Question on Simple Harmonic Motion?

Question

A body moves such that its displacement from an origin O at time t seconds is x metres, where x = 4 sin 2t. How far does the body move in the first second?

The working I got from my textbook is:

When t = 0, x = 0
When t = 1, x = 4 sin 2
= 3.64
But when t = π/4, x = 4.
Thus the distance travelled in the first second is 4+ (4-3.64) = 4.36m.

My question is: since I'm asked to find the distance of the body in the first second, should I just sub t = 1 into the equation and leave it at that? Why do I have to also sub t = π/4 into the equation? I also don't understand the rest of the working (i.e. 4+ (4-3.64) = 4.36m)? Help! Thanks in advance!

Answer 1

It's been a while for me and physics, but seeing as no one else has answered yet, I'll take a crack at it.

π/4 is the maximum distance for the pendulum's swing. π is one full revolution of the sine curve, right? So π/2 is back at the origin, on the back swing.

π/4 is the full extent of the forward swing, so you can use that to if your value for t is before it reaches the maximum extent of the swing, or if it's somewhere before or after that maximum extent.

π/4 ~= 0.78539815

0.78539815 is less than 1 second, which means that at 1 second, the pendulum has already reached the maximum extent of the swing and is on its way back.

That means that it has already reached the extent of 4m, and is now travelling back towards the origin. We know how far from the origin the pendumul is currently, which is 3.64m. But because it's travelling toward the origin, instead of away from it, we need to measure from the maximum extent of the swing. The maximum extent of the swing is 4m, so the amount it has moved SINCE the extent is 4m-3.64m=0.36m. All in all, the pendulum has moved 4m to the extent, then another 0.36m back to the center for a grand total of 4.36m.

Good luck and I hope that wasn't too confusing!

Answer 2

Hint: an object undergoing SHM retraces the same path. You need to examine the phase to determine the number of times the object has changed direction. The question asks for the total distance travelled rather than the displacement.