Law of Conservation of Energy

Law of Conservation of Energy

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Law of conservation of energy implies

The total energy of a particle in a conservative field is constant in time.

Where in the conservative force field, the total work done along the closed path by definition equals zero.

To prove:

dE/dt = 0

Where E= total energy =K.E + P.E

Let K.E=T and P.E=U then

dE/dt =dT/dt + dU/dt

We will find expressions for dT/dt and dU/dt 

For dT/dt

1. We’ll start with the definition of work done.

Work done is equal to the line interval of force.

2. Force is equal to the change of momentum which equals mass times change in velocity.

3.

We multiply and divide the F.dr expression with dt. And compare the expression with  

K.E= 1/2mv^2

On arranging we get dT/dt = F.v

Where F and v are velocity vectors. See the image for further clarification.

For dU/dt 

1. We write U in terms of its components.

2. We split the product by writing the unit vector with each term.

3. One term in the product gives gradient dell and the other gives dr.

4. We know the negative of gradient of potential energy is force. 

So we substitute the gradient in expression obtained in step 3 with negative of force. And the expression becomes 

dU/dt = -F.v

See image for clarification 

For the conservation of total energy 

dE/dt = dT/dt + dU/dt =F.v — F.v

dE/dt = 0 which implies total energy is conserved in time.