Questions: Torque and Angular Acceleration Relations

Question 1

Two identical masses hang on either side of a pulley. In which scenario does the system have lower linear acceleration?

• A. When the pulley is very light (negligible moment of inertia)
• B. When the pulley is massive (large moment of inertia)
• C. The pulley's mass doesn't affect linear acceleration — only the hanging masses matter
• D. When the rope connecting the masses is longer

Answer: (B)

A massive pulley has a large moment of inertia I, so applying τ_net = Iα, more of the net driving force goes into angularly accelerating the pulley rather than linearly accelerating the hanging masses. This is directly analogous to adding mass to a linear system: more inertia means less acceleration for the same net force. The massless-pulley approximation always overestimates linear acceleration.

Question 2

Two solid cylinders have identical total mass but different radii. Which has the greater moment of inertia about its central axis?

• A. The smaller cylinder — less volume means mass is more concentrated near the axis
• B. The larger cylinder — more of its mass sits at greater distances from the axis, and I scales with r²
• C. They are equal — same mass means same rotational inertia
• D. The larger cylinder — because it has greater surface area

Answer: (B)

Moment of inertia is I = ∫r² dm — mass at distance r from the axis contributes r² dm. A larger cylinder (same total mass spread over a larger radius) has more mass at greater r values, yielding a higher I. Equal total mass does NOT mean equal rotational inertia — distribution matters quadratically.

Question 3

The moment of inertia of a rigid body depends mainly on its total mass, not on how that mass is distributed around the rotation axis.

• T. True
• F. False

Answer: False

Moment of inertia explicitly depends on mass distribution: I = ∫r² dm. A hollow cylinder (all mass at the rim) has a greater moment of inertia than a solid cylinder of the same mass and radius, because all the mass sits at maximum r. The r² weighting means even modest redistributions of mass — like a figure skater extending their arms — have large effects on I.

Question 4

In a pulley-and-mass system where the pulley has a nonzero moment of inertia, the rope tension must be different on the two sides of the pulley.

• T. True
• F. False

Answer: True

For the pulley to angularly accelerate, there must be a net torque: τ_net = (T₁ − T₂)R = Iα. This requires T₁ ≠ T₂. If the tensions were equal, the net torque on the pulley would be zero, meaning no angular acceleration — inconsistent with the hanging masses accelerating. Unequal tension is the defining feature of a massive pulley problem and is what distinguishes it from the massless-pulley case.

Question 5

How is τ_net = Iα analogous to F_net = ma, and what does moment of inertia represent in this analogy?

Model answer: Just as F = ma says net force produces linear acceleration inversely proportional to mass, τ = Iα says net torque produces angular acceleration inversely proportional to moment of inertia. Moment of inertia is the rotational analog of mass — it quantifies resistance to changes in angular velocity. Unlike mass (a fixed scalar), I depends on mass distribution relative to the rotation axis: the r² weighting means mass farther from the axis contributes quadratically more to rotational inertia.

The analogy is exact: every linear quantity has a rotational counterpart. Force ↔ torque, mass ↔ moment of inertia, linear acceleration ↔ angular acceleration, and F = ma ↔ τ = Iα. Exploiting this structural parallel makes rotational dynamics much easier to set up and solve.