A square platform of side can rotate about a vertical axis through its center

PHYSICS · Multi-part · AP · Physics 1: Algebra-based · Energy and Momentum of Rotating Systems · Energy and Momentum of Rotating Systems

Problem

A square platform of side

2.0\,\text{m}

can rotate about a vertical axis through its center. Four workers stand at the midpoints of the edges. Each pushes tangentially on the platform edge with a force that varies with the platform’s angular position

\theta

(in radians) according to

F_i(\theta) = F_{0i}\,\sin(\theta)

, where

i = 1,2,3,4

labels the workers and the signs of

F_{0i}

are given:

F_{01} = +40\,\text{N}

F_{02} = +30\,\text{N}

F_{03} = -50\,\text{N}

F_{04} = +20\,\text{N}

The positive direction for torque is counter-clockwise. Assume the platform behaves as a thin square slab of mass

200\,\text{kg}

. (a) Express the torque contributed by each worker as a function of

\theta

, taking the moment arm into account. (The distance from the axis to each force is

1.0\,\text{m}

.) (b) Derive a simplified expression for the net torque

\tau_{\text{net}}(\theta)

acting on the platform. (c) Compute the net work done by all four workers as the platform rotates smoothly from

\theta = 0\,\text{rad}

\theta = \pi\,\text{rad}

. (d) Determine which worker transfers the greatest amount of energy to the platform over this interval and explain quantitatively why this occurs despite the presence of opposing torques. (e) Suppose a fifth worker is added at the center pushing radially outward with a variable force of magnitude

60\,\text{N}

. Discuss whether this central force alters the net work you calculated in part (c) and justify your answer.

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