Deriving position function for object in SHM

member 731016 · Dec 6, 2022

For this problem,

Textbook page on simple harmonic motion: x(t) = x_i cos ωt + (v_i/ω) sin ωt with the (v_i/ω) sin ωt term circled in red, and v(t) = −x_i ω sin ωt + v_i cos ωt, followed by part (b) energy relation v² − ax = v_i² − a_x_i = ω²A².

How did they get that formula shown?

My working is,

Handwritten-style equation: Since at t = 0, x = x_i then cos^-1(x_i / A) = phi

Equation: x_i cos(wt) - sin(wt) sin(cos^-1(x_i / A))

All the solutions wrote was,

Printed solution text labelled (a): the acceleration is a negative constant times position, so SHM with angular frequency ω; at t = 0, x = x_i and v = v_i, satisfying the requirements.

Many thanks!

haruspex · Dec 6, 2022

Your working headed off into a blind alley. You know that you need to be eliminating A, so tucking it inside an arccos function isn’t going to get you there.
Start by writing all three initial state equations: position, velocity and acceleration. Then see how you can combine them to eliminate one of the unknowns.

Orodruin · Dec 6, 2022

Please type out your work instead of attaching pictures. The pictures make it impossible to quote particular sections.

Edit: I suggest you change your unknowns from A and phi to something more direct.

member 731016 · Dec 6, 2022

haruspex said:

Your working headed off into a blind alley. You know that you need to be eliminating A, so tucking it inside an arccos function isn’t going to get you there.
Start by writing all three initial state equations: position, velocity and acceleration. Then see how you can combine them to eliminate one of the unknowns.

Thank you @haruspex and @Orodruin! I have solved the problem now.

Many thanks!

haruspex · Dec 6, 2022

Callumnc1 said:

Thank you @haruspex and @Orodruin! I have solved the problem now.

Many thanks!

You are welcome. But did you understand my comment about heading off into a blind alley? Recognising which directions cannot lead anywhere is a useful skill to develop.

member 731016 · Dec 6, 2022

haruspex said:

You are welcome. But did you understand my comment about heading off into a blind alley? Recognising which directions cannot lead anywhere is a useful skill to develop.

Thanks @haruspex! I did understand your comment, which I agree, would be very useful skill to have!

Many thanks,
Callum

Deriving position function for object in SHM

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