A dimensional formula is a mathematical formula that represents a physical quantity in terms of fundamental quantities such as mass, length, and time. By using dimensional formulae, we can see how fundamental quantities are raised to different powers to build a new physical quantity. The powers of fundamental physical quantities are referred to as dimensions.

Scientists define energy as the ability to do tasks. People utilize energy to walk and bike, drive vehicles on roads and boats on the water, cook meals on stoves, produce ice in freezers, illuminate our homes and workplaces, manufacture goods, and launch astronauts into space.

Its unit is the joule (J). Since energy is a form of work, it is expressed as

Energy = Work = Force Distance

 Energy 

Energy is defined as the capacity to do work. In other words, energy is the ability to do work. Energy can neither be created nor be destroyed, it can be converted from one form to another. It is a scalar quantity as it only has magnitude and direction. The SI Unit of energy is Joule. Energy can be found in everything in the universe, whether a living thing or a non-living thing. The energy can be divided into types i.e., Kinetic Energy and Potential Energy.

Kinetic energy: It is the form of energy of motion. The kinetic energy of an object is equal to half of its mass, times its velocity squared. 

K.E. = (½) m v²

Where m is mass, v is the velocity of the object 

Potential Energy: The energy due to rest is known as potential energy. It is the form of energy that could be used to do work. One of the most common types of potential energy is the gravitational force. 

P.E. = mgh 

where m is mass, g is gravity, and h is the height 

Differentiation Between Kinetic Energy and Potential Energy

Derivation of the Dimensional Formula of Energy 

The formula for energy is,

Energy = Force Distance

Force can be written as,

Force = mass acceleration

Acceleration is the rate of change of velocity,

Acceleration = velocitytime = m/ss

The fundamental dimensional formula of mass is [M], that of length is [L] and for time is [T].

So, Acceleration= [L1][T1][T1] = [L1][T1+1] = [L1][T2] = [L1T-2]

Substituting this in the formula for force 

Force = [M1].[L1T-2]=[M1L1T-2]

Substitute this in the formula for energy, with the dimensional formula of distance being [L]

Energy = [M1L1T-2].[L]

Combining the two terms,

Energy = [M1L2T-2]

So the dimensional formula of energy is [M1L2T-2].

Dimensional Formula

The expression indicating the powers to which fundamental units must be raised to obtain one unit of a derived quantity is the quantity’s dimensional formula.

Suppose Q is the unit of a derived quantity represented by the expression Q = MaLbTc. In that case, MaLbTc is the dimensional formula, and the exponents a, b, and c are referred to as the dimensions.

Law of Dimensional Homogeneity

• In any correct equation expressing the relationship between physical quantities, all terms must have the exact dimensions on both sides. Words denoted by a plus sign or a minus sign must have the exact dimensions.
• When a physical quantity Q has dimensions a, b, and c in length (L), mass (M), and time (T), respectively, and n1 is its numerical value in a system with fundamental units L1, M1, and T1, and n2 is its numerical value in another system with fundamental units L2, M2, and T2, respectively, then, n1 [L1,a M1b T1 c ]= n2 [L2a M2b T2 c].

Conclusion 

Fundamental quantities like length, mass, or time can be used to represent physical quantities by a mathematical formula called the dimensional formula. They represent the way in which fundamental quantities build any physical quantity when raised to different powers. These powers of fundamental physical quantities are called dimensions.

Energy, simply put, is the ability to do tasks. Energy is utilised by people in daily activities like walking,  riding a bicycle,  fueling engines, cooking food, lifting objects, illuminating rooms, air conditioning the room, launching satellites, etc.

By writing force and distance in their dimensional formulae, the dimensional formula of energy thus obtained is, [M1L2T-2].