Simple pendulum with initial velocity


I’ve attached a couple questions just because these are the type of questions I’ve had some difficulty solving since I don’t recall addressing anything too similar too similar in the past (i.e., changing velocity and determining how the amplitude varies because of this). I know to go from x(t) to v(t) you take the derivative, and that x(t) = xmcos(ωt + ø) in general. Then v(t) = -ωxmsin(ωt + ø). When looking at questions similar to 39, when I see vo is now 4vo and asking for the amplitude change to be determined, I look at xm. This means the new amplitude would have to be 4xm since all other variables are being kept constant, no? Also, once again taking the derivative to find a(t), wouldn’t the acceleration be 4 times as great once again?

Is there any other fundamental expression relating amplitude with velocity or frequency or any other variable as in question 21 or in general? Once again, I am also slightly confused on how exactly question 25 was solved too. If there any good online sources I could be referred to that would be greatly appreciated! If I am just missing something here and if someone could point out what I am not looking at, that would be amazing too!


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