Given force as a function of x, how do I find the total energy?

In summary: You're on the right track, but there's a simpler way to attack this problem.In summary, the problem involves determining the potential energy function U(x) for a system with a force equation of F = -kx + kx^3/α^2, where k and α are constants and k > 0. The first part involves finding U(x), which is given by kx^2/2 - kx^4/4α^2. Next, the problem asks what happens when E = kα^2/4, which can be solved by finding the kinetic energy T using the equation E = T + U. However, trying to find T using the equation T = 1/2mv
  • #1
GriffinC
7
4

Homework Statement


F=-kx+kx3/α2 where k and α are constants and k > 0. Determine U(x) and discuss the motion. What happens when E=kα2/4?

Homework Equations


F=ma=mv2d/dx
U=-∫Fdx

The Attempt at a Solution


The first part is easy.
U(x) = kx2/2-kx4/4α2
Now I'm looking for what happens when E=kα2/4
I know E=T+U, so T=kα2/4-kx2/2+kx4/4α2
To find T, I need to know velocity since T=1/2mv2
Solving for v, F=mv2d/dx, v=(k-3kx2/α2)/2(kx3/α2-kx)3/2

Here's where I'm not sure I'm going in the right direction. If I find T, I introduce an m term. The potential energy is independent of mass, so why would kinetic energy depend on mass? It also seems as though the whole thing is independent of time.
 
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  • #2
GriffinC said:

Homework Statement


F=-kx+kx3/α2 where k and α are constants and k > 0. Determine U(x) and discuss the motion. What happens when E=kα2/4?

Homework Equations


F=ma=mv2d/dx
U=-∫Fdx

The Attempt at a Solution


The first part is easy.
U(x) = kx2/2-kx4/4α2
Now I'm looking for what happens when E=kα2/4
I know E=T+U, so T=kα2/4-kx2/2+kx4/4α2
To find T, I need to know velocity since T=1/2mv2
Solving for v, F=mv2d/dx, v=(k-3kx2/α2)/2(kx3/α2-kx)3/2

Here's where I'm not sure I'm going in the right direction. If I find T, I introduce an m term. The potential energy is independent of mass, so why would kinetic energy depend on mass? It also seems as though the whole thing is independent of time.

Did you think of drawing a graph of ##U##?

You're right that expressing ##T = \frac12 mv^2## isn't going to help.
 

FAQ: Given force as a function of x, how do I find the total energy?

1. How do I determine the force as a function of x?

To determine the force as a function of x, you need to have a mathematical equation that relates force to position. This equation can be derived from known physical laws or experimental data.

2. Can I use any unit for force and position in the equation?

Yes, as long as the units are consistent. For example, if the force is measured in Newtons and the position is measured in meters, the equation should have units of Newton-meters.

3. What is the relationship between force and energy?

Force and energy are related, but not interchangeable. Force is a physical quantity that describes the interaction between two objects, while energy is a measure of the ability to do work. The total energy of a system is the sum of its kinetic and potential energy, which can be affected by the force acting on the system.

4. How do I find the total energy from the force as a function of x?

To find the total energy from the force as a function of x, you can use the work-energy theorem. This states that the work done by a force on an object is equal to the change in the object's kinetic energy. Therefore, integrating the force as a function of x over the displacement will give you the total work and thus the total energy.

5. Are there any other factors that can affect the total energy?

Yes, there may be other factors such as external forces, friction, and the presence of other forms of energy (such as thermal or electrical) that can affect the total energy of a system. It is important to consider all relevant factors when calculating the total energy of a system.

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