Pendulum: Energy vs. Time Question

[justadream]

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TBR Physics Book I page 253 #12

Q: "All of the following statements are true with respect to the total energy of the [pendulum] system except___"

One of the incorrect answer choices (and therefore, a TRUE statement) is:

"The potential energy and kinetic energy are equal to one another at a time that is 1/8 of the period after it had all of the gravitaitonal PE"

How is this true? Let's examine the wave diagram below. 1/8 of the period would be at the 6.25 second mark (period is 50 seconds). At the 6.25 second mark, I don't think the PE = KE. Wouldn't such a relationship only be true if the line were linear (which it clearly is not)?

 

[MedHopeful20134]

TBR Physics Book I page 253 #12

Q: "All of the following statements are true with respect to the total energy of the [pendulum] system except___"

One of the incorrect answer choices (and therefore, a TRUE statement) is:

"The potential energy and kinetic energy are equal to one another at a time that is 1/8 of the period after it had all of the gravitaitonal PE"

How is this true? Let's examine the wave diagram below. 1/8 of the period would be at the 6.25 second mark (period is 50 seconds). At the 6.25 second mark, I don't think the PE = KE. Wouldn't such a relationship only be true if the line were linear (which it clearly is not)?

At 0 displacement, all Gravitational PE is converted into Kinetic E. At 45 mm of displacement, half of PE is in the form of Kinetic, which represents 1/8 of a period.

 

[IslandStyle808]

Not too sure what to say here, I wouldn't take the graph literally. At the highest point all energy should be PE (90mm) and at the lowest point all energy should be KE (0mm or 1/4 of the period). At 1/8 of the period (+45mm displacement) it should be half KE and PE. Well, that is what I think.

 

[justadream]

I understand that at 45 mm displacement, half the energy is KE and half is PE.

My question is: how do you know that you reach 45 mm displacement at the time specified (1/8 of the period).

Doesn't that assume that displacement changes linearly with time? But here, since we have waves, the displacement does NOT change linearly.

 

[studyq]

For a pendulum the period is the time that it takes for the pendulum to return to its initial position. So if the pendulum starts its swing from the left, once it reaches its maximum height on the right it is halfway through its period.

Also, when the pendulum is at its maximum height the PE is at a maximum. KE is at a maximum at the bottom of the swing. If the pendulum is swinging from left to right the first time it passes the bottom is at a time equal to 1/4 of the period. When the pendulum has lost half of its maximum height it has lost half of its maximum potential energy and gained half of its maximum kinetic energy; the two are equal. This occurs directly between the beginning of the cycle and 1/4 of the cycle (when the pendulum reaches its minimum height and max KE) which is 1/8 of the total period.

 

[justadream]

@studyq

"When the pendulum has lost half of its maximum height it has lost half of its maximum potential energy and gained half of its maximum kinetic energy; the two are equal."
Yes, I completely understand that.

But how do you know it occurs "directly between the beginning of the cycle and 1/4 of the cycle"?

Is it because the force of gravity constant throughout? I think that's what is confusing me. I know that the restoring force is -mgsin(theta) and the tension is T= mgcos(theta) + mv^2 / r. But I guess the force of gravity (which is what is ultimately controlling the height) is constant?

Does that logic work?

 

[IslandStyle808]

I think what everyone is saying here is that this is the usual pendulum and the energy distribution will be the same. Like I have said in my previous post I don't think you should take the sine graph literally. I had a similar question in TBR on a pendulum and it was a true statement that at 1/8 of the period PE=KE.

 

[studyq]

http://forums.studentdoctor.net/threads/derivation-of-ke-pe-in-a-pendelum.1079539/
I just found this forum they explain it really well. You know that somewhere in between max KE and max PE the two are equal, since this is somewhere between o percent and 1/4 of the cycle it is approximately 1/8th. I was wrong to say directly between the more I thought about it you aren't losing height at a constant rate so you aren't going to reach half PE at exactly halfway between the two. You lose height (and therefore PE) at a quicker rate at the beginning of the swing. Not sure if that makes sense to you but you don't actually get exactly 1/8th.

 

[IslandStyle808]

http://forums.studentdoctor.net/threads/derivation-of-ke-pe-in-a-pendelum.1079539/
I just found this forum they explain it really well. You know that somewhere in between max KE and max PE the two are equal, since this is somewhere between o percent and 1/4 of the cycle it is approximately 1/8th. I was wrong to say directly between the more I thought about it you aren't losing height at a constant rate so you aren't going to reach half PE at exactly halfway between the two. You lose height (and therefore PE) at a quicker rate at the beginning of the swing. Not sure if that makes sense to you but you don't actually get exactly 1/8th.

Yep this sound right. I also did not know this (I admit), but I have realized that the MCAT is more about estimation (the reason why I said not to take the graph literally). The test is more about separating the best answer from the worst. I am not sure what the actual answer is, but it must sound extremely untrue (the untrue statement begin the right one, as mentioned in OPs first post).