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 momen

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 x 6 . The value of x is ______.

TestHub · Accepted Answer

Mass of the disc in terms of density can be written as, $m = ρ (π R^{2} t)$ .

$\Rightarrow R^{2} = \frac{m}{ρ π t}$

Now moment of inertia of the disc about the diameter is, $\frac{m R^{2}}{4}$ .

Therefore,

$I = \frac{m R^{2}}{4} = \frac{m^{2}}{4 ρ π t} \Rightarrow I \propto \frac{1}{ρ t}$

Required ratio, $\frac{I_{1}}{I_{2}} = \frac{ρ_{2} t_{2}}{ρ_{1} t_{1}} = \frac{5}{3} \times \frac{0.5}{1} = \frac{5}{6}$

Hence, $x = 5$ .

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