91Ó°ÊÓ

For any moving body the ratio of the angular momentum with the angular velocity is known as the moment of inertia.

And conservation of angular momentum shows relation between moment of inertia and angular velocity.

$$\begin{gathered} I=\frac{1}{3}M{L}^2 \\ {I}_1{\mathrm{Ó\neg }}_1=I_2{\mathrm{Ó\neg }}_2 \end{gathered}$$

Here I are the moment of inertia, $M$is the mass of the object, L angular momentum, $I_1$is the moment of inertia for the first object, ${\mathrm{Ó\neg }}_1$is the angular velocity of the first object, $I_2$is the moment of inertia for the second object,${\mathrm{Ó\neg }}_2$ is the angular velocity for the second object respectively.

Weight of gate, $M=4.5\mathrm{Kg}$

Sides of the gate, $L=1.5\mathrm{m}$

Weight of raven, $m=1.1\mathrm{Kg}$

The horizontal speed of raven, $v_1=5.0\mathrm{m}/\mathrm{s}$

Bounce back speed of raven in opposite direction, $v_2=2.0\mathrm{m}/\mathrm{s}$

From the conservation of angular momentum,

$$\begin{gathered} I_1{\mathrm{Ó\neg }}_1=I_2{\mathrm{Ó\neg }}_2 \\ m{v}_1\mathrm{â„“}+m{N}_2\mathrm{â„“}=\frac{1}{3}M{L}^2\mathrm{Ó\neg } \\ \mathrm{Ó\neg }=\frac{m\left(v_1+v_2\right)\mathrm{â„“}}{\frac{1}{3}M{L}^2} \\ \text{With}\mathrm{â„“}=L/2: \\ \mathrm{Ó\neg }=\frac{3m\left(v_1+v_2\right)}{2ML} \end{gathered}$$

Here, m is the mass of object first, $\mathrm{v}$is the mass of object second, $V_1$is the initial velocity of bar, $V_2$is the final velocity of collision, $\mathrm{â„“}$is the distance between the objects, and $\mathrm{Ó\neg }$is the angular momentum of the object respectively.

On putting values and solving it further,

$$\begin{gathered} \mathrm{Ó\neg }=\frac{3m\left(v_1+v_2\right)}{2ML} \\ =\frac{3(1.1\mathrm{Kg})(5.0\mathrm{m}/\mathrm{s}+2.0\mathrm{m}/\mathrm{s})}{2(4.5\mathrm{Kg})(1.5\mathrm{m})}âˆ\pounds \\ =1.71\mathrm{rad}/\mathrm{s} \end{gathered}$$

Hence, the angular speed of the gate will be $\mathrm{Ó\neg }=1.71\mathrm{rad}/\mathrm{s}$.

Along the pivot there is no external torques so angular momentum will be conserved but horizontally an external force on the system get applied by the pivot hence the linear momentum is not conserved.