Question

Two uniform circular discs having the same mass and the same thickness, but different radii, are made from different materials. The disc with the smaller rotational inertia is:

A
the one made from the more dense material
B
the one made from the less dense material
C
the disc with the larger angular velocity
D
the disc with the larger torque

Solution

The correct option is A the one made from the more dense materialGiven :    $$M_1 = M_2$$                   $$t_1 = t_2$$Let the densities of the two discs be $$\rho_1$$ and $$\rho_2$$ respectively.Using   $$M_1 = M_2$$ we get    $$\rho_1 \times \pi R^2_1 t_1 = \rho_2 \times \pi R^2_2 t_2$$$$\implies$$     $$\rho_1 R^2_1 = \rho_2 R^2_2$$       or,        $$\dfrac{R^2_1}{R^2_2} = \dfrac{\rho_2}{\rho_1}$$                   .............(1)Moment of inertia of circular disc      $$I = \dfrac{1}{2} MR^2$$$$\implies$$     $$\dfrac{I_1}{I_2} = \dfrac{R_1^2}{R_2^2} = \dfrac{\rho_2}{\rho_1}$$Thus   if  $$\rho_1 > \rho_2$$    $$\implies$$     $$I_1 < I_2$$Hence the disc made up of more dense material has smaller rotational inertia.Physics

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