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# Law of Conservation of Energy

Energy is the capacity of the object to do work, the energy according to physicists is always constant and can only be converted from one form to another thus, energy is always conserved. Law of conservation of energy states that for an isolated system the loss of some amount of energy is always followed by the gain of a similar amount of energy in that system. Let’s learn about them in detail in this article.

## Energy

Energy is that ability that helps in applying force in order to do some work. It is simply that force that causes things to move. The capacity to do work is known as Energy. One very important fact about energy to note is that even though, energy occurs in many forms and has so many types, from kinetic energy to potential energy to solar energy, etc. The SI Unit of energy is Joules. Apart from joules, other units of Energy are ⇢ Calories, Horsepower, Kilowatt (kW)-Power, and Kilowatt-hour (kWh). There are many different forms of energy, following list shows and describes some of the important aspects of mechanical energy:

• Kinetic Energy: Kinetic Energy is the energy that is possessed by the objects in motion. Work must be done on an object to change its kinetic energy. It is usually expressed in the form of the equation 1/2mv2.
• Potential Energy: Potential energy is defined as the energy that is possessed by an object by the virtue of its position. The potential energy is denoted by “mgh” where “h” is the height of the object.
• Mechanical Energy: This energy is the total energy associated with the position and velocity that is stored in the object. Hence, mechanical energy is the sum of kinetic energy and potential energy.
• Chemical Energy: Chemical energy is defined as the energy that is stored inside the bonds in the materials. This energy is involved in the formation or destruction of chemical bonds.
• Nuclear Energy: This energy is defined as the energy that is produced or consumed in the processes where nuclei of atoms are involved.

## What is the Law of Conservation of Energy?

It’s known that the total mechanical energy of the system remains constant if the forces working on the system are conservative in nature. Potential energy and kinetic energies keep interchanging with each other. In the case of non-conservative forces, these energies are converted to some other energy such as heat, noise, etc. In the case of a system that is isolated from the outside world, the total energy remains constant.

In an isolated system, energy can neither be created nor be destroyed. Total energy remains constant. It can be converted from one form to another form.

The principles of conservation of energy cannot be proved however no violation of this law has ever been observed. So, it is widely accepted with proof.

For an isolated system, the loss of energy in some part is followed by a gain of an equal amount of energy in some other part of the system. This principle is not yet proven but no exception of this principle is yet encountered by physicists so it is considered to be true.

The law of conservation of energy can easily be understood by the following example, here in the image given below the total energy of the cyclist at the bottom of the hill is in the form of Kinetic Energy only and at top of the hill is in form of potential energy. The total amount of energy in any system is calculated using the following equation:

UT = Ui + W + Q

where,
UT is total energy of a system
Ui is initial energy of a system
Q is heat added or removed from the system
W is work done by or on the system

Also, change in the internal energy of the system is calculated using,

ΔU = W + Q

## Law of Conservation of Energy Derivation

Law of Conservation of Energy can be derived with the help of the following example, take a ball falling from the height H and the initial velocity is zero, also the potential energy at the surface of the earth is zero.

Now at the height H from the ground,

Etotal = Epotential + Ekinetic

Etotal = 1/2mv2 + mgH

Etotal = 1/2m(0)2 + mgH

Etotal = mgH…(A)

As the ball falls to the ground, its potential energy decreases, and kinetic energy increases.

At any point B, which is at a height X from the ground, it has speed ‘v’ as it reaches point B. So, at this point, it has both kinetic and potential energy.

Etotal = Epotential + Ekinetic…(1)

Epotential = mgX…(2)

According to the third equation of motion,

v2 = 2g(H-X)

1/2mv2 = 1/2m×2g(H-X)

1/2mv2 = mg(H-X)

Ekinetic = mg(H-X)…(2)

Using (1), (2), and (3)

E = mg(H – X) + mgX

E = mg(H – X + X)

E = mgH…(B)

Also, at point C, at the ground,

Etotal = Epotential + Ekinetic…(a)

Epotential = 0…(b)  (as height is zero)

According to the third equation of motion,

v2 = 2g(H-0)

1/2mv2 = 1/2m×2g(H)

1/2mv2 = mg(H)

Ekinetic = mg(H)…(c)

Using (a), (b) and (c)

E = mg(H) + 0

E = mgH…(C)

Thus, from A, B, and C it is clear that the total energy at any point of the falling of the ball is constant (mgH).

