Differentiation of potential energy

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To find the radial force between two particles with potential energy U(r) = A/r^4, the force is calculated using F = -dU/dr. The correct derivative of U(r) leads to F = 4A/r^5, not the antiderivative previously mentioned. The confusion arises from misunderstanding the differentiation process; it is not necessary to divide U(r) by 2 since the force acts on each particle equally. A proper application of calculus is essential for accurate results. Understanding the differentiation of potential energy is crucial for solving problems involving forces in physics.
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Homework Statement



The potential energy of a system of two particles separated by a distance r is given by U(r) = (A)/(r^4), where A is a constant. Find the radial force that each particle exerts on the other. (Use A and r as appropriate in your equation.)

Homework Equations



F=-dU/dx
U is potential energy

The Attempt at a Solution



i figured out the antiderivative of U(r) is -(A)/(3*r^(3)) but it isn't the correct answer.
what else do u need to do besides the antiderivative which is equal to force. since it is 2 particles should i divide the U(r) funciton by 2?
 
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you actually want the partial derivative of r
 
ok but how do u get that? I am rather new at the whole calculus thing. wud it just b A/3r^3?
 
Think of U(r) as A*(r^-4) then try to differentiate it...
 
The book claims the answer is that all the magnitudes are the same because "the gravitational force on the penguin is the same". I'm having trouble understanding this. I thought the buoyant force was equal to the weight of the fluid displaced. Weight depends on mass which depends on density. Therefore, due to the differing densities the buoyant force will be different in each case? Is this incorrect?

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