Question:
( i ) The time period of a body having simple harmonic motion depends on the mass m of the body and the force constant k:

T =2π √m/k

A simple pendulum exhibits simple harmonic motion. Then why does the time period of a pendulum not depend upon its mass?

( ii ) For small angle oscillations, a simple pendulum exhibits simple harmonic motion ( more or less). For larger angles of oscillation, detailed analysis show that T is greater than 2π√ l/g. Explain.

( iii ) A boy with a wristwatch on his hand jumps from a helicopter. Will the wrist watch give the correct time during free fall?

( iv ) Find the frequency of oscillation of a simple pendulum that is free falling from a tall bridge.

Answer:

( i ) The time period of a simple pendulum, T =2π √m/k For a simple pendulum, k is expressed in terms of mass m, as:

k ∝ m

m/k = constant

Thus, the time period T, of a simple pendulum is independent of its mass.

( ii ) In the case of a simple pendulum, the restoring force acting on the bob of the pendulum is:

F = –mg sinθ

Where, F = Restoring force

m = Mass of the bob

g = Acceleration due to gravity

θ = Angle of displacement

For small θ, sin θ ∼ θ

For large θ, sin θ is greater than θ. This decreases the effective value of g.

Thus, the time period increases as: T = 2π√ l/g'

( iii ) As the working of a wrist watch does not depend upon the acceleration due to gravity, the time shown by it will be correct during free fall.

( iv ) As acceleration due to gravity is zero during free fall, the frequency of oscillation will also be zero.

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