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A clock which keeps correct time at 20^∘ C has a pendulum...

A clock which keeps correct time at $20^{\circ} \mathrm{C}$ has a pendulum rod made of brass. How many seconds will it gain or lose per day when temperature falls to $0^{\circ} \mathrm{C}$ ? $\left(\alpha=18 \times 10^{-6}{ }^{\circ} \mathrm{C}^{-1}\right)$ (a) $155.5 \mathrm{~s}$ (b) $15.55 \mathrm{~s}$ (c) $25.55 \mathrm{~s}$ (d) $18.55 \mathrm{~s}$

A clock which keeps correct time at $20^{\circ} \mathrm{C}$ has a pendulum rod made of brass. How many seconds will it gain or lose per day when temperature falls to $0^{\circ} \mathrm{C}$ ? $\left(\alpha=18 \times 10^{-6}{ }^{\circ} \mathrm{C}^{-1}\right)$
(a) $155.5 \mathrm{~s}$
(b) $15.55 \mathrm{~s}$
(c) $25.55 \mathrm{~s}$
(d) $18.55 \mathrm{~s}$

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Key Concepts

Small Change Approximation

For small variations in the pendulum length, the resulting fractional change in the period can be approximated by taking half of the fractional change in length. This approximation allows one to estimate the cumulative effect on time accuracy over a long period, such as determining how many seconds per day a clock will lose or gain.

Simple Pendulum Period

The period of a simple pendulum is given by the formula T = 2??(L/g), where L is the length of the pendulum and g is the acceleration due to gravity. This formula demonstrates that any change in the length of the pendulum, such as through thermal expansion or contraction, alters its oscillation period and thus its timekeeping accuracy.

Thermal Expansion

Thermal expansion refers to the tendency of materials to change in dimensions (expand or contract) with temperature changes. For a pendulum made of brass, a decrease in temperature leads to a contraction of the rod. The change in length is proportional to the original length, the coefficient of linear expansion, and the temperature change, thereby affecting the pendulum’s period.

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