Question

What is simple pendulum? Show that the motion of simple pendulum is shm and hence deduce the expression for the time period of pendulum?

Answer 1

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ȯ The motion of Simple Pendulum as Simple Harmonic Motion. When we pull a simple pendulum from its equilibrium position and then release it, it swings in a vertical plane under the influence of gravity. It begins to oscillate about its mean position. Therefore, the motion is periodic and oscillatory.

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Answer 2

Consider a simple pendulum with mass m and length l.

Let y and θ be its linear and angular displacements respectively.

θ = y/l

At extreme position,

Tangential acceleration is-

a = gsinθ

Torque about point of suspension is-

τ = -Fl

τ = -l(mgsinθ)

M.I. is related by-

τ = Iα

-l(mgsinθ) = Iα

When, θ is small, sinθ = θ,

α = -lmgθ/I

Comparing this with std eqn of SHM ω = √(mgl/I),

We can say that pendulum shows SHM.

● Period of oscillation-

T = 2π/ω

T = 2π/√(mgl/I)

Here, I = ml^2

T = 2π/√(mgl/ml^2)

T = 2π√(l/g)