Discussion
A spring-mass system is viscously damped with a viscous damping constant c. The energy dissipated per cycle when the system is undergoing a harmonic vibration \(XCos\omega_{d}t\) is given by
Answer: A
The energy dissipated per cycle is \(\pi c\omega_{d}X^{2}\).
The damping force is \(c\dot{x}\), so the energy lost over one cycle of \(x = X\cos\omega_{d}t\) is
\[ \begin{aligned} W_{d} &= \oint c\dot{x}\,dx = \int_{0}^{2\pi/\omega_{d}} c\dot{x}^{2}\,dt \\ &= c\omega_{d}^{2}X^{2}\int_{0}^{2\pi/\omega_{d}}\sin^{2}\omega_{d}t\,dt \\ &= \pi c\omega_{d}X^{2} \end{aligned} \]
Energy must scale with amplitude squared and with the damping constant, so \(\pi\omega_{d}X^{2}\) (no c) and the two forms linear in X cannot be right, and \(\omega_{d}^{2}\) has the wrong dimensions for energy.
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