# How can these conservation of energy and work energy theorems be related to each other? 2. Work-Energy Theorem: a. What physical condition must be true for the Work-Energy Theorem to become a...

How can these conservation of energy and work energy theorems be related to each other?

2. **Work-Energy Theorem**: a. What physical condition must be true for the Work-Energy Theorem to become a statement of **Conservation of Energy**?

b. Use your answer from part (a) to derive a statement of Conservation of Energy from the Work-Energy Theorem.

c. Use your answer from part (b) to demonstrate that energy is conserved by passing back and forth between Kinetic Energy and Potential Energy.

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The work-theorem energy states that **the net work done on an object or a system equals to the change of kinetic energy of that object or the system. **The net work means the work of ALL of the forces acting on the object.

`W_(NET) = K_f - K_i=DeltaK`

The net work can be broken down to the sum of the work done by two different types of forces: conservative and non-conservative, so the work-energy theorem can be rewritten as

`K_f - K_i = W_(cons) + W_(nc)`

The conservative forces, by definition, are the forces such that the work done by them on the object equals the negative change of the potential energy of that object:

`W_(cons) = -DeltaU = -(U_f - U_i)`

If this is plugged into the work-energy theorem, it becomes

`DeltaK = -DeltaU + W_(nc)` , or

`DeltaK + DeltaU = W_(nc)`

This means that the change of the TOTAL energy of the system equals the work of ONLY the non-conservative forces.

When there is no non-conservative forces present, then `W_(nc) = 0` , so the work-energy theorem becomes the statement of the conservation of energy:

` ` `DeltaK +DeltaU=0` , which means that energy of the system is conserved when there is no external non-conservative forces acting on it.

It can also be written as

`K_f + U_f = K_i + U_f` , the sum of kinetic and potential energies in the final state equals the sum of kinetic and potential energies in the initial state. This means that if potential energy decreased, the kinetic energy must increase, and vice versa: if kinetic energy decreased, the potential energy must increase. In other words, in the absence of external forces, the energy can pass back and forth between kinetic and potential energy.