(C) The time period of a simple pendulum is given by $T = 2\pi \sqrt{\frac{L}{g}}$.Taking the natural logarithm on both sides: $\ln T = \ln(2\pi) + \f

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(C) The time period of a simple pendulum is given by $T = 2\pi \sqrt{\frac{L}{g}}$.Taking the natural logarithm on both sides: $\ln T = \ln(2\pi) + \frac{1}{2} \ln L - \frac{1}{2} \ln g$.Differentiating with respect to temperature $\theta$: $\frac{1}{T} \frac{dT}{d\theta} = \frac{1}{2L} \frac{dL}{d\theta}$.Since $\frac{dL}{d\theta} = L\alpha$,we have $\frac{1}{T} \frac{dT}{d\theta} = \frac{1}{2L} (L\alpha) = \frac{\alpha}{2}$.Therefore,the slope $S = \frac{dT}{d\theta} = \frac{T\alpha}{2}$.Given $T = 2 \ s$,we get $S = \frac{2\alpha}{2} = \alpha$.

Class 11 Physics 10-1.Thermometry, Thermal Expansion and Calorimetry Thermal Expansion for Solid

Difficult

$A$ simple pendulum made of a bob of mass $m$ and a metallic wire of negligible mass has a time period $2 \ s$ at $T = 0 \ ^oC$. 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 the metal is $\alpha$,then the value of $S$ is:

JEE MAIN 2016 All JEE MAIN

A
$\frac{\alpha}{2}$
B
$2\alpha$
C
$\alpha$
D
$\frac{1}{\alpha}$

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10-1.Thermometry, Thermal Expansion and Calorimetry

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Thermal Expansion for Solid

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On what does the value of the coefficient of linear expansion depend?

Medium

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$A$ metal tape is calibrated at $25^{\circ} C$. On a cold day when the temperature is $-15^{\circ} C$,the percentage error in the measurement of length is (Coefficient of linear expansion of metal $= 1 \times 10^{-5} {}^{\circ} C^{-1}$) (in $\%$)

MediumAP EAMCET 2022

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The temperature of a metal strip having coefficient of linear expansion $\alpha$ is increased from $T_1$ to $T_2$ resulting in an increase of its length by $\Delta L_1$. The temperature is further increased from $T_2$ to $T_3$ such that the increase in its length is $\Delta L_2$. Given $T_3 + T_1 = 2T_2$ and $T_2 - T_1 = \Delta T$, the value of $\Delta L_2$ is . . . . . . .

DifficultJEE MAIN 2026

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If the length of a cylinder on heating increases by $2\%$,the area of its base will increase by ....... $\%$

Medium

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$A$ uniform metal rod of length $L$ and mass $M$ is rotating about an axis passing through one of the ends and perpendicular to the rod with angular speed $\omega$. If the temperature increases by $t^{\circ} C$, then the change in its angular velocity is proportional to which of the following? (Coefficient of linear expansion of rod $= \alpha$)

MediumTS EAMCET 2001

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On what does the value of the coefficient of linear expansion depend?

$A$ metal tape is calibrated at $25^{\circ} C$. On a cold day when the temperature is $-15^{\circ} C$,the percentage error in the measurement of length is (Coefficient of linear expansion of metal $= 1 \times 10^{-5} {}^{\circ} C^{-1}$) (in $\%$)

If the length of a cylinder on heating increases by $2\%$,the area of its base will increase by ....... $\%$

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