Q1: A weight is attached to a spring and is suspended from a ceiling so that it

Question

Q1: A weight is attached to a spring and is suspended from a ceiling so that it lies 30 cm from the ceiling when it is at rest. The spring is stretched so that the weight is 38 cm from the ceiling and is then released. It bounces up and down for a while. Each time it comes to the bottom of its bounce, the length of the stretch is 75% of the previous stretch. The period of each oscillation ( bounce ) of the weight is 1/2 second. Write an equation that represents the distance from the weight to the ceiling.

Explanation / Answer

Let's consider the damping and the cycling separately.
Since, in 1/2 second, it becomes 75% of the previous stretch, we can write
e-r(1/2) = .75

Taking logs,

-1/2 r = ln(.75)

r = -2 ln(.75) = 0.575364

Next, have the oscillation by using a cos cycle. Note that the question asks about distance from the ceiling, so being stretched is a positive. Cos is a maximum at 0. Since the period is 1/2 second, = 2/period =  4.

Thus, the cycle will be cos(4t) and the amplitude will be 8 e -0.575364t (8 because 38 - 30 = 8 is the initial amplitude of the cycle). 30 is the constant because this is displacement from 30.

Let distance at time t be d(t). Then, d(t) = 30 + 8 e -0.575364t cos(4t)

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