## Law of Conservation of Energy Examples

Various events or examples we see in our daily life support the Law of Conservation of Energy. Some of the examples to prove the law of Conservation of Energy are discussed below in this article.

• Engines convert chemical energy into mechanical energy.
• Electric motor converts electrical energy into mechanical energy.
• Electric bulb converts electric energy into light and heat energy.
• Hydroelectric power plants convert the potential energy of water into the kinetic energy of the turbine, which is further converted into electrical energy.
• Loudspeaker converts electrical energy into sound energy.
• Electrochemical cell converts chemical energy into electric energy and vice versa.

## Energy Conservation

From the above examples, it is clear that the total energy of the system is always conserved it can only be converted from one form to other. It wastes lots of our energy by converting one form of energy into another form. So we must use our energy resources widely this is called energy conservation.

## Work and Power

Often a change in energy is accompanied by the work done, usually work done is required but sometimes the rate of work being done also becomes important to realize a physical process. This rate of work being done is also termed power. For a block of mass “M”, a force F produces a displacement of “r” in the block. In this case, the work done by the force on the object is defined by the equation given below.

For a constant force and the displacement . The work done is defined by, This is the dot product between two vectors, so if the Force makes an angle θ with the displacement. Then the work done will be given by,

W = |F||r|cosθ

Power is defined as the rate of work being done. The average power is defined by the total energy transferred or work done per unit of time.

P = W/T

Where W presents the net work done and T represents the total time taken.

In case the rate of work done is changing, the instantaneous power is given by,

P = dW/dt

Also, Check

## Solved Examples on Conservation of Energy

Example 1: Find the work done when a force of F = x + 3 produces a displacement of 3 m.

Solution:

Work done by a variable force is given by,

W = ∫Fdx

F(x) = x + 3

Calculating the work done.

W = Here, the displacement is x = 3

W = x2/3 + 4x   (at x = 3)

W = 32/3 + 4(3)

W = 15 J

Example 2: The work being done on a system is given by the following equation,

W = 3t2

Calculate the instantaneous power at t = 4.

Solution:

Instantaneous power is given by,

P = dW/dt

Given: W = 3t2

Calculating power, P = dW/dt P = 6t

At, t = 4

P = 6(4)

P = 24 J

Example 3: The work being done on a system is given by the following equation,

W = t3 + 5t + 10

Calculate the instantaneous power at t = 2.

Solution:

Instantaneous power is given by,

P = dW/dt

Given:

W = t3 + 5t + 10

Calculating power,

P = dW/dt P = 3t2 + 5

At, t = 2

P = 3(t2) + 5

P = 3(22) + 5 J

P = 3(4) + 5

P = 17 J

Example 4: An object is kept at a height of 20m. It starts falling towards the ground. Find the velocity of the object just before it touches the ground.

Solution:

Potential energy at the start will be equal to the kinetic energy just before touching the ground.

P. E = K. E

mgh  = 1/2mv2

Given:

g = 10

h = 20 m

Plugging the values inside the equation,

mgh = 1/2mv2

2gh = v2

v = v = v = 20 m/s.

## FAQs on the Law of Conservation of Energy

### Question 1: State the law of conservation of energy.

Law of conservation of energy states, that foe a closed system the total energy of the system is always conserved.

### Question 2: Law of conservation of energy examples.

Various examples explaining the law of conservation of energy include,

• Hydroelectric power plants convert potential energy to electrical energy.
• The loudspeaker converts electrical energy into sound energy.
• An electric motor converts electric energy into mechanical energy, etc.

### Question 3: Who discovered the law of conservation of energy?

Julius Robert Mayer was credited with the discovery of the law of conservation of energy.

### Question 4: What is Energy?

The ability of a body to do work is called energy.