- #1

mrpolar

- 5

- 0

I looked around for hours but it seems like I'm the only one who finds it confusing.

I understand the concept of potential energy and work, but have a problem with the equations.

Here is what I don't understand:

The work energy theorem states that K

_{2}-K

_{1}=W.

W = ∫Fdx from evaluated between x

_{2}and

_{1}.

The Force done by a spring is -kx, so when only the spring acts on the body the equation should be:

K

_{2}-K

_{1}=-½kx

_{2}

^{2}+ ½kx

_{1}

^{2}

K

_{2}+½kx

_{2}

^{2}=K

_{1}+½kx

_{1}

^{2}

But now, since W=-U(x) (potential energy) the equation turns out to be:

K

_{2}-½kx

_{2}

^{2}=K

_{1}-½kx

_{1}

^{2}

For the total energy balance (kinetic + potential). But the lectures and examples I saw, and also the logical thing as I understand it, is that the equation should be K

_{2}+½kx

_{2}

^{2}=K

_{1}+½kx

_{1}

^{2}for the kinetic + potential energy balance.

Why do I get the signs wrong?

Here's an example for a question which made me really confused:

http://oyc.yale.edu/physics/phys-200/lecture-5#ch3 (minute 40:30).

In short, a mass is hanged from a spring attached the ceiling, so the mass has 2 forces acting on it: gravity and the spring.

In order to find the kinetic energy - potential energy equation, we have ΔKinetic_Energy = the negative of the work (-∫Fdx) of the total forces.

Now I know that gravity acts downwards, so the force is -mg, and the spring's force is always -kx, so the equation should look something like this:

K

_{2}-K

_{1}= -∫F(gravity)dx - ∫F(spring)dx evaluated between x

_{2}and

_{1}. The negative signs before integrals is because we want to find the potential energy which is the negative of the work done.

K

_{2}-K

_{1}= -∫-mgdx - ∫-kxdx = ∫mgdx + ∫kxdx

K

_{2}-K

_{1}= mgx

_{2}+½kx

_{2}

^{2}- mgx

_{1}+½kx

_{1}

^{2}

K

_{2}-mgx

_{2}-½kx

_{2}

^{2}= K

_{1}-mgx

_{1}-½kx

_{1}

^{2}

which is the total energy equation.

In class, they had the potential energy signs all opposite like this: K

_{2}+mgx

_{2}+½kx

_{2}

^{2}= K

_{1}+mgx

_{1}+½kx

_{1}

^{2}which of course looks more sensible and logical.

The problem is that I can't see where I got the maths wrong. I follow the rules of and get it all messed up time and time again.

Please, help me out here, this is driving me nuts!

Ben.