Question 22: Centre of Mass and Rotational Motion

Question

Four holes of radius 5 cm are cut from a thin square plate of 20 cm and mass 1 kg. The moment of inertia of the remaining portion about Z-axis is

Accepted Answer

Area mass density, $$\sigma  = {M \over {16{R^2}}}$$ ($$\because$$ Area = $$4R 	imes 4R = 16{R^2}$$)
Mass of each hole $${m_1} = \sigma \pi {R^2} = {M \over {16{R^2}}}\pi {R^2} = {{\pi M} \over {16}}$$
Distance between centre of plate and centre of hole
$$x = {{\sqrt {{{(2R)}^2} + {{(2R)}^2}} } \over 2} = {{2\sqrt 2 R} \over 2}$$
Moment of inertia of one hole at about Z-axis
$${I_1} = {1 \over 2}{m_1}{R^2} + {m_1}{x^2} = {{5\pi } \over {32}}M{R^2}$$
Moment of inertia of whole plate about Z-axis
$$I = {{M{{(4R)}^2}} \over 6} = {8 \over 3}M{R^2}$$
Required moment of inertia
$${I_0} = I - 4{I_1},$$ $$ = \left[ {{8 \over 3} - 4\left( {{{5\pi } \over {32}}} ight)} ight]M{R^2} = \left[ {{8 \over 3} - {{5\pi } \over 8}} ight]M{R^2}$$
Given, R = 5 cm and M = 1 kg
So, $${I_0} = \left[ {{8 \over 3} - {{5\pi } \over 8}} ight]1 	imes 25 	imes {10^{ - 4}} = 0.0017$$ kg-m
2

Answer

15 kg-m
2

Answer

0.37 kg-m
2

Answer

0.0017 kg-m
2

Answer

0.08 kg-m
2

Centre of Mass and Rotational Motion

Four holes of radius 5 cm are cut from a thin square plate of 20 cm and mass 1 kg. The moment of inertia of the remaining portion about Z-axis is

Choose the correct answer: