The quantities of heat required to raise the temperature of two solid copper spheres of radii $r_{1}$ and $r_{2}$ ( $r_{1} = 1.5 r_{2})$ through $1 K$ are in the ratio :

\frac{3}{2}

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\frac{5}{3}

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\frac{27}{8}

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\frac{9}{4}

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Solution

The correct option is C $\frac{27}{8}$
Given:

$r_{1} = 1.5 r_{2}; Δ T = 1 K$

Let $Q_{1}$ and $Q_{2}$ be the heat supplied to the sphere $1$ and $2$ respectively.

Using formula $Q = m s Δ T$

$\Rightarrow \frac{Q_{1}}{Q_{2}} = \frac{m_{1}}{m_{2}}$

( $∵$ both spheres are made up of same material and $Δ T$ is also same)

$∴ \frac{Q_{1}}{Q_{2}} = \frac{ρ V_{1}}{ρ V_{2}} (∵ m = ρ V$ )

$\Rightarrow \frac{Q_{1}}{Q_{2}} = \frac{r_{1}^{3}}{r_{2}^{3}} (∵ V \propto r^{3})$

$\Rightarrow \frac{Q_{1}}{Q_{2}} = \frac{{1.5}^{3}}{1^{3}} = \frac{3.375}{1}$

$\Rightarrow \frac{Q_{1}}{Q_{2}} = \frac{27}{8}$

Hence, $(c)$ is the correct answer.

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