Question

State the principle of homogeneity and derive the expression for (T) time period of a
simple pendulum in terms of length of pendulum and acceleration due to gravity by
method of dimensions.​

Answer 1

Answer:

Principle of Homogeneity states that dimensions of each of the terms of a dimensional equation on both sides should be the same. This principle is helpful because it helps us convert the units from one form to another.

Time Period of Simple Pendulum Derivation

Using the equation of motion, T – mg cosθ = mv^2L

The torque tending to bring the mass to its equilibrium position,

τ = mgL × sinθ = mgsinθ × L = I × α

For small angles of oscillations sin ≈ θ,

Therefore, Iα = -mgLθ

α = -(mgLθ)/I

– ω0^2 θ = -(mgLθ)/I

ω0^2 = (mgL)/I

ωo = √(mgL/I)

Using I = ML2, [where I denote the moment of inertia of bob]

we get, ω0 = √(g/L)

Therefore, the time period of a simple pendulum is given by,

T = 2π/ω0 = 2π × √(L/g)

Energy of Simple Pendulum

From the above figure, the potential energy (PE) of the bob at B with respect to A is,

m×g×h = mg × (L – L cosθ)h = L – L cosθ

= mgL × (1 – cosθ)cosθ = 1 – 2 sin2θ/2

= mgL [1 – (1 – 2 sin2 θ/2)]

For small angles sin (θ/2) ≈ θ/2

= mg × L2(θ/2)2= 1/2 (mgLθ2)

Potential Energy of a simple pendulum is 1/2 (mgLθ2).

Kinetic Energy of the Bob:

1/2 (Fω2) = 1/2 (mL2) (dθ/dt)2 [ω = dθ/dt]

=1/2 (ML2) [θ20 ω02 cos2 (ω0t)]

=1/2 (ML2) ω02 [θ02 (1 – sin2 ω0t}]

=1/2 (ML2) × (g/L) × [θ02 – θ2]

∴ Kinetic Energy of a simple pendulum is [1/2 × mgLθ02] – 1/2 mgLθ2θ0 – Amplitude

Mechanical Energy of the Bob:

E = KE + PE= 1/2 mg Lθ02 =constant

The energy of the simple pendulum is conserved and is equal in magnitude to the potential energy at the maximum amplitude.