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Question:

A simple pendulum made of a bob of mass m and a metallic wire of negligible mass has time period 2s at T=0. If the temperature of the wire is increased and the corresponding change in its time period is plotted against its temperature, the resulting graph is a line of slope S. If the coefficient of linear expansion of metal is α, then the value of S is?

1α

α

α²

2α

Solution:

The correct option is C α
Time period of a pendulum is given by
T=2π√(l/g)
where l: length and g: acceleration due to gravity
T²=4π²l/g
At temperature θ, length=l₀(1+α(θ−θ₀))
∴T²=4π²l₀(1+α(θ−θ₀))/g
Differentiating with respect to temperature,2T(dT/dθ)=4π²l₀α/g
dT/dθ=αT²/2T=αT/2=α