Calculating Force from a Given Potential Energy

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To determine the force associated with the given potential energy function U=(x^3 y^2 z^4i - x^2 y^3 z j+x^3 y^3 k)J, the force components can be calculated using the negative gradient of the potential energy. The equation Fx=-dU/dx is a starting point for finding the force in the x-direction. Participants express uncertainty about the differentiation process and how to apply it correctly. Clarification on the derivation steps and the application of the gradient in three dimensions is needed to solve the problem effectively. Understanding the relationship between potential energy and force is crucial for accurate calculations.
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potential energy?

Homework Statement




A potential is given by U=(x^3 y^2 z^4i - x^2 y^3 z j+x^3 y^3 k)J. Determine the force associated with this potential.

Homework Equations



Fx=-dU/dx?

The Attempt at a Solution

I am not sure if i would just derive or how to approach this probelm
 
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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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