Dimensional Formula of Rotational Kinetic Energy

The energy gained due to the motion of the mass is known as kinetic energy. It is present in any piece of mass that moves, be it vertically or horizontally. Kinetic energy has various types such as vibrational (energy produced by vibrational motion), rotational (energy produced by rotational motion), and translational (energy produced by translational motion, i.e. the energy due to motion from one location to another). The rotation of an object generates rotational energy, which is one of the forms of kinetic energy. The moment of inertia is noticed when the rotational energy is observed on the axis of the rotation of the object. Rotational energy is often known as angular kinetic energy.

Rotational Motion

A rotational motion is defined as the movement of a point in a circular path with an axis of rotation that cannot be altered. In rotational dynamics, the causes of rotational motion are taken into account along with its attributes, while in rotational kinetics, rotational motion is evaluated without addressing its causes.

Examples of Rotational Motion

1. In our daily lives, we see examples of rotational motion. Day and night cycles are created by the earth’s rotation on its axis.
2. The rotatory motion of helicopter blades is also rotatory motion.
3. A door that opens and closes by swivelling on its hinges.
4. A Ferris wheel in an amusement park with a spinning top.

Dimensional Formula

The dimensional formula of any bodily amount is defined as the expression that represents how and which of the bottom portions are protected in that amount. It is denoted through enclosing the symbols for base portions with suitable strength in rectangular brackets i.e. [ ].

An example is the dimension formula of mass which is given as (M). 

Example to Write Dimensions

Let’s take the formula of the area of the rectangle:

Area of the rectangle = length x breadth

= l x l ( where breadth is also showing the length of the side)

= [L1] X [L1]

= [L2]

Here, we can see the length to the power of 2 and we cannot find the dimension of mass and time. Hence, the dimension of the area of a rectangle is written as [M0 L2 T0].

Let’s take the formula of speed:

Speed = Distance / Time

The distance can be written in length [L]

Time can be written as [T]

The dimensional formula would be [ M0 L1 T-1]

Hence, we can conclude that the speed is dependent on only length and time, not mass.

Dimensional Equation

Physical quantity is equated to the dimensional formula, to get the dimensional equation. 

Example:

Velocity =  [ M0 L1 T-1]

Here, velocity is the physical quantity, which is equated to the dimensional formula.

Dimensional Formula of Rotational Kinetic Energy

A rolling piece of mass has kinetic energy in both translation and rotation. The product of rotational inertia and the square of the angular velocity magnitude results in rotational kinetic energy. 

The mathematical formula for the calculations can be written as:

KR = 1 / 2.(l).(ω)2 

Where:

KR: Rotational kinetic energy

l: a moment of inertia

ω: Angular velocity

The dimensional formula of the rotational kinetic energy can be written as:

[M1L2T-2]

Where:

M = Mass

L = Length

T = Time

Derivation for the dimensional formula of rotational kinetic energy:

Rotational Kinetic Energy = 1 / 2 . (Moment of Inertia). (Angular velocity)2

• I.e KR = 1 / 2.(l).(ω)2

• The moment of inertia (MOI) = (Mass).(Radius of Gyration)2

Therefore, the dimension of MOI can be written as

[M1L2T0]

• The angular velocity = (Δθ).(t-1)

Therefore , the dimensional formula of angular velocity can be written as

[M0L0T-1]

From the equations I, II, and III:

Rotational Kinetic Energy = 1 / 2 . (Moment of Inertia). (Angular velocity)2

Or 

KR = [M1L2T0].[M0L0T-1]2 = [M1L2T-2]

Hence, the dimensional formula of the rotational kinetic energy is

[M1L2T-2]

Conclusion

Rotational kinetic energy is one of the forms of kinetic energy, like vibrational or translational. It is the energy gained due to the circular movement of the piece of the mass followed by various factors, such as a moment of inertia and rotational velocity. There are various factors that influence the rotational kinetic speed, like the speed of the object, the mass of the object and the position of the object (near or far) from the rotational axis. The value of rotational kinetic energy cannot be negative and the energy gained during the process is always transformed into other forms and cannot be conserved. The dimensional formula of the rotational kinetic energy is [M1L2T-2].