Question

The diameter of a solid metallic sphere is equal to the inner diameter of a hollow metallic sphere. If inner diameter of hollow sphere is two thirds of its outer diameter. Both the spheres are made of same metal. If an equal amount of heat is supplied the both the spheres. Find the ratio of increase in temperature of the solid sphere to that of the hollow one?

Answer 1

Given: The diameter of a solid metallic sphere is equal to the inner diameter of a hollow metallic sphere. If inner diameter of hollow sphere is two thirds of its outer diameter. Both the spheres are made of same metal. If an equal amount of heat is supplied the both the spheres.

To find the ratio of increase in temperature of the solid sphere to that of the hollow one

Solution:

We know that,

Heat transfer = $Q=mc\Delta T$

Where, m=mass, c=specific heat capacity, $\Delta T$=temperature.

$Q=\rho Vc\Delta T$ (where $\rho$=density, V=volume, as $m=\rho V$)

Now if there are two bodies to which equal amount of heat supplied and are of same metal.

$Q_1={\rho }_1{V}_1{c}_1\Delta T_1$

$Q_2={\rho }_2{V}_2{c}_2\Delta T_2$

If same metal $({\rho }_1={\rho }_2,c_1=c_2)$

Ratio of increase in temperature

$\frac{\Delta T_1}{\Delta T_2}=\frac{V_2}{V_1}.....\left(i\right)$

Now in question,

Let the diameter of solid sphere be = $d$

Inner diameter of hollow sphere = $d$

Outer diameter of hollow sphere =$\frac{3}{2}d$

temperature rise in solid sphere = $\Delta T_1$

temperature rise in hollow sphere=$\Delta T_2$

According to equation (i)

$\frac{\Delta T_1}{\Delta T_2}=\frac{V_2}{V_1}$

Volume of hollow sphere $=\frac{4}{3}\pi (r_2^3-r_1^3)$

$r_2$=outer radius

$r_1$= inner radius

Volume of solid sphere $=\frac{4}{3}\pi r^3$

$\frac{{\Delta T}_1}{{\Delta T}_2}=\frac{\frac{4\pi }{3}\left[(\frac{27d^3}{64}-\frac{d^3}{8})\right]}{\frac{4\pi }{3}\left[\frac{d^3}{8}\right]}=\frac{\frac{27-8}{64}}{\frac{1}{8}}⟹\frac{{\Delta T}_1}{{\Delta T}_2}=\frac{19}{64}\times 8⟹\frac{{\Delta T}_1}{{\Delta T}_2}=\frac{19}{8}$

is the the ratio of increase in temperature of the solid sphere to that of the hollow one