(C) The volume of a sphere of radius $R$ is given by $V = \frac{4}{3}\pi R^3$.Taking the differential of both sides with respect to $T$, we get $\frac

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(C) The volume of a sphere of radius $R$ is given by $V = \frac{4}{3}\pi R^3$.Taking the differential of both sides with respect to $T$, we get $\frac{dV}{dT} = \frac{4}{3}\pi (3R^2) \frac{dR}{dT} = 4\pi R^2 \frac{dR}{dT}$.For linear expansion, the change in radius is given by $\Delta R = R \alpha \Delta T$, which implies $\frac{dR}{dT} = R \alpha$.Substituting this into the volume differential equation: $\Delta V \approx 4\pi R^2 (R \alpha \Delta T)$.Therefore, $\Delta V = 4\pi R^3 \alpha \Delta T$.

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

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The temperature of a metallic sphere of radius $R$ is increased by a small amount $\Delta T$. If the linear coefficient of thermal expansion of the metal is $\alpha$, the approximate increase in the volume of the sphere is :

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A
$2\pi R^3 \alpha \Delta T$
B
$3\pi R^3 \alpha \Delta T$
C
$4\pi R^3 \alpha \Delta T$
D
$6\pi R^3 \alpha \Delta T$

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The temperature of a metallic sphere of radius $R$ is increased by a small amount $\Delta T$. If the linear coefficient of thermal expansion of the metal is $\alpha$, the approximate increase in the volume of the sphere is :

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Two rods of same area of cross-section have lengths $L$ and $2 \,L$ and coefficients of linear expansion $2 \alpha$ and $\alpha$ respectively. If they are welded to form a composite rod of length $3 \,L$, then the coefficient of linear expansion of the composite rod is

Two rods of different metals have coefficients of linear expansion $\alpha_1$ and $\alpha_2$ respectively. Their respective lengths are $L_1$ and $L_2$. At all temperatures,$(L_2 - L_1)$ remains the same. The correct relation is:

$A$ steel meter scale is to be ruled so that millimeter intervals are accurate within about $5 \times 10^{-5} \,m$ at a certain temperature. The maximum temperature variation allowable during the ruling is (Coefficient of linear expansion of steel $= 10 \times 10^{-6} \,K^{-1}$) (in $^{\circ} C$)

Find the ratio of the length of a steel rod and a copper rod if the steel rod is $4 \ cm$ longer than the copper rod at any temperature. $[$The coefficients of linear expansion for steel and copper are $1.1 \times 10^{-5} /^{\circ} C$ and $1.7 \times 10^{-5} /^{\circ} C$ respectively$]$

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