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{9}{4}

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\frac{3}{2}

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\frac{5}{3}

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\frac{27}{8}

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Solution

The correct option is D $\frac{27}{8}$
We know that,
$Δ Q (change in heat; heat needed) = m a s s \times s p e c i f i c h e a t c a p a c i t y \times c h a n g e i n t e m p e r a t u r e$

Specific heat capacity will be same for both. Let it be $S$ .

$Δ T = 1 k$ (given : change in temperature is 1k)

$d e n s i t y = m a s s / v o l u m e$

as both are made up of copper they have the same density. let it be $D$ .

the volume of the first sphere $= \frac{4}{3} π r_{1}^{3}$

the volume of the second sphere $= \frac{4}{3} π r_{2}^{3}$
taking the ratio of both delta Q
$\frac{Δ Q_{1}}{Δ Q_{2}} = \frac{D \times V_{1} \times 1 k}{D \times V_{2} \times 1 k}$

$\frac{Δ Q_{1}}{Δ Q_{2}} = \frac{V_{1}}{V_{2}} = \frac{(4 / 3 π r_{1}^{3})}{(4 / 3 π r_{2}^{3})}$

$\frac{Δ Q_{1}}{Δ Q_{2}} = {(\frac{r_{1}}{r_{2}})}^{3}$

$\frac{Δ Q_{1}}{Δ Q_{2}} = {(\frac{3}{2})}^{3}$

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

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