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# 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. - CBSE (Science) Class 11 - Physics

ConceptSome Systems Executing Simple Harmonic Motion

#### Question

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 is greater than 2pisqrt(1/g)  Think of a qualitative argument to appreciate this result.

#### Solution 1

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)

#### Solution 2

In the case of a simple pendulum, the restoring force acting on the bob of the pendulum is given as:

F = –mg sinθ

Where,

= 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

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Solution 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. Concept: Some Systems Executing Simple Harmonic Motion.
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