Question

Two discs of same mass and different radii are made of different materials such that their thicknesses are 1 cm and 0.5 cm respectively. The densities of materials are in the ratio 3:5. The moment of inertia of these discs respectively about their diameters will be in the ratio of \(\frac{x}{6}\). The value of x is ____.

Hint:

The moment of inertia of a disc depends on its mass and radius, and changes in the density and thickness affect both.

Answer 1

Correct Answer: 5

The moment of inertia \( I \) of a disc about its diameter is given by: \[ I = \frac{1}{2} m r^2 \] where \( m \) is the mass of the disc and \( r \) is its radius. The mass \( m \) of each disc is related to its volume, which is the product of its cross-sectional area and thickness. Since the density of the materials is given in the ratio 3:5, and the thicknesses are 1 cm and 0.5 cm, we can write the mass of each disc as: \[ m_1 \propto \rho_1 r_1^2 \times 1 \quad \text{and} \quad m_2 \propto \rho_2 r_2^2 \times 0.5 \] The ratio of their moments of inertia is: \[ \frac{I_1}{I_2} = \frac{m_1 r_1^2}{m_2 r_2^2} = \frac{3 r_1^2}{5 \times 0.5 r_2^2} = \frac{6 r_1^2}{5 r_2^2} \] Thus, \( \frac{x}{6} = \frac{6 r_1^2}{5 r_2^2} \), and the value of \( x \) is 5.