# Section 3.7 – Mechanical and Electrical Vibrations

Directions. The following are review problems for the section. It is recommended you work the problems yourself, and then click "Answer" to check your answer. If you do not understand a problem, you can click "Video" to learn how to solve it.
1. A mass weighing 3 lb stretches a spring 3 in. If the mass is pushed upward, contracting the spring a distance of 1 in, then set in motion with a downward velocity of 2 ft/s, and if there is no damping, find the position $$u$$ of the mass at any time $$t$$. Determine the frequency, period, amplitude and phase angle of the motion.

$$u(t)=-\dfrac{1}{12}\cos(8\sqrt{2}t)+\dfrac{1}{4\sqrt{2}}\sin(8\sqrt{2}t)$$
Tthe frequency $$w_0$$ is $$8\sqrt{2}\ \dfrac{\mbox{rad}}{\mbox{s}}$$
The period $$T_0$$ is $$\dfrac{\pi}{4\sqrt{2}}$$ seconds.
The amplitude $$R$$ is $$\dfrac{\sqrt{22}}{24} \mbox{ ft }$$
The phase angle of motion $$\delta$$ is $$\arctan(-\dfrac{3}{\sqrt{2}})+\pi$$ radians.

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2. A spring is stretched 10 cm by a force of 3 N. A mass of 2 kg is hung from the spring and is also attached to a viscous damper that exerts a force of 3 N when the velocity of the mass 5 m/s. If the mass is pulled down 5 cm below its equilibrium position and given an initial velocity of 10 cm/s, determine its position $$u$$ at any time. Find the quasifrequency of the motion.

$$u=e^{-\frac{3}{20}t}\left[0.05\cos(\mu t)+\dfrac{2.15}{\sqrt{5991}}\sin(\mu t)\right]$$ where the quasifrequency is $$\mu =\dfrac{\sqrt{5991}}{20}$$

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