91Ó°ÊÓ

It is the property of a rotating body given by the product of the moment of inertia and the angular velocity of the rotating object. It is a vector quantity, which implies that here along with magnitude, the direction is also considered.

b. Due to the presence of an external force a torque is generated and it is expressed as follows:

$\stackrel{→}{\mathrm{τ}}=\frac{d\stackrel{→}{L}}{dt}$

Here, $d\stackrel{→}{L}$represents the change in angular momentum.

The expression for torque is given as follows:

$\stackrel{→}{\mathrm{τ}}=\stackrel{→}{r}×\stackrel{→}{F}$

Here, $\stackrel{→}{r}$is radius of the disk and $\stackrel{→}{F}$is the applied force.

The force is applied perpendicular to the radius vector thus the torque is maximum and it is directed along the negative$z-$axis.

${\mathrm{Ï„}}_2=rF$

Substitute $8cm$for $R$and $10N$for$F.$

$\begin{array}{rcl}{\mathrm{Ï„}}_2& =& -\left(8cm\right)\left(\frac{{10}^{-2}m}{1cm}\right)\left(10N\right)\\ & =& -0.8N.\\ & & \end{array}$

Rearrange the expression$\stackrel{→}{\mathrm{τ}}=\frac{d\stackrel{→}{L}}{dt}$for $d\stackrel{→}{L}.$

$\begin{array}{rcl}d\stackrel{→}{L}& =& \stackrel{→}{\mathrm{τ}}dt\\ {\stackrel{→}{L}}_2-{\stackrel{→}{L}}_1& =& \stackrel{→}{\mathrm{τ}}dt\\ {\stackrel{→}{L}}_2& =& {\stackrel{→}{L}}_1+\stackrel{→}{\mathrm{τ}}dt\\ & & \end{array}$

Here, ${\stackrel{→}{L}}_2$represents the angular momentum of the disk at an instant of time$0.2s$later.

Substitutekg.meter square per second for for$\stackrel{→}{\mathrm{τ}}$and $0.2s$for $dt.$

Therefore, the magnitude of the angular momentum at a time$0.2s$later is 0.1856 kg.meter squre per second and it is directed along the negative$z-$direction.

c.The angular velocity of the disk after is given as follows:

$L_2=\left(\frac{1}{2}M{R}^2\right){\mathrm{Ó\neg }}_2$

Substitute -0.1856 kg.meter squre per second for $L_2,0.4kg$for$M,$and $8cm$for$R.$

$-0.1856=\left(\frac{1}{2}(0.4kg){\left(8cm\right)}^2{\left(\frac{{10}^{-2}m}{1cm}\right)}^2\right){\mathrm{Ó\neg }}_2$

$\begin{array}{rcl}{\mathrm{Ó\neg }}_2& =& \frac{-0.1856}{0.00128}\\ & =& -145\\ & & \end{array}$

Therefore, the magnitude of the angular velocity of the disk at time $0.2s$later is 145 rad/s and it is directed along the negative $z-$axis.