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A particle's position obeys x = A*cos(omega*t - phi) under the action of the force F = F0*cos(omega*t). Construct four options and identify the average power delivered by this force over a cycle. [Numerical options constructed since the originals were blank.]

1. (1/2)*F0*A*omega*sin(phi)
2. (1/2)*F0*A*omega*cos(phi)
3. F0*A*omega*sin(phi)
4. (1/2)*F0*A*omega

Correct answer: (1/2)*F0*A*omega*sin(phi)

Solution

Velocity v = dx/dt = -A*omega*sin(omega*t - phi). Instantaneous power P = F*v = -F0*A*omega*cos(omega*t)*sin(omega*t - phi). Expand sin(omega*t - phi) = sin(omega*t)*cos(phi) - cos(omega*t)*sin(phi). Averaging over a cycle: <cos(omega*t)*sin(omega*t)> = 0 and <cos²(omega*t)> = 1/2. P_avg = -F0*A*omega*[0*cos(phi) - (1/2)*sin(phi)] = (1/2)*F0*A*omega*sin(phi).

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