## Physics Tutorial

*Conservative Forces and Conservation of Energy (Proof)*

#### Intro

In this short section we will

a) define conservative forces and

b) prove that such forces conserve a quantity we call energy for constant mass systems.

A conservative force is defined as:

(1)

Where is a function of the coordinates.Now we multiply both sides by . We get:

(2)

Where we used Newton’s second law in the first equal sign. It is left to the reader to prove:

(3)

Where is the velocity. Using the equations above we have:

(4)

Defining energy to be the sum of the kinetic energy and the potential energy we have:

(5)

In other words , the standard conservation of energy.

#### Sample Problem

Prove

(1)

#### Solution

For the first equality we have:

For the second equality we have:

(1)

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