How Do You Calculate Force from a Two-Dimensional Potential Energy Function?

In summary: The potential is the potential energy, not the kinetic energy. You haven't given any context so I have no idea what you are talking about.
  • #1
Thermon
2
0

Homework Statement


A potential energy function for a two-dimensional force is of the form U = 3x3 * y - 7x.
Find the force that acts at the point (x, y).[/B]

Homework Equations


In a 1-dimensional case:
ΔU = -∫Fx dx
dU = -Fx dx
Fx = -dU/dx

The Attempt at a Solution


I know how to find the force in a 1-dimensional case; it's the gradient at the given x.

But I can't wrap my head around it when there are two variables.

Could it perhaps be the sum of the derivatives; Fx = -(dU/dx + dU/dy)?[/B]
 
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  • #2
Thermon said:

Homework Statement


A potential energy function for a two-dimensional force is of the form U = 3x3 * y - 7x.
Find the force that acts at the point (x, y).[/B]

Homework Equations


In a 1-dimensional case:
ΔU = -∫Fx dx
dU = -Fx dx
Fx = -dU/dx

The Attempt at a Solution


I know how to find the force in a 1-dimensional case; it's the gradient at the given x.

But I can't wrap my head around it when there are two variables.

Could it perhaps be the sum of the derivatives; Fx = -(dU/dx + dU/dy)?[/B]
The force will be a vector. If ##\phi(x,y)## is a potential for ##\vec F(x,y)## then$$
\vec F(x,y) = \langle \phi_x,\phi_y\rangle$$
 
  • #3
LCKurtz said:
The force will be a vector. If ##\phi(x,y)## is a potential for ##\vec F(x,y)## then$$
\vec F(x,y) = \langle \phi_x,\phi_y\rangle$$

So it's the combined vector of Vy and Vx? Correct me if I'm wrong, but does that means that the force at (x, y) would be the net vector of the Epot up and downwards against Ekin?
Finding those two would involve finding the integral of both Fx and Fy
 
  • #4
LCKurtz said:
The force will be a vector. If ##\phi(x,y)## is a potential for ##\vec F(x,y)## then$$
\vec F(x,y) = \langle \phi_x,\phi_y\rangle$$

Thermon said:
So it's the combined vector of Vy and Vx? Correct me if I'm wrong, but does that means that the force at (x, y) would be the net vector of the Epot up and downwards against Ekin?
Finding those two would involve finding the integral of both Fx and Fy

I don't know what you mean by "combined vector" and "net vector" and "upwards and downwards". It is a force vector field having two components or a magnitude and direction. And I don't know what Fx and Fy you are talking about. You get the vector field from the potential by taking the partials of the potential, not integrating, as I gave in the formula above.
 
  • #5


I would approach this problem by first recognizing that potential energy is a scalar quantity and that forces are vector quantities. This means that we cannot simply add the derivatives of U with respect to x and y to find the force at a given point.

Instead, we can use the gradient operator to find the force at a given point (x, y). The gradient operator is defined as ∇ = (∂/∂x, ∂/∂y), so in this case, we can write the force as F = -∇U = (-∂U/∂x, -∂U/∂y).

Using the given potential energy function, we can find the partial derivatives with respect to x and y as follows:
∂U/∂x = 9x^2 - 7
∂U/∂y = 3x^3

Therefore, the force at the point (x, y) is F = (-9x^2 + 7, -3x^3).

It is important to note that this is the force at a specific point (x, y), and it may vary at different points depending on the potential energy function. In general, the force at a point (x, y) is given by the negative gradient of the potential energy function at that point.
 

FAQ: How Do You Calculate Force from a Two-Dimensional Potential Energy Function?

What is force?

Force is a physical quantity that describes the strength or magnitude of an interaction between two objects. It is typically measured in units of Newtons (N) and can be thought of as a push or pull on an object.

What is potential energy?

Potential energy is the energy that an object possesses due to its position or arrangement in a system. It is typically measured in units of Joules (J) and can be thought of as stored energy that has the potential to do work.

How are force and potential energy related?

Force and potential energy are related through the concept of work. Work is done when a force causes an object to move a certain distance. This work can either increase or decrease the potential energy of the object, depending on the direction of the force.

What is the difference between gravitational potential energy and elastic potential energy?

Gravitational potential energy is the potential energy an object has due to its position in a gravitational field, such as the energy an object has at the top of a hill. Elastic potential energy, on the other hand, is the potential energy stored in an object that is stretched or compressed, such as a spring.

How is potential energy converted into kinetic energy?

When an object is in motion, it has both potential and kinetic energy. As the object moves, potential energy is converted into kinetic energy. This is because as the object's position changes, its potential energy decreases while its kinetic energy increases. The total energy of the object remains constant, but it can be converted between potential and kinetic forms.

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