Question

Two solid spheres are made of copper. The radius of second sphere is $3$ times that of the first. The quantities of heat required to raise temperature of each sphere by $2\text{K}$ are in the ratio of

• A. $\frac{1}{9}$
• B. $\frac{9}{4}$
• C. $\frac{1}{18}$
• D. $\frac{1}{27}$

Answer 1

The correct option is D $\frac{1}{27}$

Heat required to raise the temperature of a substance by $\Delta T$
$H=m\times S\times \Delta T$,
where $S$ is the specific heat capacity.

Here,
$S_1=S_2$
$\frac{H_1}{H_2}=\frac{m_1\Delta T_1}{m_2\Delta T_2}=\frac{\frac{4}{3}\pi R_1^3\rho \times \Delta T_1}{\frac{4}{3}\pi R_2^3\rho \times \Delta T_2}$
$={\left(\frac{R_1}{R_2}\right)}^3\times \left(\frac{\Delta T_1}{\Delta T_2}\right)$
$={\left(\frac{1}{3}\right)}^3\times \left(\frac{2}{2}\right)=\frac{1}{27}$