A light rigid bar of length pivots about one end in a horizontal plane

PHYSICS · Multi-part · CBSE · Physics · System of Particles and Rotational Motion · System of Particles and Rotational Motion

Problem

A light rigid bar of length 0.60 m0.60\,\text{m} pivots about one end in a horizontal plane. Three small motors are attached along the bar, each capable of exerting a constant torque about the pivot. When the bar is released from rest, the following torques (about the pivot, positive for counter-clockwise) are applied simultaneously: • Motor 1: +2.5 N⋅m+2.5\,\text{N}\cdot\text{m} from θ=0 rad\theta = 0\,\text{rad} to θ=π/4 rad\theta = \pi/4\,\text{rad} • Motor 2: −1.8 N⋅m-1.8\,\text{N}\cdot\text{m} from θ=π/6 rad\theta = \pi/6\,\text{rad} to θ=π/2 rad\theta = \pi/2\,\text{rad} • Motor 3: +3.0 N⋅m+3.0\,\text{N}\cdot\text{m} from θ=0 rad\theta = 0\,\text{rad} to θ=π/2 rad\theta = \pi/2\,\text{rad} (a) Determine the algebraic expression for the net torque as a function of angular position θ\theta for 0≤θ≤π/2 rad0 \le \theta \le \pi/2\,\text{rad}. Represent the net torque in piecewise form if necessary. (b) Using your expression from part (a), calculate the total rotational work done on the bar as it moves from θ=0 rad\theta = 0\,\text{rad} to θ=π/2 rad\theta = \pi/2\,\text{rad}. (c) If the bar has a moment of inertia of 0.12 kg m20.12\,\text{kg}\,\text{m}^2 about the pivot, determine its angular speed at θ=π/2 rad\theta = \pi/2\,\text{rad}. (d) Suppose Motor 2 is reprogrammed to switch on only after θ=π/3 rad\theta = \pi/3\,\text{rad}. Without detailed calculation, state whether the final angular speed at θ=π/2 rad\theta = \pi/2\,\text{rad} would increase, decrease, or remain the same. Justify your answer briefly.

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