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

The correct option is A the one made from the more dense material
Given :    $$M_1 = M_2$$                   $$t_1  = t_2$$
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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