Answer:
The angular velocity is ![w = 1.43\ rad/sec](https://tex.z-dn.net/?f=w%20%3D%201.43%5C%20%20rad%2Fsec)
Explanation:
From the question we are told that
The mass of wooden gate is ![m_g = 4.5 kg](https://tex.z-dn.net/?f=m_g%20%3D%204.5%20kg)
The length of side is L = 2 m
The mass of the raven is ![m_r = 1.2 kg](https://tex.z-dn.net/?f=m_r%20%3D%201.2%20kg)
The initial speed of the raven is ![u_r = 5.0m/s](https://tex.z-dn.net/?f=u_r%20%3D%205.0m%2Fs)
The final speed of the raven is ![v_r = 1.5 m/s](https://tex.z-dn.net/?f=v_r%20%3D%201.5%20m%2Fs)
From the law of conservation of angular momentum we express this question mathematically as
Total initial angular momentum of both the Raven and the Gate = The Final angular momentum of both the Raven and the Gate
The initial angular momentum of the Raven is ![m_r * u_r * \frac{L}{2}](https://tex.z-dn.net/?f=m_r%20%2A%20u_r%20%2A%20%5Cfrac%7BL%7D%7B2%7D)
Note: the length is half because the Raven hit the gate at the mid point
The initial angular momentum of the Gate is zero
Note: This above is the generally formula for angular momentum of square objects
The final angular velocity of the Raven is ![m_r * v_r * \frac{L}{2}](https://tex.z-dn.net/?f=m_r%20%2A%20v_r%20%2A%20%5Cfrac%7BL%7D%7B2%7D)
The final angular velocity of the Gate is ![\frac{1}{3} m_g L^2 w](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B3%7D%20m_g%20L%5E2%20w)
Substituting this formula
![m_r * u_r * \frac{L}{2} = \frac{1}{3} m_g L^2 w + m_r * v_r * \frac{L}{2}](https://tex.z-dn.net/?f=m_r%20%2A%20u_r%20%2A%20%5Cfrac%7BL%7D%7B2%7D%20%20%3D%20%20%20%5Cfrac%7B1%7D%7B3%7D%20m_g%20L%5E2%20w%20%2B%20m_r%20%2A%20v_r%20%2A%20%5Cfrac%7BL%7D%7B2%7D)
![\frac{1}{3} m_g L^2 w = m_r * v_r * \frac{L}{2} - m_r * u_r * \frac{L}{2}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B3%7D%20m_g%20L%5E2%20w%20%20%20%3D%20%20%20%20m_r%20%2A%20v_r%20%2A%20%5Cfrac%7BL%7D%7B2%7D%20-%20%20%20m_r%20%2A%20u_r%20%2A%20%5Cfrac%7BL%7D%7B2%7D)
![\frac{1}{3} m_g L^2 w = m_r * \frac{L}{2} * [u_r - v_r]](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B3%7D%20m_g%20L%5E2%20w%20%20%20%3D%20%20%20%20m_r%20%2A%20%20%5Cfrac%7BL%7D%7B2%7D%20%2A%20%5Bu_r%20-%20v_r%5D)
Where
is the angular velocity
Substituting value
![\frac{1}{3} (4.5)(2)^2 w = 1.2 * \frac{2}{2} * [5 - 1.5]](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B3%7D%20%284.5%29%282%29%5E2%20%20w%20%20%20%3D%20%20%20%201.2%20%2A%20%20%5Cfrac%7B2%7D%7B2%7D%20%2A%20%5B5%20-%201.5%5D)
![6w = 4.2](https://tex.z-dn.net/?f=6w%20%3D%204.2)
![w = \frac{6}{4.2}](https://tex.z-dn.net/?f=w%20%3D%20%5Cfrac%7B6%7D%7B4.2%7D)
![w = 1.43\ rad/sec](https://tex.z-dn.net/?f=w%20%3D%201.43%5C%20%20rad%2Fsec)