# Confused about energy in a system (Understanding)

Take for example, a ball falling on a spring. Before touching the spring, the ball has velocity v. Then x metres of spring is compressed.

Total energy that is lost in the spring=1/2 mv2+mgx

When the spring is compressed for x m, it has 1/2 (F) (x) J of energy, right?

So to find the average F exerted by spring, should I combine the two equations together to find f?

However, my teacher said that the first equation should be linked to Fx as work is done by the ball. Thats another way of looking at it though, but the answer is different obviously. How do you explain it? I know it is my misunderstanding, can anyone help me see the light?

Thanks.

## Answers and Replies

Total energy that is lost in the spring=1/2 mv2+mgx

This would be the total energy absorbed by the spring, not lost.

When the spring is compressed for x m, it has 1/2 (F) (x) J of energy, right?

No, I don't think so. If you take away the 1/2 then that will be the relationship for work done on the spring. In other words,

Work = F x X

But work is not the same as energy, do you know the difference?

So to find the average F exerted by spring, should I combine the two equations together to find f?

Combine which two equations? Ideally, you would find the average force by applying the mean value theorem on F = -kx. But since this is a simple and ideal problem you could just find the total change in force on the spring and divide by 2. Or,

Favg = (Ff-Fi)/2

However, my teacher said that the first equation should be linked to Fx as work is done by the ball. Thats another way of looking at it though, but the answer is different obviously. How do you explain it? I know it is my misunderstanding, can anyone help me see the light?

Should be linked to Fx in what way? The equation you had up there looks correct to me. To total potential energy in the spring should the kinetic energy that was in the ball along with any potential energy from a gravitational force.

PEspring = 0.5*m*V^2 + m*g*x = 0.5*-k*x^2

Doc Al
Mentor
Take for example, a ball falling on a spring. Before touching the spring, the ball has velocity v. Then x metres of spring is compressed.

Total energy that is lost in the spring=1/2 mv2+mgx
OK. That's the energy stored in the spring when compressed to the max from the ball.

When the spring is compressed for x m, it has 1/2 (F) (x) J of energy, right?
Not sure what this is. What's F? Are you trying to calculate the work done? You could say that the energy stored = F_ave * x, where F_ave is the average force exerted by the spring.

Usually one expresses the energy stored in the spring in terms of the spring constant (which determines the force as a function of x) as well as x. Do you know that result?

Maybe you can rephrase your question.

This would be the total energy absorbed by the spring, not lost.

No, I don't think so. If you take away the 1/2 then that will be the relationship for work done on the spring. In other words,

Work = F x X

But work is not the same as energy, do you know the difference?

Combine which two equations? Ideally, you would find the average force by applying the mean value theorem on F = -kx. But since this is a simple and ideal problem you could just find the total change in force on the spring and divide by 2. Or,

Favg = (Ff-Fi)/2

Should be linked to Fx in what way? The equation you had up there looks correct to me. To total potential energy in the spring should the kinetic energy that was in the ball along with any potential energy from a gravitational force.

PEspring = 0.5*m*V^2 + m*g*x = 0.5*-k*x^2

Ahh yes. My mistake typo there. I think it is energy lost by the ball to the spring. But isnt the energy equation for a spring is 1/2 F x, where F is the average Force and x is the distance compressed? I thought by conservation of energy, PE+k.e lost by ball=1/2 Fx, which my teacher said that it should be PE+KE=Fx.

BTW, Im trying to find the average force and F is the force, Doc Al.

Doc Al
Mentor
But isnt the energy equation for a spring is 1/2 F x, where F is the average Force and x is the distance compressed?
No. Why would you need the 1/2? As I said earlier, the work done to compress the spring--and thus the energy stored in the spring--is F_ave * x.
I thought by conservation of energy, PE+k.e lost by ball=1/2 Fx, which my teacher said that it should be PE+KE=Fx.
Your teacher is correct.

In the case of a spring, the average force is one half of the maximum force, thus: F_ave = 1/2 kx. Combining this with the earlier equation gives you the energy stored in the spring: E = F_ave*x = (1/2 k x)*x = 1/2kx^2.

where does the formula 1/2 Fx come from - i need to find out where the half comes from for some coursework of mine :S