Question

Two spheres made of the same substance have diameters in the ratio 1:2. Their thermal capacities are in the ratio of
A) 1:2
B) 1:8
C) 1:4
D) 2:1

Answer 1

In order to solve the question, we will first use the relation of thermal capacity then we will substitute the value of mass with product of volume and density so as to find the direct relation with radius so as to use the ratio given the question to find the ratio of thermal capacity

Formula used:
${\text{thermal capacity = mass }} \times {\text{ specific heat}}$
Volume = $\dfrac{4}{3}\pi {({r_1})^3}$
Mass of sphere one = volume $\times$density

Complete step by step answer:
In the question we are given that there are two spheres which are made up same substances and we have to find the ratio of the thermal capacities of both the spheres
Ratio of the diameters as given in question = 1:2
As both the spheres are made of same material then they will have same specific heat
To find the thermal capacity we will use the relation
${\text{thermal capacity = mass }} \times {\text{ specific heat}}$
Thermal capacity of sphere one = ${C_1} = {m_1}S$
Mass of sphere one = volume $\times$density
Volume = $\dfrac{4}{3}\pi {({r_1})^3}$
Substituting the value of mass in thermal capacity
${C_1} = \dfrac{4}{3}\pi {({r_1})^3}\rho S$
Thermal capacity of sphere one = ${C_2} = {m_2}S$
Mass of sphere two = volume $\times$density
Volume = $\dfrac{4}{3}\pi {({r_2})^3}$
Substituting the value of mass in thermal capacity
${C_2} = \dfrac{4}{3}\pi {({r_2})^3}\rho S$
Thermal capacities are in ratio
$\dfrac{{{C_1}}}{{{C_2}}} = \dfrac{{\dfrac{4}{3}\pi {{({r_1})}^3}\rho S}}{{\dfrac{4}{3}\pi {{({r_2})}^3}\rho S}}$
All the constants will be cancelled and we will get
$\dfrac{{{C_1}}}{{{C_2}}} = \dfrac{{{{({r_1})}^3}}}{{{{({r_2})}^3}}}$
Now we will be substituting the ratio of radius that 1:2 in the above equation
$\dfrac{{{C_1}}}{{{C_2}}} = \dfrac{{{{(1)}^3}}}{{{{(2)}^3}}}$
Opening the cube
$\dfrac{{{C_1}}}{{{C_2}}} = \dfrac{1}{8}$
Hence, the correct option is B) 1:8

Note: Many of the people will may confuse how the ratio of diameter is directly use in the answer this because as radius is half of diameter and we have been given in the ratio so both half will cut down so it doesn’t matter ratio of both radius and diameter are same in the ratio.