Derivation of KE = PE in a pendelum

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May 14, 2014
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In TBR they tell you that the kinetic energy will equal the potential energy in a pendulum system at 1/8th of a period.

There is also a question:

19. At what point in the cycle of a pendulum does the kinetic energy equal the potential energy?

B. t= pi/2
C. t= (3pi)/4
D. t=(3pi)/2

Choice C is the best answer. At t = 0, the system is motionless, so it has KE = 0 and PE at a maximum. This eliminates choice A. At pi/2 the bob is at its lowest point, moving left. At 3pi/2 the bob is at its lowest point, moving right In both cases, KE is maximum and PE is minimum. Because the pendulum has the same energetics at pi/2 as it has at 3pi/2, choices B and D are eliminated. Choice C is the best answer, because it is the only remaining answer. At 3pi/4 the bob is half way between its lowest point and highest point, moving left This falls half way between points where KE is maximum and PE is maximum. The best answer is choice C.

I am having a hard time understanding this question and the idea the KE = PE at 1/8th period. Can someone please explain both conceptually and mathematically?

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The problem with this question is they are asking you to solve conceptually. Mathematically would not get you an exact answer of 3pi/4. Thats why they say "Choice C is the best answer because the rest are very wrong".

Conceptually if you pull a pendulum to max height it has full PE. If you let it swing all the way to the other side and back for a full period (2pi) it is back where it started at max height.
At 1 pi it is at max height but on the opposite side from where it started.
At 1/2 pi it is at minimum height and has no potential energy and all kinetic energy.
At 1/4 pi OR 3/4 pi, the pendulum would have approximately half KE half PE. (Also true for 5/4, 7/4 etc)
To answer the bold question, 1/8 of a period is between when you drop it t=0, and 1/4 period when it reaches the lowest point of the swing.

Mathematically this answer is not perfectly true because the pendulum only approximates simple harmonic motion. So when time is directly between PEmax and KEmax you are going to be closer to one or the other because angular velocity is changing throughout the motion of the pendulum. So if you solve for height at exactly t=pi/4 it would not be 1/2h.
Choice C is simply the best because all the others are wrong and C is the only one that has both PE and KE at the same time.
I guess I am pretty screwed for the MCAT because I have a hard time with conceptual physics questions :(

Thank you though.
For this, if you used math you could eliminate the wrong choices the same way the answer did. If you got a problem like this and tried to find an answer during an exam, you would run out of time doing unnecessary calculations. Here the other thing to consider is the options are given in variable form.

The take home is that the period of a pendulum is 2pi, and you need to know how PE and KE are related to each other. And KE=PE at a point somewhere between PE max and KE max, which is T=1/8 period = (1/4)pi or (3/4)pi or (5/4)pi etc.