If the ratio of the diameters of two spheres | vedclass

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(B) The heat capacity $C$ of an object is given by $C = mc$,where $m$ is the mass and $c$ is the specific heat capacity.Since both spheres are made of

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$1:8$

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(B) The heat capacity $C$ of an object is given by $C = mc$,where $m$ is the mass and $c$ is the specific heat capacity.Since both spheres are made of the same material,their specific heat capacity $c$ is the same.Mass $m$ can be expressed as $m = V \rho$,where $V$ is the volume and $\rho$ is the density.For a sphere,$V = \frac{4}{3} \pi r^3$.Therefore,the ratio of heat capacities is $\frac{C_1}{C_2} = \frac{m_1 c}{m_2 c} = \frac{m_1}{m_2} = \frac{V_1 \rho}{V_2 \rho} = \frac{V_1}{V_2}$.Substituting the volume formula,$\frac{C_1}{C_2} = \frac{\frac{4}{3} \pi r_1^3}{\frac{4}{3} \pi r_2^3} = \left( \frac{r_1}{r_2} \right)^3$.Given the ratio of diameters is $1:2$,the ratio of radii is also $r_1:r_2 = 1:2$.Thus,$\frac{C_1}{C_2} = \left( \frac{1}{2} \right)^3 = \frac{1}{8}$.The ratio of their heat capacities is $1:8$.

If the ratio of the diameters of two spheres made of the same material is $1:2$,what is the ratio of their heat capacities?

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