Oscillations about equilibrium

Question

A mass moves back and forth in simple harmonic motion with amplitude A and period T.

 

a) In terms of T, how long does it take for the mass to move through a total distance of 4A?

 

b) How long does it take for the mass to move through a total distance of 7A?

Answer 1

Draw a picture! ;)

 

a) If you've drawn the right picture, you'll see that the oscillating mass starts at +A, moves through zero to -A, then moves back through zero to +A again. I count 4 A's. What do you get? I guess, then, that it took time T (the period) to make that one round trip, traveling a distance of 4A.

 

b) So, 3 more A's, huh? Well, here we go again . From +A through zero to -A, then back to zero and STOP. That was 3A's right there. 3/4 T. Total time T+3/4 T= 7T/4.

Answer 2

The problem with the second question is that simple harmonic motion velocity is not uniform across time, instead velocity moves in a sine wave. If I remember correctly, starting at any locations other than the ones I've mentioned, the duration of 7A is not going to be the same.    

 

Moving distance A from center to any maximum or vice versa will take exactly T/4, but going A from any other point in the cycle will vary.   The fastest motion will be from 1/2 A to -1/2 A and vice versa, the slowest will be from 1/2 A to A to 1/2A and -1/2A to -A to -1/2 A.

 

So, the answer is 7/4 * T for those 4 special cases.

 

For everything else, the answer is 7/4 * T plus or minus some factor that I'd have to figure out, probably based upon the derivative of the cos between pi/4 and 3pi/4 and between -1pi/4 and pi/4.