Why Is Force Defined as the Negative Gradient of Potential Energy?

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SUMMARY

The relationship between force and potential energy is defined by the equation F_x = -dU/dx, where F_x represents the force and U represents potential energy. This definition arises from the work done by a conservative force when moving an object at constant velocity, which is expressed as a line integral of the force. The negative gradient indicates that force acts in the direction of decreasing potential energy. This concept is rooted in fundamental calculus principles, making it intuitive for understanding conservative forces.

PREREQUISITES
  • Understanding of basic calculus, particularly derivatives and integrals.
  • Familiarity with the concept of conservative forces in physics.
  • Knowledge of potential energy and its relation to work.
  • Basic grasp of Lagrangian mechanics is beneficial but not necessary.
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  • Study the concept of conservative forces and their properties.
  • Learn about line integrals and their applications in physics.
  • Explore the fundamental theorem of calculus and its implications in physics.
  • Investigate the principles of Lagrangian mechanics for a deeper understanding of energy dynamics.
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Students of physics, educators explaining mechanics, and anyone interested in the mathematical foundations of force and energy relationships.

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F_x= (-dU)/dx

He used dn in place of dx in a different example.

My professor wrote this on the board, and I think he tried to explain why but I didn't get it. Normally he's pretty good, but I'm not understanding the relationship here. Why is this true?
 
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Do you know anything about Lagrangian mechanics?
 
I do not.
 
You don't need to know Lagrangian mechanics to understand this.

This is essentially a definition.

You define the potential energy U of a conservative force F at two different points in space by the amount of work needed to move an object from one location to another at constant velocity.

Thus, the potential energy U is defined as a line integral of F (since work is force times distance).

The result with the derivative is simply an application of the fundamental law of calculus.

Of course, you could also define it the other way around if you wanted, but this integral definition tends to be more intuitive.
 

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