# Conservation of Mechanical Energy

## Mechanical Energy

Mechanical energy is the sum of kinetic energy and potential energy in an object that is used to do a particular work. In other words, it describes the energy of an object because of its motion or position, or both.

Consider the case of an ideal simple pendulum (friction-less). We can say that this system's mechanical energy is a combination of its kinetic and gravitational potential energy. As the pendulum swings back and forth, there is a constant exchange of kinetic and potential energy. When the bob reaches its maximum height, the system's potential energy is the highest, while its kinetic energy is zero. The kinetic energy is greatest at the mean position, and the potential energy is zero.

## Classification of Mechanical Energy

Mechanical energy is classified into two types:

• Potential energy and
• Kinetic energy.

The sum of the two is known as total mechanical energy. In nature, mechanical energy is limitless.

### Potential Energy

Potential energy is the force that a body has the potential to develop if it is moved. Potential energy is not the same as movement energy. Instead, it is the energy stored in a body as a result of its physical properties, such as mass or position.

Example:

The best example of potential energy is gravitational potential energy.

Consider throwing a basketball high into the air. The ball's trajectory is straightforward: it rises, reaches its peak, pauses for a moment, and then begins to fall. The ball has the most potential energy at its highest point. It has no kinetic energy at a point in time when it is not moving (however brief that moment may be).

The potential energy of an object increases when it is subjected to gravitational force. Potential energy can be classified into elastic potential energy, gravitational potential energy, electric (electromagnetic) potential energy, and nuclear potential energy.

The mgh formula is used to calculate the object's potential energy and maximum kinetic energy:

PE = mgh, where

PE = Potential Energy

m = mass of the object

g = gravity (9.8 m/s2) taken as a net force

h = height of the object

An object's mechanical energy is thus proportional to its mass, object height, or vertical position.

### Kinetic Energy

Kinetic energy, as opposed to potential energy, is the mechanical energy of motion or energy of motion, rather than position. The higher the kinetic energy, the faster the movement. The highest kinetic energy that a body can develop while moving is its top speed.

Example:

Returning to our basketball example, the ball rises, reaches its highest point, and then falls back down. As the ball rises, its speed decreases due to gravity, and its kinetic energy decreases. It gains momentum as it falls back down, and as its speed increases, so does its kinetic energy. When it hits the ground, its kinetic energy reaches its peak before vanishing (assuming it does not bounce back).

We can see from these two extreme points that the system has both kinetic and potential energy, the sum of which is constant. These observations reveal a lot about mechanical energy conservation. But how can we demonstrate this for every other system? In the following section, we will learn more about mechanical energy conservation through the use of an appropriate example.

Images Coming soon

## Conservation of Mechanical Energy

According to the principle of Mechanical energy,

The total mechanical energy of a system is conserved, which means that it cannot be created or destroyed; it can only be internally converted from one form to another if the forces acting on the system are conservative in nature.

Example:

Consider an example of one-dimensional motion of a system to better understand this statement. If a body is displaced by Δx due to the action of a conservative force F, we can deduct from the work-energy theorem that the network done by all the forces acting on a system is equal to the change in the kinetic energy of the system.

Mathematically, $\mathrm{\Delta KE =F(x)\Delta x}$

Where, $\mathrm{\Delta K}$ is the change in kinetic energy of the system. Considering only conservative forces are acting on the system $\mathrm{W_{net}=W_c}$.

Thus, $\mathrm{W_c\:=\:\Delta KE}$

Furthermore, when conservative forces perform work in a system, the system loses potential energy equal to the work performed. As a result, $\mathrm{W_c\:=\:-PE}$

This means that if only conservative forces are involved in the process, the total kinetic energy and potential energy of the system remain constant.

KE + PE = constant

$$\mathrm{KE_{i+}PE_i\:= \:KE_{f+}PE_f}$$

Where denotes the initial values and f denotes the final values of KE and PE.

This law only applies when the forces are conservative in nature. The total kinetic energy plus the total potential energy is defined as the system's mechanical energy. In a system consisting only of conservative forces, each force is associated with a type of potential energy, and the energy only changes between kinetic energy and different types of potential energy, so the total energy remains constant.

## Total Mechanical Energy of the System

Images Coming soon

Example:

Let's look at an example to better understand this principle. Assume a ball of mass m is dropped from a height H cliff, as shown above.

At height H:

Potential Energy(PE) = mxgxH

Kinetic Energy(KE) = 0

Total mechanical energy = mgH

At height h:

Potential energy(PE) = mxgxh

Kinetic Energy(KE) = $\mathrm{1/2(mv^2)}$

The velocity $\mathrm{v_1}$ at a height h of an object of mass m falling from a height H can be written using the equations of motion as:

$$\mathrm{v_1\sqrt{2g(H - h)}}$$

Hence, the kinetic energy can be given as,

$$\mathrm{\frac{1}{2}m(\sqrt{2g(H - h)})^2= mgH−mgh}$$

Total mechanical energy = (mgH−mgh)−mgh = mgH

Hence, the kinetic energy can be given as,

$\mathrm{Kinetic \:Energy = \frac{1}{2}m(\sqrt{2gH})^2=mgH}$

Total mechanical energy = mgH

## FAQs

Q1. Explain the principle of mechanical energy conservation.

Ans: The total mechanical energy of a system is conserved, which means that it cannot be created or destroyed; it can only be internally converted from one form to another if the forces acting on the system are conservative in nature.

Q2. Define mechanical energy of the system.

Ans: The total kinetic energy plus the total potential energy is defined as the system's mechanical energy.

Q3. Is kinetic energy conserved in elastic collisions?

Ans: Yes, kinetic energy is conserved in elastic collisions.

Q4. Name the device which converts electrical energy into mechanical energy

Ans: Electric motor

Q5. Give an example where heat energy is converted into mechanical energy?

Ans: Heat engine