JEE MAIN - Physics (2011 - No. 24)
A thin horizontal circular disc is rotating about a vertical axis passing through its center. An insect is at rest at a point near the rim of the disc. The insect now moves along a diameter of the disc to reach its other end. During the journey of the insect, the angular speed of the disc.
continuously decreases
continuously increases
first increases and then decreases
remains unchanged
설명
Here no external force is applied on the disc so Torque ($$\tau $$) = 0.
So angular momentum is conserved.
That means $${I_1}{\omega _1} = {I_2}{\omega _2}$$
$$ \Rightarrow $$ $${\omega _2} = {{{I_1}{\omega _1}} \over {{I_2}}}$$
$$\therefore$$ Angular speed is inversely proportional to Moment of inertia.
For disc $$I = {1 \over 2}M{R^2}$$
$$\therefore$$ Moment of Inertia is proportional to Mass.
As insect moves along a diameter, the effective mass of disc first decreases then increases and hence the moment of inertia first decreases then increases so from principle of conservation of angular momentum, angular speed, first increases then decreases.
So angular momentum is conserved.
That means $${I_1}{\omega _1} = {I_2}{\omega _2}$$
$$ \Rightarrow $$ $${\omega _2} = {{{I_1}{\omega _1}} \over {{I_2}}}$$
$$\therefore$$ Angular speed is inversely proportional to Moment of inertia.
For disc $$I = {1 \over 2}M{R^2}$$
$$\therefore$$ Moment of Inertia is proportional to Mass.
As insect moves along a diameter, the effective mass of disc first decreases then increases and hence the moment of inertia first decreases then increases so from principle of conservation of angular momentum, angular speed, first increases then decreases.