Potential energy of a particle in a system

AI Thread Summary
The discussion revolves around calculating the potential energy function U(x) for a particle influenced by a conservative force defined as F = (-Ax + Bx^6)ihat N. The user initially attempted to derive U(x) but received incorrect results, leading to confusion about the correct approach. It is clarified that the potential energy function is related to the force through the gradient, with the correct formulation being φ(x) = (A/2)x^2 - (B/7)x^7 + C, where C is a constant. The user is still uncertain about the steps to find the change in potential and kinetic energy as the particle moves between specified positions. The conversation emphasizes the importance of correctly interpreting the force and its implications for potential energy calculations.
Claudia Sanchez
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1. A single conservative force acting on a particle within a system varies as
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= (− Ax + Bx6)ihat N, where A and B are constants,
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is in Newtons, and x is in meters.
(a) Calculate the potential energy function U(x) associated with this force, taking U = 0 at x = 0.
(b) Find the change in potential energy and change in kinetic energy as the particle moves from
x = 1.30 m to x = 3.60 m.

2. I got Ax^2/2 - b^7/2 but it was wrong so I'm really confused on how to go about this problem

Thanks!
 
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The potential energy corresponding to force \vec{F} is a scalar function \phi(x,y) such that \nabla \phi= -\vec{F}. Of course, \nabla \phi is defined as \frac{\partial \phi}{\partial x}\vec{i}+ \frac{\partial \phi}{\partial y}\vec{j}. But it is not clear to me what "F" is- is "N" a vector? If so, what vector?

If you mean that \vec{F}= (-Ax+ Bx^6)\vec{i} then we must have
\frac{\partial \phi}{\partial x}= Ax- Bx^6
\frac{\partial \phi}{\partial y}= 0.

From \partial \phi/\partial x= Ax- Bx^6, we have
\phi(x,y)= \frac{A}{2}x^2- \frac{B}{7}x^7+ p(y)
But then
\frac{\partial \phi}{\partial y}= p'(y)= 0
so that p(y) is actually a constant:
\phi(x, y)= \frac{A}{2}x^2- \frac{B}{7}x^7+ C
 
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Hi, thanks for the help! that was the same answer I got but it was wrong and I don't really understand what I'm supposed to do
 
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