A pendulum clock loses 12 s a day if the temperature is $40^{\circ} \mathrm{C}$ and gains 4 s a day if the temperature is $20^{\circ} \mathrm{C}$. The temperature at which the clock will show correct time, and the co-efficient of linear expansion ($\alpha$) of the metal of the pendulum shaft are respectively :

(D) $25^{\circ} \mathrm{C} ; \alpha=1.85 \times 10^{-5} /{ }^{\circ} \mathrm{C}$

(A) $60^{\circ} \mathrm{C} ; \alpha=1.85 \times 10^{-4} /{ }^{\circ} \mathrm{C}$

(B) $30^{\circ} \mathrm{C} ; \alpha=1.85 \times 10^{-3} /{ }^{\circ} \mathrm{C}$

(C) $55^{\circ} \mathrm{C} ; \alpha=1.85 \times 10^{-2} /{ }^{\circ} \mathrm{C}$

JEE Main · 2016 · 03 Apr 2016 · Offline · Heat & Thermodynamics

A pendulum clock loses 12 s a day if the temperature is

4 0^{\circ} C

and gains 4 s a day if the temperature is

2 0^{\circ} C

. The temperature at which the clock will show correct time, and the co-efficient of linear expansion (

α

) of the metal of the pendulum shaft are respectively :

(A) $6 0^{\circ} C; α = 1.85 \times 1 0^{- 4} /^{\circ} C$
(B) $3 0^{\circ} C; α = 1.85 \times 1 0^{- 3} /^{\circ} C$
(C) $5 5^{\circ} C; α = 1.85 \times 1 0^{- 2} /^{\circ} C$
(D) $2 5^{\circ} C; α = 1.85 \times 1 0^{- 5} /^{\circ} C$ ✓ Correct

Answer: (D)

Solution

A pendulum clock runs fast when cool and slow when warm, and the daily error is proportional to the temperature difference from the correct-time temperature

θ_{0}

. Losing 12 s at

4 0^{\circ}

C and gaining 4 s at

2 0^{\circ}

C means the error is zero somewhere between, in the ratio

\frac{12}{40 - θ _{0}} = \frac{4}{θ _{0} - 20}

; cross-multiplying,

3 (θ_{0} - 20) = 40 - θ_{0}

, so

θ_{0} = 2 5^{\circ}

C. The fractional daily error is

\frac{Δ N}{N _{0}} = \frac{1}{2} α Δ θ

with

N_{0} = 86400

s; using the 12 s loss over

Δ θ = 1 5^{\circ}

α = \frac{2 \times 12}{86400 \times 15} \approx 1.85 \times 1 0^{- 5}^{\circ} C^{- 1}

JEE Main · 2016 · 03 Apr 2016 · Offline · Single correct option

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JEE Main · 2016 · 03 Apr 2016 · Offline · Heat & Thermodynamics

Solution

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