TBR: Height in Pendulum Motion

[justadream]

TBR Physics I page 158 #13

The passage present an example of an object being dropped (but attached to a string). Imagine the object a distance X from the structure that is holding the string.

As you would expect, upon dropping the object, the object undergoes a pendulum-like motion with energy conservation.

I could very well be incorrect but I believe TBR implies that regardless of the magnitude of distance X, the height that the object obtains after being dropped is the same. I draw this conclusion from TBR's discussion about how the amount of "slack" rope has no effect on the height reached.

But this result seems unintuitive to me.

Here is a depiction:

So in the second scenario, I should expect the object to reach the initial height?

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[Cawolf]

Yes - as you stated the length of the rope does not matter - the energy will be conserved (in an ideal pendulum).

 

[justadream]

@Cawolf
I just realized my depiction is somewhat inaccurate. New picture below:

The rope length is the same for both situations. Just that it is taut in the first one, but not in the second one.

 

[Cawolf]

That's interesting - I have not come across anything like that.

Intuitively it seems like the motion would not be constrained to circular motion - so there would be motion in other directions - and it would not go as high. That is my initial thought.

I don't know quantitatively though - was there a passage dealing with this?

 

[justadream]

@Cawolf

TBR Physics I page 158 #13

From my OP: "I could very well be incorrect but I believe TBR implies that regardless of the magnitude of distance X, the height that the object obtains after being dropped is the same. I draw this conclusion from TBR's discussion about how the amount of "slack" rope has no effect on the height reached."

I'd appreciate it if you could check whether I have misinterpreted it.

 

[Cawolf]

I happen to have the book handy as I am on my day 1 of SN2ed - learning translational motion!

I agree with the answer choice - by conservation of energy.

It makes sense that if we are considering an ideal rope (inextensible and without mass) in a frictionless system - that the rope cannot do work on rider. In this case, after the freefall, the rope would become taut and constrain the motion to the circular path. The passage also states that rider takes on pendulum like motion - further enforcing the idea that the slack should have no effect.

I have not encountered this before - but if we keep the concepts simple, and assume energy is conserved - then A is the most reasonable choice. Though I do agree that it is slightly counter intuitive.

 

[justadream]

@Cawolf

Do you think there is some type of constraint (e.g., minimum distance) that governs whether something will undergo circular motion?

I mean if you have a rope that is 9999999999999999999m long and you place the object .0000000000000000000001mm from the structure and then drop it, is it really going to go circular motion and reach the same height? I experimented with my headphones (bad example, I know) but it didn't come anywhere close.

BTW: I have a feeling that by the time you do many of these TBR questions, many of the questions/answers will be oddly familiar lol.

 

[Cawolf]

Well - every time I answer a question on here, I read and take the questions - so I am not really missing out on the learning opportunity. Though I bet I will have seen a few of them before!

This is an ideal (ie. non-real) system, so I don't think any physical model we can make at home will be a good example.

The energy has to go somewhere, and the only place it can go is to the height of the rider - so unless we are saying energy is lost to friction/heat/work/etc, then yes - it would go to the same height. I do agree it seems "weird", but in this "ideal" scenario - it is the only thing that makes sense.