Work Done without using the Conservation of Energy

In summary, a small mass is pulled over the top of a frictionless half cylinder by a massless cord. The work done in moving the mass from the bottom to the top of the cylinder is found using the conservation of energy.
  • #1
veronicak5678
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0

Homework Statement



A small mass m is pulled over the top of a frictionless half cylinder of radius R by a massless cord passing over the top. Without using the conservation of energy, find the work done in moving the mass from the bottom to the top of the cylinder at a constant speed, in terms of m, g, and R.

Homework Equations



?


The Attempt at a Solution



Really not sure how to approach this... Can I somehow set up an integral?
 
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  • #2
Yes, you'll need to set up an integral of F*ds along the cylinder. Figure out the force that must be applied as a function of position along the cylinder. Hint: Measure position in terms of angle, from 0 to ∏/2 radians.
 
  • #3
I understand the concept, but I am still having trouble setting up the function. I guess I can disregard normal force, so the only force doing work is the tension.

So the work of tension is Force of T * displacement * cos angle. Since the tension is always in the same direction as the displacement, the angle will always be cos 0, or 1, so I can leave that out. The displacement will be R * angle. I don't know how to figure out the force of the tension at any given point since the weight is sometimes pulling down (like at 0 degrees) and sometimes at a right angle, where I think it should then be doing no work.
 
  • #4
veronicak5678 said:
I understand the concept, but I am still having trouble setting up the function. I guess I can disregard normal force, so the only force doing work is the tension.
Yes. We only care about the work done by the tension force.
So the work of tension is Force of T * displacement * cos angle. Since the tension is always in the same direction as the displacement, the angle will always be cos 0, or 1, so I can leave that out. The displacement will be R * angle.
At any given position, the infinitesimal displacement will be Rdθ.
I don't know how to figure out the force of the tension at any given point since the weight is sometimes pulling down (like at 0 degrees) and sometimes at a right angle, where I think it should then be doing no work.
Since the speed is constant, the tension must be just enough to balance out the tangential component of the weight. Find that as a function of θ.
 
  • #5
Not sure I know how to do that...

I think the integral should be from 0 to pi/ 2 of Rdtheta * wcos theta?
 
  • #6
Exactly!

Now just do that integral.
 
  • #7
Also: check your answer. you should be able to use conservation of energy to check it at least!

I did a similar problem (frictionless half-pipe) with some AP teachers in outreach last month.
 
  • #8
So the work done is just wR [sin theta]0 to pi/2 =

wR sin(pi/2) - wR sin(0) = wR ?
 
  • #9
Also, how can I check using the conservation of energy if I don't know the speed?
 
  • #10
veronicak5678 said:
So the work done is just wR [sin theta]0 to pi/2 =

wR sin(pi/2) - wR sin(0) = wR ?
Yep.
 
  • #11
veronicak5678 said:
Also, how can I check using the conservation of energy if I don't know the speed?
The speed doesn't change.
 
  • #12
OK. I get it. Thanks to all of you for your help!
 

1. What is work done without using the Conservation of Energy?

Work done without using the Conservation of Energy refers to the process of performing a task or exerting a force, without taking into account the principle of energy conservation. This means that energy is not conserved or transferred, and the work done is not equal to the change in energy of the system.

2. How is work done without using the Conservation of Energy different from traditional work?

Traditional work involves the transfer or conversion of energy from one form to another, while work done without using the Conservation of Energy does not involve any change in energy. In traditional work, the work done is equal to the change in energy of the system, but in the absence of Conservation of Energy, this is not the case.

3. Can work be done without using the Conservation of Energy?

Yes, work can technically be done without using the Conservation of Energy, but it is not a sustainable or accurate approach. The Law of Conservation of Energy is a fundamental principle in physics, and ignoring it can lead to incorrect results and misunderstandings about the behavior of energy.

4. What are some examples of work done without using the Conservation of Energy?

One example of work done without using the Conservation of Energy is pushing a stationary object on a frictionless surface. In this scenario, no energy is transferred or conserved, and the work done is not equal to the change in energy of the object. Another example is a person holding a weight at a constant height, as there is no change in energy even though work is being done.

5. Why is it important to understand work done without using the Conservation of Energy?

While work done without using the Conservation of Energy is not a valid approach in physics, it is still important to understand it as it can help us identify potential flaws or inaccuracies in our calculations. Additionally, it can also help us appreciate the significance of the Law of Conservation of Energy and how it governs the behavior of energy in the universe.

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