Potential Energy Homework: Determine F & Evaluate Work

[imy786]

Homework Statement

Define the potential energy function V(r) for a particle which is acted on by
a conservative force F(r), given that V = 0 at r = to. Also, explain how F
can be determined if V is known.

A particle moving in the x - y plane is subject to a conservative force F(x, y)
whose potential function is V = Kx^3y^ 2, where K is a constant.
Evaluate F(x, y). Also, determine the work done on the particle by this force
in moving it from the origin, x = O, y= O, to the point x = 2, y= 4.

Homework Equations

P=mgh

The Attempt at a Solution

need help here to start this off

 

[HallsofIvy]

I am constantly amazed at how many people post "homework" problems here when they appear to be saying they have never seen such a problem before! Surely. if you have been given a problem like this you must be expected to know something about it!

To start with, "p= mgh" is NOT a relevant equation. That is the equation for the potential energy in a very specific instance: The change in gravitation potential energy through a short change in height. This problem does not say the force is gravitational, it says the force is a function of x and y (or a function of r- you seem to have two different problems here). Do you know the general definition of "potential energy" in a conservative force field? Do you know how to do the opposite problem- given the potential energy function, find the force at any point?

 

[imy786]

$$F = -kx$$ <--- force of a spring

$$U(x) = -W$$

$$U(x) = -\int_0^{x} -kxsingle-quote dxsingle-quote$$

$$U(x) = k (\frac{1}{2}xsingle-quote^2)_0^{x}$$

$$U(x) = \frac{1}{2} kx^2$$
-- potential energy of spring

 

[Rainbow Child]

This is good enough for 1 dimension. What about more dimensions?

And about HallsofIvy's question:

Do you know how to do the opposite problem- given the potential energy function, find the force at any point?

 

[imy786]

using differntiation..to find the force

 

[Rainbow Child]

Actually for 1 dimension! For more

$$\vec{f}=-\vec{\nabla}\,U$$

Thus for your problem the force is ...

 

[imy786]

so how is F can be determined if V is known.

 

[Rainbow Child]

$$\vec{\nabla}=\left(\frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z}\right), \quad U=K\,x^3\,y^ 2$$

 

[imy786]

can you please explain this expression.

 

[Rainbow Child]

The force is given by

$$\vec{f}=-\vec{\nabla}\,U$$

Thus for your problem

$$\vec{f}=-\vec{\nabla}\,U\Rightarrow \vec{f}=-\left(\frac{\partial U}{\partial x},\frac{\partial U}{\partial y},\frac{\partial U}{\partial z}\right)$$

Replace your $U$ and you have the answer.

 

[malawi_glenn]

imy786: why did you make a new thread with the same problem stated?

are you familiar with vectors?

 

[imy786]

there was 2 parts to this problem...should i delete the other part

 

[malawi_glenn]

"Potential Energy part 2

--------------------------------------------------------------------------------

1. Homework Statement

A particle moving in the x - y plane is subject to a conservative force F(x, y)
whose potential function is V = Kx^3y^ 2, where K is a constant.
Evaluate F(x, y). Also, determine the work done on the particle by this force
in moving it from the origin, x = O, y= O, to the point x = 2, y= 4."

You posted there, it has the same content as the problem you posted here.

Also say what it is that you don't understand, what have you tried? have you tried do get the force from the potential? What does your course book say about it? What does your course book say how to find work done by a force?

In order to get help from us, you must help us to help you.

 

[enricfemi]

hey,every guys!very glad to join this topic.
as i read this thread i found the problem:
imy786 maybe is in high school now,and i dout if he really understand nable operator.

is this really your homework,my dear imy786!(can't image this high school homework now)

 

[imy786]

im at university not in high school...dopey enricfemie

 

[imy786]

$$\vec{\nabla}=\left(\frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z}\right), \quad U=K\,x^3\,y^ 2$$

w= V(r1) - V(r2)= change in PE.

at x=2 and y=4.

how do i determine the work done from this info? please help