Given force, find the change in kinetic energy

[oneplusone]

Homework Statement

You are given Force as a function of distance (F vs x). Find the change in kinetic energy from $x=a $ to $x=b$.

Homework Equations

see attempt solution
conservation of energy equations

The Attempt at a Solution

Since F = -dU/dx we get the integral of F = U.

Then do U(b)-U(a) to get the answer.

Is this right?

 

[UltrafastPED]

Yes ... the Work-Energy theorem.

 

[oneplusone]

Change in Kinetic energy

If you are given a function which describes force applied to the object, and want to find the change in kinetic energy, how would you do it?
The two approaches i thought of were:
1. U(a)-U(b) (integrate f)
2. integral of Force from a to b

Basically, I don't get which is correct, and WHY it's correct.
Can someone explain please?

 

[Nugatory]

If you are given a function which describes force applied to the object, and want to find the change in kinetic energy, how would you do it?
The two approaches i thought of were:
1. U(a)-U(b) (integrate f)
2. integral of Force from a to b

Basically, I don't get which is correct, and WHY it's correct.
Can someone explain please?

I assume that by $U(a)$ in the first one, you mean the potential. If so, the quick answer is: Second formula always works, first formula is a special case of the second.

The second approach is always correct, as long as you remember to include all the forces in the integral. For example, if you're applying a force on the object, but there's an equal frictional force in the opposite direction, the speed will be constant, no acceleration, no change in kinetic energy. (Much work will still be done by the two forces, but it will end up as some form of energy other than kinetic, usually heat).

When you apply the second formula to a certain class of problems, you get the first formula. We use that method whenever we can because it's usually much easier, but we can only use it if all the forces are "conservative", meaning that the value of the potential at a given point is the same no matter how you got there. Gravity is an example of a conservative force; the kinetic energy lost as a ball rolls uphill comes back when the ball rolls back downhill. Friction is a example of a non-conservative force; what it takes it never gives back.

 

[oneplusone]

Why doesn't the first one work? Or what case would it work?
And thanks.

 

[cepheid]

Why doesn't the first one work? Or what case would it work?
And thanks.

Nugatory already explained this, so all I can do is repeat it in a slightly different way. The equation F = -dU/dx is only true for conservative forces. Conservative forces are forces that have the property that the work they do on an object when it moves from point a to point b depends ONLY on the locations of those endpoints a and b. It does NOT depend on path taken to get from a to b. So, something like gravity is a conservative force. The work done by gravity between two points depends only on the height difference between those two points, not on the path the object took to get between them. In contrast, something like friction is not a conservative force, because the work done by friction on an object moving from point a to point b is going to be highly dependent on the path taken between those two points. Potential energy is not a meaningful concept for non-conservative forces: if an object loses kinetic energy in moving from point a to point b because of negative work done by a non-conservative force, then you cannot "get that energy back" simply by moving back from point b to point a. The energy lost is not stored as potential energy for later recovery. It's gone (into some other form that is neither kinetic nor potential). Mechanical energy (kinetic + potential) is not conserved when non-conservative forces are acting. So for a non-conservative force, F is not equal to -dU/dx, and that's why this method doesn't work.

The work-energy theorem (W = ΔKE) is always true, regardless of whether the force doing the work is conservative or non-conservative. That's why this method for calculation always works.

 

[oneplusone]

Yes, now i get it! Thank you very much everyone!
One final question: is there an easy way to check if a function of potential energy is conservative? Like given an arbitrary function, how can you tell if it's conservative or not?

 

[cepheid]

Yes, now i get it! Thank you very much everyone!
One final question: is there an easy way to check if a function of potential energy is conservative? Like given an arbitrary function, how can you tell if it's conservative or not?

When you say "a function of potential energy", do you mean some U(x)?

Again, U(x) is only defined and meaningful for conservative forces. IF you have a conservative force, THEN it can be expressed as the derivative of some potential energy function U(x).

If the force is not conservative, then it CANNOT be expressed as the derivative of a potential energy function, and so there is NO U(x).