## anonymous one year ago 5 owlbucks 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.

1. BTaylor

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

2. BTaylor

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 $$c$$.

3. anonymous

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

4. anonymous

Do you concur?

5. BTaylor

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

6. BTaylor

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

7. anonymous

Cool, thank you.

8. anonymous

5 Owlbucks sent!