An iron sphere having diameter $D$ and mass $M$ is immersed in hot water so that the temperature of the…
An iron sphere having diameter $D$ and mass $M$ is immersed in hot water so that the temperature of the sphere increases by $\delta T$. If $\alpha$ is the coefficient of linear expansion of the iron then the change in the surface area of the sphere is
Given, diameter of sphere $=D$
Initial surface area, $A=4 \pi R^2$
$=4 \pi\left(\frac{D}{2}\right)^2$
$=\pi D^2$ ...(i)
Surface area after heating by temperature $\delta T$,
$A^{\prime}=4 \pi\left(\frac{D^{\prime}}{2}\right)^2=\pi\left(D^{\prime}\right)^2$ ...(ii)
where $D^{\prime}$ is the new diameter.
From the equation of linear expansion, we have
$D^{\prime}=D(1+\alpha \delta T)$ ...(iii)
putting the value of $D^{\prime}$ from eq. (iii) in eq. (ii) we get
$A^{\prime}=\pi D^2(1+\alpha \delta T)^2$
$=\pi D^2\left(1+\alpha^2 \delta T^2+2 \alpha \delta T\right)$ ...(iv)
$\begin{aligned} & =\pi D^2\left(1+\alpha^2 \delta T^2+2 \alpha \delta T\right)-\pi D^2 \\ & =\pi D^2\left[1+\alpha^2 \delta T^2+2 \alpha \delta T-1\right] \\ & =\pi D^2\left[\alpha^2 \delta T^2+2 \alpha \delta T\right]\end{aligned}$
Change in surface area $=A^{\prime}-A \doteq \pi D^{\prime 2}-\pi D^2$
$=\pi D^2 \alpha \delta T(\alpha . \delta T+2)$