Question

What are the characteristics of simple harmonic motion. Show that the motion of simple pendulum is simple harmonic
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Answer 1

Answer:

The motion of a simple pendulum is not a simple harmonic motion, because the differential equation which governs that motion is nonlinear:

d2θdt2+gℓsinθ=0

What happens is that, for very small angular displacements ( θ≈0 ), we can approximate sinθ≈θ and so the approximate (linearized) equation of the pendulum motion becomes:

d2θdt2+gℓθ=0

This is the second order linear differential equation of harmonic motion, whose solution is a sinusoidal function. If θ(0)=θ0 and θ′(0)=0 , the solution is

θ(t)=θ0cos(gℓ−−√t)

with ω0=gℓ−−√ being the oscillation frequency.

So, to summarize, the simple harmonic motion x(t) is the one which complies with the differential equation of harmonic motion:

d2xdt2+ω20x=0

PLS MARK IT AS BRAINLIST

Answer 2

Explanation:

These are Basic Conditions and characteristics for a body to exhibit SHM : 1- A restoring force must act on the body. 2- Body must have acceleration in a direction opposite to the displacement and the acceleration must be directly proportional to displacement. 3- The system must have inertia (mass).