Question

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An object attached to a coiled spring is pulled down a distance 10 units from its rest position and then released. Assuming that the motion is simple harmonic with a period T= 8 seconds, write an equation that relates the displacement d of the object from its rest position after t seconds. Also assume that the direstion of positive motion is up.

Answer 1

Because it is simple harmonic motion, we know we will have a sinusoidal (ie., sine or cosine) function. It will take the form \(y(t) = a \sin(bt + c) \).

Answer 2

In this equation \(y(t) = a \sin(bt + c)\):
\(a\) is the amplitude of the motion. Since you are pulling it down a distance of 10 units, the object will oscillate between -10 and +10 units, so the amplitude is 10.
\(b\) modifies the period of the motion. It follows the formula \(\text{period} = 2 \pi \times b\). Since we know the period is 8, you can find \(b\) by dividing 8 by \(2 \pi\).
\(c\) modifies the phase shift. Once you've found \(a\) and \(b\), you can plug in your initial value (position = -10 @ t = 0) to find \(c\).

Answer 3

This is what I got and I believe it is correct. \( \large d = -10 cos(\frac{\pi}{4}t) \)

Answer 4

Do you concur?

Answer 5

That is correct. Also, good choice using cosine instead of sine. Removes the whole phase shift issue.

Answer 6

In the future, I would graph it (desmos.com/calculator) to make sure it matches all your requirements.

Answer 7

Cool, thank you.

Answer 8

5 Owlbucks sent!