The temperature of a pendulum, the time period of which is...

The temperature of a pendulum, the time period of which is $t$, is raised by $\Delta T$. The change in its time period 15:
(a) $\frac{\overline{1}}{2} \alpha f \Delta T$
(b) $2 \alpha t \Delta T$
(c) $\frac{1}{2} \alpha \Delta T$
(d) $2 \alpha \Delta T$

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Key Concepts

Coefficient of Linear Expansion

The coefficient of linear expansion (?) quantifies how much a material's length changes per unit temperature change. In the context of a pendulum, an increase in temperature causes the pendulum’s rod or string to expand, thereby increasing its length. This slight increase in length, governed by thermal expansion, directly influences the pendulum’s time period.

Small Change Approximations (Binomial Approximation)

For small changes, such as a slight increase in temperature, the binomial approximation allows for a simplified evaluation of expressions of the form (1 + x)^n. When applied to the pendulum’s period formula, this approximation helps to estimate the relative change in the period due to the small fractional increase in length, simplifying the relationship to terms linear in ?T.

Simple Pendulum Period

The time period of a simple pendulum is determined by its length and the acceleration due to gravity, encapsulated in the formula T = 2??(L/g). This relationship shows that any change in the length of the pendulum will affect its period, making it a crucial concept when analyzing variations in the pendulum’s behavior under different conditions.

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