Simple Harmonic Motion: Displacement after s seconds

mexqwerty

The question is:

A 56.0 kg bungy-jumper hangs suspended from her bungy-cord, at rest. She is displaced from this position by 15.0 m downward, and then released. She bounces up and down, with a period of 5.800 s. Assume the woman undergoes simple harmonic motion, described by
y(t) = X cos(2 π t / T + φ)

Where is the woman after 43.20 s of bouncing? (enter a negative value if she is below her rest position).

The answer is 14.2m.

I've been using x=Xcos(2*pi*t/T) --> 15cos(2pi43.2/5.8) = 10.27m. This equation was in my book. What did I do wrong? Am I supposed to use y(t) = X cos(2 π t / T + φ? In that case, I don't know what φ is...

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Doc Al

Mentor
Are you using your calculator properly? (Radians, not degrees.)

mexqwerty

Oooh, thanks. I did not know you had to use radians. So do you have to always have to use radians when dealing with simple harmonic motion?

Doc Al

Mentor
Oooh, thanks. I did not know you had to use radians. So do you have to always have to use radians when dealing with simple harmonic motion?
Generally, yes. But you can always convert from one to the other.

Note what's going on in the equation y(t) = X cos(2 π t / T). That ratio t/T tells you what fraction of a period you are dealing with. The 2π tells you that you are dealing with radians, since one complete period is 2π radians.

mexqwerty

Oh, that make sense. Thanks, that was really helpful. =)

booooo

how do you know if the woman is above or below her rest position?

Doc Al

Mentor
how do you know if the woman is above or below her rest position?
Set up properly, the equation will tell you.

Or you can just see what fraction of a full period she ends up at.

booooo

Do you mean by using this equation ?y(t) = X cos(2 π t / T + φ)
btw what isφ

Doc Al

Mentor
Do you mean by using this equation ?y(t) = X cos(2 π t / T + φ)
Yes.

btw what isφ
φ is a phase factor. But since the motion starts at maximum displacement below the rest position, you may not need it.

What final equation would you use to describe the position as a function of time?

booooo

simply using y(t) = X cos(2 π t / T) ?
is that right

Doc Al

Mentor
simply using y(t) = X cos(2 π t / T) ?
is that right
Since she starts out below the rest position, I'd put a minus sign in front of that. Then you'd be fine.

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