, even with the same total mass. This is because I = Σmᵢrᵢ²: the ring has all its mass at radius R (contributing maximum r² per unit mass), while the solid cylinder's mass spans all radii 0 to R, averaging to a smaller effective r². Mass close to the axis barely resists rotation; mass far from the axis resists strongly. Moment of inertia depends on mass distribution, not just total mass.
Question 3 True / False
When a rigid body rotates about a fixed axis, most particles in the body share the same angular velocity and therefore also share the same linear velocity.
TTrue
FFalse
Answer: False
All particles in a rigidly rotating body do share the same angular velocity ω — this is the defining feature of rigid-body rotation about a fixed axis. But linear speed v = rω depends on both ω and the particle's distance r from the axis. Particles farther from the axis have larger r and therefore larger linear speed. Only the angular quantities (ω, α) are uniform throughout the body; the linear quantities (v, tangential acceleration) vary with distance from the axis. A rim point and a hub point on the same wheel have the same ω but very different v.
Question 4 True / False
Redistributing a rotating body's mass farther from the rotation axis — while keeping total mass constant — increases its moment of inertia and makes it harder to change its rotational speed.
TTrue
FFalse
Answer: True
Moment of inertia I = Σmᵢrᵢ² has an r² weighting: a particle at radius 2r contributes four times as much to I as the same particle at radius r. Moving mass farther from the axis therefore increases I substantially. From τ = Iα, a larger I means a given torque produces less angular acceleration — the body is harder to spin up or slow down. This is why flywheels are designed with mass concentrated at the rim (high I, resistant to speed changes), and why a figure skater who extends their arms rotates more slowly.
Question 5 Short Answer
A figure skater spins with arms outstretched, then pulls them tightly to their body and immediately spins faster. Explain this using moment of inertia and angular momentum.
Think about your answer, then reveal below.
Model answer: Angular momentum L = Iω is conserved when no external torque acts (friction from the ice is negligible). When the skater pulls their arms inward, mass moves closer to the rotation axis, decreasing moment of inertia I (since I = Σmᵢrᵢ², smaller r means smaller I). Because L = Iω must remain constant, a decrease in I requires a proportional increase in ω: the skater spins faster. This is a direct consequence of I depending on mass distribution — the same total mass in a compact configuration (small r, small I) spins much faster than when spread out.
The three key ideas: (1) I = Σmᵢrᵢ² — decreasing r decreases I; (2) conservation of angular momentum L = Iω — with no external torque, L is constant; (3) therefore ω must increase when I decreases. The figure skater is a canonical demonstration because the acceleration is immediately felt, making the abstract r² dependence of moment of inertia viscerally intuitive. It also connects fixed-axis rotation to the broader principle of angular momentum conservation.