Q5 · Rotation

Problem

Uniform rod pivoted at O rotates horizontally with angular speed $\omega$. Insect walks from O to end at constant speed v (relative to rod), reaches end at t = T. Angular speed stays $\omega$. Plot |τ| vs t.

Key insight: Torque = rate of change of angular momentum. Insect adds time-varying I as it walks outward.

Setup

flowchart TB
  R["Rod ω = const"] --> I["I = m(vt)² increases"]
  I --> L["L = Iω ∝ t²"]
  L --> Tau["τ = dL/dt ∝ t"]

1. Angular momentum

Insect at distance r = vt from pivot.

$$L = m(vt)^2 \omega$$

2. Differentiate

ω constant.

$$\tau = \frac{dL}{dt} = 2mv^2\omega t$$

3. Graph

τ ∝ t — linear through origin.

$$\Rightarrow (B)$$

Deep dive

System angular speed ω stays constant (given), but moment of inertia increases as insect moves: I = m(vt)². So L = Iω = m(vt)²ω ∝ t². Torque τ = dL/dt ∝ t — a straight line through the origin.

Common pitfalls

• Assuming τ is constant
• Using F×r instead of dL/dt
• Thinking ω changes

Hints

• Torque = dL/dt. Insect adds I = m(vt)².

Answer

(B)