Problem 15 - Entrance Test

A constant net torque τ is applied to a rigid body that is initially at rest. The body rotates about a fixed axis with an angular acceleration α. How does the instantaneous power delivered to the body vary with time t?

A. A) Power is constant.B. B) Power is proportional to t.C. C) Power is proportional to t².D. D) Power is proportional to sqrt(t).E. E) Power is proportional to 1/t.

Correct: B

The instantaneous power (P) delivered to a rotating rigid body is given by the product of the net torque (τ) and its instantaneous angular speed (ω): P = τω We are given that a constant net torque τ is applied. According to Newton's second law for rotation (τ = Iα), if the torque is constant and the moment of inertia (I) is constant, then the angular acceleration (α) must also be constant. Since the body starts from rest, its initial angular speed (ω₀) is 0. For constant angular acceleration, the angular speed at any time t is given by the kinematic equation: ω = ω₀ + αt Since ω₀ = 0, we have: ω = αt Now, substitute this expression for ω into the power equation: P = τ (αt) We also know that α = τ/I (from τ = Iα). Substitute α back into the power equation: P = τ * (τ/I) * t P = (τ²/I)t Since τ and I are constants, the term (τ²/I) is a constant. Therefore, the instantaneous power P is directly proportional to time t. P ∝ t The correct choice is B.