**Conservation of Mechanical Energy **

### Key Concepts

- Mechanical energy
- The formula of mechanical energy
- Law of conservation of energy

### Introduction

**Gravitational potential energy:**

Gravitational potential energy can be defined as the energy possessed by the body with respect to other bodies in the gravitational field.

Gravitational potential energy is given by

U = mgh

Where,

U= gravitational potential energy

m = mass

g = acceleration due to gravity

h = heigh at which body experiencing potential energy

**Elastic potential energy:**

The energy is possessed by anybody which has elastic property (spring). The potential energy created in the body before the release, or the relaxation of the body is called elastic potential energy.

The elastic potential energy is given by the magnitude of the force in the ideal spring.

F = Kx

Where,

F = magnitude of the force

K = spring constant

X = displacement of the spring

The elastic potential energy is given by

U = ½ k x^{2}

### Explanation

**Mechanical energy:**

Mechanical energy is the sum of kinetic energy and potential energy of the body.

M.E = K.E + P. E

**The formula for the mechanical energy:**

As we discussed that the mechanical energy is the sum of kinetic energy and potential energy,

I.e., Mechanical energy = potential energy + kinetic energy

M.E = ½ mv^{2}+ mgh

**Law of conservation of energy:**

Law of conservation of energy states that energy is neither created nor destroyed but changes from one form to another.

**Example:**

Consider a ball that is at height **h** and is at rest.

To prove a law of energy conservation, we have to show that the energy in each case is equal.

**Case 1:**

Consider ball is at heigh H.

Here the balls have potential energy and kinetic energy is zero.

P.E = mgH

K.E = 0

P.E + K.E = 0 + mgH

M.E = mgH

**Case 2:**

Consider a ball falling freely from the height H.

Here the ball has both potential energy and kinetic energy.

P.E = mgh

K.E = ½ mv^{2}

But v = √2g(H−h)

Sub v in K.E

M.E = K.E + P.E

M.E = 1/2 m(√2g(H−h))^{2} + mgh

M.E = mgH

**Case 3:**

Consider that a ball reached to the ground from the height H.

Here the potential energy is zero, and the ball has only kinetic energy.

P.E = 0

K.E = ½ mv^{2}

But v =√2gH

Sub v in K.E

M.E = K.E + P.E

M.E = 1/2 m(√2gH)^{2 }+ 0

M.E = mgH

## Summary

**Mechanical energy:**Mechanical energy is the sum of kinetic energy and potential energy of the body.

M.E = K.E +P.E

**The formula for the mechanical energy:**As we discussed that the mechanical energy is the sum of kinetic energy and potential energy,

I.e., Mechanical energy = potential energy + kinetic energy

**M.E = 1**

_{/2}mv^{2}+ mgh**Law of conservation of energy:**Law of conservation of energy states that energy is neither created nor destroyed but changes from one form to another.

K=spring constant

X = displacement of the spring

The elastic potential energy is given by

**U= 1**

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