# Law of Conservation of Energy

Energy is the capacity of the object to do some work, intuitively also without energy, it is very hard for any human being to be productive and do some work. In the language of physics, if an object stores any energy, it can be converted into different forms. A stationary object can contain potential energy which can be converted into kinetic energy and an object can into motion. This kind of interconversion between different forms of energies is very common and lays the ground for the law of conservation of energy. Let’s look at these concepts of different energies and the law of conservation of energy in detail.

### 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 soo 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 ⇢ Calorie, Horsepower, Kilowatt (kW)-Power, 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 1/2mv^{2}.**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**Chemical Energy:**Chemical**Nuclear Energy:**This energy is defined as the energy that is produced or consumed in the processes where nuclei of atoms are involved.

### 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 the proof. This concept of conservation transcends various fields of science, for example, life sciences, engineering, and chemistry.

### 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 as 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 forceand 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 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**

**Sample Problems **

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

**Solution: **

The 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 = x

^{2}/3 + 4x⇒ W = 32/3 + 4(3)

⇒ W = 15 J

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

**W = 3t ^{2}**

**Calculate the instantaneous power at t = 4. **

**Solution: **

Instantaneous power is given by,

P = dW/dt

Given:

W = 3t

^{2}Calculating power,

P = dW/dt

⇒

⇒ P = 6t

At t = 4

P = 6(4)

⇒ P = 24 J

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

**W = t ^{3} + 5t + 10**

**Calculate the instantaneous power at t = 2. **

**Solution: **

Instantaneous power is given by,

P = dW/dt

Given:

W = t

^{3}+ 5t + 10Calculating power,

P = dW/dt

⇒

⇒ P = 3t

^{2}+ 5At t = 2

P = 3(t

^{2}) + 5⇒ P = 3(2

^{2}) + 5 J⇒ P = 3(4) + 5

⇒ P = 17 J

**Question 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. **

**Answer: **

If earth and the object are considered as a system. Then gravitational force between them can be considered as internal forces. In that case, law of conservation of energy can be applied.

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

P. E = K. E

⇒ mgh = 1/2mv

^{2}Given:

g = 10

h = 20m

Plugging the values inside the equation,

gh = 1/2v

^{2}⇒ 2gh = v

^{2}⇒ v =

⇒ v =

⇒ v = 20 m/s.

**Question 5: An object is kept at a height of 100m. It starts falling towards the ground. Find the velocity of the object just before it touches the ground. **

**Answer: **

If earth and the object are considered as a system. Then gravitational force between them can be considered as internal forces. In that case, law of conservation of energy can be applied.

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

P. E = K. E

⇒ mgh = 1/2mv

^{2}Given:

g = 10

h = 100m

Plugging the values inside the equation,

gh = 1/2v

^{2}⇒ 2gh = v

^{2}⇒ v =

⇒ v =

⇒ v = 10√20 m/s.