91Ó°ÊÓ

The rotating analogue of linear momentum is angular momentum (also known as moment of momentum or rotational momentum). Because it is a conserved quantity—the total angular momentum of a closed system remains constant—it is a significant quantity in physics.

Radius of the uniform-density disk,$R=0.6m$

Mass of the disk,$M=5kg$

Mass of clay,$m=0.4kg$

Location of the clay,

Initial speed of the clay,$v=8m/s$

Initial angular speed of the disk,${\mathrm{Ó\neg }}_0=0.51rad/s$

A rotating uniform-density disk of radius $R$is mounted in the vertical plane is shown in the below figure.

In vector notation, initial velocity of the clay,

Initial angular velocity of the disk,

Initial linear momentum of the clay is,

Initial angular momentum of the clay is

Moment of inertia of the uniform-density disk of mass $M$and $R$is given by

$I=\frac{1}{2}M{R}^2$

Initial angular momentum of the disk can be calculated as

Therefore, the angular momentum of the combined system of wheel plus clay about the centre, just before impact is

The direction of the initial angular momentum of the system is out of the page.

The net external torque acting on the system is zero, so the angular momentum of the system is conserved.

$$\begin{gathered} {\stackrel{→}{\mathrm{τ}}}_{net}=\frac{d\stackrel{→}{L}}{dt} \\ 0=\frac{d\stackrel{→}{L}}{dt} \\ {\stackrel{→}{L}}_f={\stackrel{→}{L}}_i \end{gathered}$$

Therefore, the angular momentum of the combined system of wheel plus clay about the centre, just after the impact is

The final moment of inertia of the system is equal to the sum of the moment of inertias of the clay and disk about the centre of the disk.

$$\begin{gathered} {\stackrel{→}{I}}_f={\stackrel{→}{I}}_{elay}+{\stackrel{→}{I}}_{disk} \\ =m{R}^2+\frac{1}{2}M{R}^2 \\ =\left(m+\frac{M}{2}\right)R^2 \\ =\left(0.4kg+\frac{5kg}{2}\right)(0.6m{)}^2 \\ =1.044kg.m^2 \end{gathered}$$

The final rotational angular momentum of the system is

${\stackrel{→}{L}}_f=I_f{\stackrel{→}{\mathrm{Ó\neg }}}_f$

The final angular velocity of the wheel can be calculated as

Option D is correct, because the axle exerts an upward force in the direction opposite to the linear momentum.