Using center of mass for loop de loop

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Discussion Overview

The discussion revolves around the application of the center of mass in solving the loop de loop problem in physics. Participants explore the implications of using the center of mass for calculating potential energy (PE) in this context, examining both theoretical and practical considerations.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants recall a professor's suggestion to use the center of mass for calculating height in the potential energy formula, questioning its necessity.
  • Others argue that while the size of the object is often negligible, using the center of mass makes sense for calculating changes in gravitational potential energy.
  • There is a discussion about the potential energy at different heights, with some suggesting that it could be non-zero at ground level, which raises questions about the reference point for potential energy.
  • Participants discuss the accuracy required in calculations and the relevance of the center of mass in various scenarios, particularly in classic problems involving centripetal acceleration.
  • Mathematical expressions are presented to compare potential energy calculations with and without accounting for the center of mass, leading to further inquiries about the derivation of these expressions.
  • One participant introduces the idea of irregularly shaped objects, such as a rain cloud, to illustrate the complexities of defining a center of mass in different contexts.

Areas of Agreement / Disagreement

Participants express varying opinions on the necessity and implications of using the center of mass in potential energy calculations. No consensus is reached on when it is essential to consider the center of mass, and multiple viewpoints remain on the topic.

Contextual Notes

Limitations in the discussion include assumptions about the size of objects, the arbitrary nature of reference points for potential energy, and the dependence on specific problem contexts. Some mathematical steps remain unresolved, particularly regarding the derivation of potential energy expressions.

Who May Find This Useful

This discussion may be useful for students and educators in physics, particularly those interested in the nuances of potential energy calculations and the role of the center of mass in various physical scenarios.

Sho Kano
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A quick question, I remember my professor said to use the center of mass for solving the classic loop de loop problem. For example, he wanted us to use the distance from the surface of the loop to the center of the object (instead of just from the loop to the surface of the block) for the height in PE = mgh. Thinking back on this, it seems pretty weird that this was done. What do you guys think?
 
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Often the size of the object is assumed negligible compared to other distances in the problem. But when you calculate changes in the gravitational PE of an object, it makes sense to use changes in the center of mass position.
 
Doc Al said:
Often the size of the object is assumed negligible compared to other distances in the problem. But when you calculate changes in the gravitational PE of an object, it makes sense to use changes in the center of mass position.
So let's say the mass is at rest at the highest position, and is just about to fall. So its potential energy would be say mg(51) instead of mg(50)? And at the zero reference point (ground), its potential energy is still mg(1)?
 
Sho Kano said:
So let's say the mass is at rest at the highest position, and is just about to fall. So its potential energy would be say mg(51) instead of mg(50)? And at the zero reference point (ground), its potential energy is still mg(1)?
If you measure PE from ground level, when the object is resting on the ground its PE = mgd (say). Then when it's at the highest point, resting on top of the track at height h, its PE would be mg(h+d). For some things it doesn't matter, since the change in PE will still be mgh.

It depends on exactly what you are trying to calculate.
 
Doc Al said:
If you measure PE from ground level, when the object is resting on the ground its PE = mgd (say). Then when it's at the highest point, resting on top of the track at height h, its PE would be mg(h+d). For some things it doesn't matter, since the change in PE will still be mgh.

It depends on exactly what you are trying to calculate.
I see, I just find it weird that its potential energy would be non-zero at ground level. When would one want to use the center of mass to calculate PE?
 
Sho Kano said:
I see, I just find it weird that its potential energy would be non-zero at ground level.
Realize that the zero level is arbitrary. Only changes in gravitational PE have meaning.

Sho Kano said:
When would one want to use the center of mass to calculate PE?
Depends on how accurate you want to be and what you are asked to calculate.
 
Doc Al said:
Realize that the zero level is arbitrary. Only changes in gravitational PE have meaning.Depends on how accurate you want to be and what you are asked to calculate.
In the classic problem, where it asks to calculate the energy needed to get pass the loop, is this required? I'm having trouble thinking up with situations where it would want to be done. What is the size difference to make this matter?
 
Sho Kano said:
In the classic problem, where it asks to calculate the energy needed to get pass the loop, is this required?
Think about it. What's the PE initially? What's the PE of the object when it passes the top of the loop? (Note that the object is now below the track.) Since centripetal acceleration is involved, what radius will you use?

Sho Kano said:
I'm having trouble thinking up with situations where it would want to be done. What is the size difference to make this matter?
It depends how accurate you want to be. Many versions of the problem don't even give the dimensions of the object, so you are expected to ignore it.
 
Doc Al said:
Think about it. What's the PE initially? What's the PE of the object when it passes the top of the loop? (Note that the object is now below the track.) Since centripetal acceleration is involved, what radius will you use?
For without accounting for the center of mass:
y = 2r + 0.5r

For accounting for the center of mass,
y = 2r + 0.5r - 2x

(y is the minimum height needed to pass the loop, r is radius of the loop, and x is the distance to the center of mass)
So it would matter right?
 
  • #10
Sho Kano said:
For without accounting for the center of mass:
y = 2r + 0.5r
OK.

Sho Kano said:
For accounting for the center of mass,
y = 2r + 0.5r - 2x
How did you arrive at this?

Sho Kano said:
So it would matter right?
Sure. As to how much it matters, that depends on how x compares to r.
 
  • #11
Doc Al said:
How did you arrive at this?
mg(y+x) = mg(2r-x) + 0.5mrg
y + x = 2r - x + 0.5r
 
  • #12
Sho Kano said:
mg(y+x) = mg(2r-x) + 0.5mrg
y + x = 2r - x + 0.5r
Check that 0.5mrg term. Recall my comment about what radius to use.
 
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  • #13
Doc Al said:
Check that 0.5mrg term. Recall my comment about what radius to use.
0.5mrg should be 0.5m(r-x)g?
 
  • #14
Sho Kano said:
0.5mrg should be 0.5m(r-x)g?
Right.
 
  • #15
Perhaps it helps to think about the case where the object is an irregular shape or even flexible with no obvious "surface" to measure from. Something like a rain cloud...

Cloud.png
 
  • #16
CWatters said:
Perhaps it helps to think about the case where the object is an irregular shape or even flexible with no obvious "surface" to measure from. Something like a rain cloud...
That will certainly make things more interesting!

I, of course, was speaking of the simpler case of a rigid object (a cart, for example) with a well-defined center of mass.
 
  • #17
With a cloud the centre of mass might be the only think you can estimate the position of.
 

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