Applications of second order differential equations
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Mechanical Vibrations
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 ?- 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 if the hydraulic shock has a damping constant of , with an initial positions of 0.75 meters, and an initial velocity of 0 ?
- What is the position of the mass at time if the hydraulic shock has a damping constant of , with an initial position of 0.5 meters, and an initial velocity of 5 ?
- What is the position of the mass at time if the hydraulic shock has a damping constant of , with an initial position of 0.3 meters, and an initial velocity of -0.3 ?
Electrical Circuits
Find the charge at time for an electrical circuit with a resistor that has a resistance of , an inductor with , a capacitor with , and a battery with charge . The initial charge is coulombs, and the initial current is amps.