Re: conservation of energy problem(?)

In summary, the student is struggling with how to find the center of mass for a quarter disc, and is looking for help from an expert. The expert provides a trigonometric solution that is almost correct, but the coordinates are incorrect.
  • #1
yugeci
61
0

Homework Statement



ZmHYeN7.png
(apologies for the bad picture, but it was the only one I could find online)

Homework Equations



Conservation of energy, KE1 + PE1 = KE2 + PE2
PE = mgh
KE = 0.5mv²

The Attempt at a Solution



I'm not sure where to begin. I know the answer but I don't know how you get it. Can't you simply use conservation of energy here? At the top PE = mgr and KE = 0... and at the bottom PE = 0 and KE = 0.5mv². If I do that v = √2gr, but the answer is actually √(gr(π/2 + 4/π)). I'm not sure what to do with the mass per unit length (density?) either and how it's relevant to this equation.

Steps on how to solve this problem would be appreciated.
 
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  • #2
So you have an expression for the energy from the gravitational force on the TOP link, are there any other forces acting on the top link?
 
  • #3
yugeci said:
Can't you simply use conservation of energy here?
Why not?

yugeci said:
At the top PE = mgr and KE = 0...
That would be true if all the mass were concentrated at a point at the top, but that's not the case. The mass is spread out over the channel.

Hint: How would you measure the gravitational PE of an extended object?
 
  • #4
mgY where Y is the Y-coordinate of the center of mass. So that would mean

initial PE = mg(r/2)
initial KE = 0 (?) (KE isn't affected by length?)
final PE = mg(-π/4)
final KE = 0.5mv²

Took the bottom right corner of the quadrant as the datum. When using the equation initial PE + initial KE = final KE + final PE I still don't get the right answer here. I am assuming my expressions for the KE is wrong.
 
  • #5
yugeci said:
mgY where Y is the Y-coordinate of the center of mass.
Good.

yugeci said:
So that would mean

initial PE = mg(r/2)
Are you sure that's where the center of mass is?
 
  • #6
On second thought, no. Would it be halfway down the mass? Is this diagram correct?
R4R9aov.png


So doing some trigonometry there I get the Y coordinate as r sin (π/4) which ends up being r sqrt(2) upon 2, but I still get the wrong answer when substituting that in my equation above.
 
  • #7
yugeci said:
On second thought, no. Would it be halfway down the mass? Is this diagram correct?
R4R9aov.png


So doing some trigonometry there I get the Y coordinate as r sin (π/4) which ends up being r sqrt(2) upon 2, but I still get the wrong answer when substituting that in my equation above.
No, that's not the right way to find the centre of mass either. Have you not been taught how to find a centre of mass?
 
  • #8
Um, no. Not really? I just assumed it would be half-way down the length of the string. Where would it be?

Interesting my answer is almost correct (but not quite and probably just by luck).
 
  • #9
yugeci said:
I just assumed it would be half-way down the length of the string. Where would it be?
To find the center of mass, you'll have to use a bit of calculus. (Or look it up somewhere.)

Since the chain is curved, the center of mass won't be on the chain itself.
 
  • #10
OK so after applying some integration (never studied this in class for center of mass, weird) + verifying it online the coordinates seem to be (-4R/3pi, 4R/3pi). Putting the Y coordinate into my equation I wrote above and making v the subject I get:

v = √gr((π/2) + (8π/3))

But the answer is √gr((π/2) + (4/π))
 
  • #11
yugeci said:
OK so after applying some integration (never studied this in class for center of mass, weird) + verifying it online the coordinates seem to be (-4R/3pi, 4R/3pi).
Dare I ask you to show how you got that? It doesn't seem right. How did you verify it? (All you care about is the y-coordinate.)
 
  • #12
yugeci said:
OK so after applying some integration (never studied this in class for center of mass, weird) + verifying it online the coordinates seem to be (-4R/3pi, 4R/3pi). Putting the Y coordinate into my equation I wrote above and making v the subject I get:

v = √gr((π/2) + (8π/3))

But the answer is √gr((π/2) + (4/π))
You seem to have found the centre of mass of a quarter disc, not a quarter circle arc.
 
  • #13
Yep. Whoops. That was silly of me. I found the arc center of mass as -2r/π, 2r/π and I seem to get the right answer with it now.

Thanks for all of the help.
 

Related to Re: conservation of energy problem(?)

1. What is the conservation of energy problem?

The conservation of energy problem refers to the concept that energy cannot be created or destroyed, but can only be transformed from one form to another. This means that the total amount of energy in a closed system remains constant over time.

2. Why is the conservation of energy important?

The conservation of energy is important because it is a fundamental principle of physics that helps us understand and predict the behavior of physical systems. It allows us to make accurate calculations and models of energy interactions in various systems, from microscopic particles to the entire universe.

3. How is the conservation of energy related to the laws of thermodynamics?

The conservation of energy is closely related to the first law of thermodynamics, which states that energy cannot be created or destroyed, only transformed. This law is a direct result of the conservation of energy principle. The second law of thermodynamics also follows from the conservation of energy, as it states that the total entropy (a measure of energy dispersal) of a closed system always increases over time.

4. Are there any exceptions to the conservation of energy?

No, there are no known exceptions to the conservation of energy. While energy can be converted from one form to another, the total amount remains constant. However, in certain situations such as at the quantum level, energy can appear to be created or destroyed, but this is due to limitations in our current understanding of energy and its behavior.

5. How is the conservation of energy applied in everyday life?

The conservation of energy is applied in numerous ways in our everyday lives. For example, it is used in designing energy-efficient buildings and vehicles, understanding the energy flow in our bodies for optimal health, and in renewable energy systems such as solar panels and wind turbines. It also plays a crucial role in the functioning of electronic devices and the production of food and other goods.

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