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Question
Answer the following questions:
The motion of a simple pendulum is approximately simple harmonic for small angle oscillations. For larger angles of oscillation, a more involved analysis shows that T is greater than `2pisqrt(1/g)` Think of a qualitative argument to appreciate this result.
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Solution 1
In the case of a simple pendulum, the restoring force acting on the bob of the pendulum is given as:
F = –mg sinθ
Where,
F = Restoring force
m = Mass of the bob
g = Acceleration due to gravity
θ = Angle of displacement
For small θ, sinθ = θ
For large θ, sinθ is greater than θ.
This decreases the effective value of g.
Hence, the time period increases as:
`T = 2pi sqrt(1/g)`
Where, l is the length of the simple pendulum
Solution 2
The restoring force for the bob of the pendulum is given by
`F = -mg sintheta`
if `theta` is small thensin `theta = theta = y/l` `:. F = -(mg)/l y`
i.e the motion is simple harmonic and time period is` T = 2pi sqrt(1/g)`
Clearly, the above formula is obtained only if we apply the approximation `sin theta ~~ theta`
For large angles this approximation is not valid and T is greater than `2pi sqrt(1/g)`
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