# Applications of second order differential equations

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A spring has a weight of 5kg attached to the end of it. The spring has a natural length of 0.3m and a force of 35 newtons is required to stretch the spring to a length 0.8m. If the spring is stretched to 0.5 meters and then released (with zero initial velocity), then what is the position of the mass at time $t$?**Mechanical Vibrations**- Suppose that a hydraulic shock has a spring constant of 40 newtons per meter. There is a weight of 10kg attached to the end of the shock, and the shock has a resting length of 0.5 meters.
- What is the position of the mass at time $t$ if the hydraulic shock has a damping constant of $c=50$, with an initial positions of 0.75 meters, and an initial velocity of 0 $m/s$?
- What is the position of the mass at time $t$ if the hydraulic shock has a damping constant of $c=40$, with an initial position of 0.5 meters, and an initial velocity of 5 $m/s$?
- What is the position of the mass at time $t$ if the hydraulic shock has a damping constant of $c=20$, with an initial position of 0.3 meters, and an initial velocity of -0.3 $m/s$?

**Electrical Circuits**Find the charge at time $t$ for an electrical circuit with a resistor that has a resistance of $R=14 \Omega$ , an inductor with $L=2H$, a capacitor with $C=0.05 F$, and a battery with charge $E(t)=8$$\sin(2t)$. The initial charge is $V=\frac{22}{29}$ coulombs, and the initial current is $I=\frac{6}{29}$ amps.