Potential energy of a particle in a system

In summary, the potential energy function associated with the given conservative force is given by U(x) = (Ax^2)/2 - (Bx^7)/7. The change in potential energy from x = 1.30 m to x = 3.60 m is -23.22 Joules, while the change in kinetic energy is 0 Joules. The confusion may arise due to the unclear definition of the force and its vector form.
  • #1
Claudia Sanchez
4
0
Warning! Posting template must be used for homework questions.
1. A single conservative force acting on a particle within a system varies as
Farrowbold.gif
= (− Ax + Bx6)ihat N, where A and B are constants,
Farrowbold.gif
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!
 
Last edited:
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  • #2
The potential energy corresponding to force [itex]\vec{F}[/itex] is a scalar function [itex]\phi(x,y)[/itex] such that [itex]\nabla \phi= -\vec{F}[/itex]. Of course, [itex]\nabla \phi[/itex] is defined as [itex]\frac{\partial \phi}{\partial x}\vec{i}+ \frac{\partial \phi}{\partial y}\vec{j}[/itex]. But it is not clear to me what "F" is- is "N" a vector? If so, what vector?

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

From [itex]\partial \phi/\partial x= Ax- Bx^6[/itex], we have
[tex]\phi(x,y)= \frac{A}{2}x^2- \frac{B}{7}x^7+ p(y)[/tex]
But then
[tex]\frac{\partial \phi}{\partial y}= p'(y)= 0[/tex]
so that p(y) is actually a constant:
[tex]\phi(x, y)= \frac{A}{2}x^2- \frac{B}{7}x^7+ C[/tex]
 
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  • #3
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
 

1. What is potential energy?

Potential energy is the energy possessed by an object or particle due to its position or configuration within a system. It is a form of energy that is stored and can be converted into other forms of energy.

2. How is potential energy related to a particle in a system?

In a system, a particle's potential energy is determined by its position relative to other particles or objects in the system. It is affected by factors such as the distance between particles, their masses, and the forces acting on them.

3. What factors affect the potential energy of a particle?

The potential energy of a particle is influenced by the particle's mass, its position within the system, and the forces acting on it. The type of potential energy, such as gravitational or electric, also plays a role.

4. How is potential energy calculated for a particle in a system?

The potential energy of a particle in a system can be calculated using the equation PE = mgh, where m is the mass of the particle, g is the acceleration due to gravity, and h is the height or distance the particle is from the reference point.

5. What is the relationship between potential energy and kinetic energy?

Potential energy and kinetic energy are both forms of energy that an object or particle can possess. Potential energy can be converted into kinetic energy when the object or particle is in motion, and vice versa. This relationship is described by the law of conservation of energy.

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