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Old Oct 20th 2014, 10:12 AM   #1
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Energy Conservation Questions

A force increases linearly with time, F = c*t where c is a constant, and acts on an object of mass m. The object is at rest at t=0 and you have to calculate its velocity at a later time T. No other forces are acting on the object. No other information is given to you. Which of the following statements is true and describes a viable strategy to tackle this problem?

A. Energy for this system is conserved and you can easily calculate the final kinetic energy after first finding the potential energy that belongs to F.


B. The work-energy theorem is valid. You simply calculate the work done by F between t=0 and t=T to get the final kinetic energy.


C. You can use Newton's Second Law to calculate the acceleration a = F(t)/m. Then you integrate once with respect to time to obtain the final velocity.


D. Both B and C are valid strategies which are easily executed.


I thought it was A, is that correct?
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Old Oct 20th 2014, 10:34 AM   #2
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Originally Posted by NSB3 View Post
I thought it was A, is that correct?
No. You don't know anything about potential energy for this system.

One way to approach this problem is to actually try each approach and see if you get an answer. Hint: work is the integral of the force acting over distance, not time.
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Old Oct 20th 2014, 10:47 AM   #3
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Okay thanks! So does that mean that this one would be D. Because you can use work energy theorem because you have positions and you can find potential energy also? I know that B is correct but am not completely sure about A. I know that C is incorrect because it is F(x) not t.

This is a 1-D problem. A force increases linearly with the x-coordinate, F = c*x, where c is a constant, and acts on an object of mass m. The object starts from rest at x=0 and you have to calculate its velocity at later position x=A. No other forces are acting on the object. This is all the information given to you. Which of the following statement is true and describes a viable strategy to tackle this problem?

A.
Energy is conserved. You can easily find the potential energy for F and calculate the final kinetic energy.

B.
The work-energy theorem is valid. You integrate F with respect to x between 0 and A and thus obtain the kinetic energy at A.

C.
Newton's Second Law gives you the acceleration as a=F(x)/m. Now it is very easy to integrate with respect to time to obtain the velocity.

D.
Both A and B are valid strategies and easily executed.
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Old Oct 20th 2014, 11:01 AM   #4
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I'm confused - why did you change the problem from F(t)=ct to F(x)= cx? And if F(x)=cx starting with initial condition x = 0, then F =0 and the object never moves. In the problem as you first wrote it indeed a = f(t)/m = ct/m, which you can integrate easily to get velocity as a function of time.

Last edited by ChipB; Oct 20th 2014 at 11:08 AM.
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Old Oct 20th 2014, 11:07 AM   #5
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I am sorry for the confusion. I figured out the first one I posted which is Newtons Second Law; however, the second one is a completely different problem just similar aspect but it changed it from F(t) to F(x). I thought when it is changed that the answer would be both A and B.
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Old Oct 20th 2014, 11:33 AM   #6
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OK, I see. I agree it's A and B, but I'm curious what you get for the object's PE as a function of x? Be careful with signs!
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Old Oct 20th 2014, 03:12 PM   #7
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The first problem:

Originally Posted by NSB3 View Post
A force increases linearly with time, F = c*t where c is a constant, and acts on an object of mass m. The object is at rest at t=0 and you have to calculate its velocity at a later time T. No other forces are acting on the object. No other information is given to you. Which of the following statements is true and describes a viable strategy to tackle this problem?

A. Energy for this system is conserved and you can easily calculate the final kinetic energy after first finding the potential energy that belongs to F.


B. The work-energy theorem is valid. You simply calculate the work done by F between t=0 and t=T to get the final kinetic energy.


C. You can use Newton's Second Law to calculate the acceleration a = F(t)/m. Then you integrate once with respect to time to obtain the final velocity.


D. Both B and C are valid strategies which are easily executed.


I thought it was A, is that correct?
I am going to call "valid" strategies that can be performed without any heroic amounts of difficulty on an exam.

A is out: It is not obvious to me how to determine if a force is conservative if it is given as a function of time, not of position.

B is out: The work-energy theorem will, of course, work in any situation where there are no external (that is to say neglected) forces involved. However we have to calculate the work, which is found by Integral( F(t)dx ). It is not clear to me how to determine the x dependence of t without resorting to Newton's 2nd Law.

C is in: It is easy to calculate v as a function of t.

I would say the only valid strategy would be C. Though B and C certainly can both be done, B clearly gives difficultes.

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Old Oct 20th 2014, 03:18 PM   #8
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The second problem.

Originally Posted by NSB3 View Post
This is a 1-D problem. A force increases linearly with the x-coordinate, F = c*x, where c is a constant, and acts on an object of mass m. The object starts from rest at x=0 and you have to calculate its velocity at later position x=A. No other forces are acting on the object.
Just to be clear, ChipB is correct. If it is not moving at x = 0, the force F(0) = 0 which says that the object won't move. Is there a typo?

-Dan
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Old Dec 25th 2014, 11:30 AM   #9
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Originally Posted by NSB3
A. Energy for this system is conserved and you can easily calculate the final kinetic energy after first finding the potential energy that belongs to F.
The energy for this system is not necessarily conserved. I haven't calculated the Hamiltonian so I don't know. I'll leave that to you. It's not that hard. Just look it up on the internet using Google. It's a misconception in mechanics to assume that just because the force can be derived from a potential that the energy is conserved. It must also be true that the potential must not be an explicit function of time. Only if the kinetic energy and potential energy functions are functions of time and the sum is constant then energy is conserved. The best way to do this is to find the Hamiltonian of the system and see if its a function of time. If it is then the energy of the system is conserved.
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