Q130 / 160 · TS EAMCET 2007 · Physics

Two solid spheres ( $A$ and $B$ ) are made of metals of different densities $\rho_A$ and $\rho_B$ respectively. If their masses are equal, the ratio of their moments of inertia $\left(I_B / I_A\right)$ about their respective diameter is

  • Correct answer
    A
    $\left(\frac{\rho_B}{\rho_A}\right)^{2 / 3}$
  • B
    $\left(\frac{\rho_A}{\rho_B}\right)^{2 / 3}$
  • C
    $\frac{\rho_A}{\rho_B}$
  • D
    $\frac{\rho_B}{\rho_A}$

Solution

As two solid spheres are equal in masses, so $m_A=m_B$ $\begin{aligned} \Rightarrow & \frac{4}{3} \pi R_A^3 \rho_A & =\frac{4}{3} \pi R_B^3 \rho_B \\ \Rightarrow & \frac{R_A}{R_B} & =\left(\frac{\rho_B}{\rho_A}\right)^{1 / 3}\end{aligned}$ The moment of inertia of sphere about diameter $I=\frac{2}{5} m R^2$ $\begin{array}{ll}\Rightarrow \quad & \frac{I_A}{I_B}=\left(\frac{R_A}{R_B}\right)^2 \quad\left(\text { as } m_A=m_B\right) \\ \Rightarrow \quad & \frac{I_A}{I_B}=\left(\frac{\rho_B}{\rho_A}\right)^{2 / 3}\end{array}$

Hence, the correct answer is option: 'A'.

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