Mechanical equivalent of heat can be defined as the amount of work done to produce a unit quantity of heat. In other words whenever a mechanical work is completely transformed into heat, the amount of heat produced is directly proportional to the amount of work done.

W ∝ Q  —— Where W = Amount of Work and Q = units of heat.
W = J Q

In the above equation constant J is Known as Mechanical equivalent of heat or Joule’s equivalent.
From above equation we get,

Mechanical equivalent of heat (J) = Amount of Work (W) / units of Heat (Q).

Dimensional Formula of Work = M¹L²T^-2
Dimensional Formula of Heat = M¹L²T^-2

substituting these values we get,

Dimensional Formula of Mechanical equivalent of heat (J) = M⁰L⁰T^-0 . we can also say Mechanical equivalent of heat (J) is dimensionless quantity.
SI unit of Mechanical equivalent of heat (J) is Joules/calorie (J calorie^-1 ).

Internal energy (U) is defined as the energy which is related to atomic, sub atomic and molecular energy of the system. Internal energy comprises of the kinetic and potential energy related with transitional , rotational and vibrational motion of the molecules, it also includes the energy of electromagnetic interactions of molecules , atomic and sub atomic constituents of molecules.

Mathematically,
Internal energy (U) = Energy.
Dimensional formula of Energy = M¹L²T^-2

So Dimensional Formula of Internal Energy (U) = M¹L²T^-2
SI unit of Internal Energy (U) is Joule (J)

Specific heat capacity is defined as the quantity of heat required to increase the temperature of a unit mass of fluid by one temperature unit.

Mathematically,
Specific heat capacity = Q /m Θ

Where Q= Heat, m = mass  and Θ = Temperature

Dimensional formula of Heat(Q) = M¹L²T^-2
Dimensional formula of mass = M¹L⁰T⁰
Dimensional formula of Temperature(Θ) = M⁰L⁰T⁰K¹

Putting these values in above equation we get,
Dimensional formula of Specific Heat Capacity = M⁰L²T^-2K^-1
SI unit of Specific Heat Capacity is Joule kg-1 K^-1

Consider a body of mass (M) and radius (R) rolling down (without slipping) a smooth inclined plane making an angle of inclination (Θ). When a body rolls down its Potential Energy ( resting at the top of inclined plane) is converted into the Kinetic energy of translation as well as rotation. As we know that body is rolling down a smooth inclined plane this means there will be no loss of energy due to friction. So the loss in Potential Energy is same as gain in kinetic energy.

Loss in potential Energy = Gain in kinetic energy

Mgh = ½ (M v²) + ½ (I ω ²)

From the above fig we know that h = ℓ sin Θ, substituting this in the above equation we get,
Mg ℓ sin Θ = ½ (M v²) + ½ (I ω ²)
Mg ℓ sin Θ = ½ (M v²) + ½ (M K² .(v² / R ²) )
Where K = radius of gyration, ω = v/R and I = M K²

Mv² / 2 (1 + (K² / R²)) = Mg ℓ sin Θ

v² = 2 g ℓ sin Θ / (1 + (K² / R²)) but v² = 2aℓ

So Formula for acceleration of a body rolling down a smooth inclined plane,

a = g sin Θ / (1 + (K² / R²))

Law of conservation of angular momentum is stated as “If the total external torque acting on a body is zero, the total angular momentum of that body remains constant or conserved”. This means that the angular momentum of that body remains same in absence of external torque.

According to the Law of conservation of angular momentum

         Angular momentum = I₁ω_{1 =} l₂ω₂= constant

I₁ and ω₁ are the initial moment of inertia and angular velocity of a rotating body

I₂ and ω₂ are new moment of inertia and angular velocity of the body