Exploring the Effects of Equal Water Depths on Gravitational Potential Energy

In summary, the student is trying to find the change in gravitational potential energy due to the addition of water to a vessel. He is using the center of gravity to calculate the change, but he is not considering the gain in gravitational potential energy in the vessel Y.
  • #1
toforfiltum
341
4

Homework Statement


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Homework Equations


GPE=mgh

The Attempt at a Solution


I don't understand why the answer is B. My approach is assigning h a random value, say 10m. When the depths of water in both vessels are equal, the height is 5m. Therefore, since the mass of water in both vessels are still m, shouldn't the loss in GPE be half? I don't know why I'm wrong.
 
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  • #2
For an extended object, the gravitational potential energy is determined by the location of the center of gravity of the object. Consider the height of the center of gravity before and after.
 
  • #3
I still don't get one quarter. I will only get that answer if I don't consider the gain in GPE in vessel Y.
 
  • #4
What is the initial height of the center of gravity? The final height?
 
  • #5
1/2h and 1/4h?
 
  • #6
Yes. So, what are the initial and final values of gravitational PE?
 
  • #7
1/2mgh - 1/4mgh?
 
  • #8
OK, that would be the loss of gravitational PE.
 
  • #9
But how about the gain in GPE in vessel Y?
 
  • #10
toforfiltum said:
But how about the gain in GPE in vessel Y?
That is already taken care of. If you consider only vessel X, then the change in PE would be

PEf - PEi = (m/2)g(h/4) - mg(h/2) = (1/8)mgh - (1/2)mgh = -(3/8)mgh.

Vessel Y gains PE in the amount (m/2)g(h/4) = (1/8)mgh.

Overall, there is a change of -(3/8)mgh + (1/8)mgh = -(1/4)mgh.

But you don't need to consider each vessel individually. Just consider the entire volume of water and work with the center of gravity of the entire volume.
 
  • #11
Oh, thanks. I didn't know that I should use centre of gravity to solve this question. And what do you mean by an extended object?
 
  • #12
Extended just means that the mass is spread out over some region rather than being concentrated at a point.
 
  • #13
Ok, thanks again:oldsmile:
 

FAQ: Exploring the Effects of Equal Water Depths on Gravitational Potential Energy

What is potential energy?

Potential energy is the energy that an object possesses due to its position or state. It is the energy that can be converted into other forms, such as kinetic energy, and is typically measured in joules (J).

What causes loss of potential energy?

The loss of potential energy is caused by a change in the position or state of an object. This could be due to external forces, such as gravity or friction, or internal factors, such as a change in the object's shape or composition.

How is potential energy calculated?

The potential energy of an object can be calculated using the formula PE = mgh, where m is the mass of the object, g is the acceleration due to gravity, and h is the height or vertical distance from the reference point.

What are some examples of potential energy being lost?

Examples of potential energy being lost include a roller coaster reaching the bottom of a hill, a ball rolling down a hill, a stretched spring returning to its original shape, and a book falling off a shelf. In each of these cases, the object's position or state has changed, resulting in a decrease in potential energy.

Can potential energy be converted into other forms of energy?

Yes, potential energy can be converted into other forms of energy, such as kinetic energy, thermal energy, or electrical energy. This conversion often occurs when an object is in motion or when external forces act upon it.

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