Rotational Kinetic Energy can be defined as the kinetic energy associated with rotational motion of an object.  kinetic energy = 1/2 (mv²)  we all know that every particle in an object has same angular speed ω, and tangential speed depends on the distance r from the axis of rotation. Total kinetic Energy of rotating object is the sum of kinetic energy of all individual particles of an object.

so Rotational kinetic energy (K_R) = ∑ 1/2 (mv²) = 1/2 ∑ (m r² ω²) = 1/2 (∑ m r²) ω².

We can also write it as K_R = 1/2 Iω².——- (Moment of Inertia (I)= ∑ m r² ).

Rotational kinetic energy (K_R) = 1/2 Iω²

Mathematically.

Rotational kinetic energy (K_R) = 1/2 (Moment of Inertia x (Angular velocity)²)

Now we know,

Dimensional Formula of Moment of Inertia= M¹L²T⁰
Dimensional Formula of Angular Velocity = M⁰L⁰T^-1

Putting these values in above equation we get

So Dimensional Formula of Rotational kinetic energy = M¹L²T^-2
SI unit of Rotational kinetic energy is Joule (J).

Surface Potential  is defined as work done per unit mass.

Mathematically,
Surface Potential = Work / Mass

Dimensional Formula of Work= M¹L²T^-2
Dimensional Formula of Mass= M¹L⁰T⁰

Putting this value in the above equation we get,

Dimensional Formula of Surface Potential= M⁰L²T^-2
SI unit of Surface Potential is m²s^-2.

Refractive Index is defined as the relative speed at which light moves through a material with respect to its speed  in vacuum.

Mathematically,
Refractive Index (n) = speed of light in a vacuum (c) /speed of light in a medium (v)

As both the values belong to same  quantity (speed), units gets nullified. So Refractive Index is dimensionless quantity.

Dimensional Formula of Refractive Index= M⁰L⁰T⁰.
Refractive Index is dimensionless quantity.

Specific volume is defined as volume per unit mass of fluid. Specific volume is reciprocal of mass density.

Mathematically,
Specific volume = volume of fluid / mass of fluid.

Dimensional formula of volume = M⁰L³T⁰
Dimensional formula of mass = M¹L⁰T⁰

Putting this value in the above equation we get,

Dimensional Formula of Specific volume= M^-1L³T⁰.
SI unit of Specific volume  is m³/kg .

Kinetic energy can be defined as the energy possessed by the body by virtue of its motion.

Mathematically,
Kinetic energy (K.E) = 1/2 [Mass (m) X Velocity (v)²]

Dimensional formula of  Mass = M¹L⁰T⁰
Dimensional formula of velocity= M⁰L¹T^-1

Putting the values in the above equation we get,

Dimensional formula of Kinetic energy (K.E) = M¹L⁰T⁰ X ( M⁰L¹T^-1)²

Dimensional formula of Kinetic energy (K.E) = M¹L⁰T⁰ X M⁰L²T^-2

Dimensional Formula of Kinetic energy= M¹L²T^-2.
SI unit of Kinetic energy is Joule (J).

Note : In the above equation ½  is a constant and  constants do not affect the dimensional formula of any quantity.