Answer:
The angular velocity is [tex]w = 1.43\ rad/sec[/tex]
Explanation:
From the question we are told that
The mass of wooden gate is [tex]m_g = 4.5 kg[/tex]
The length of side is L = 2 m
The mass of the raven is [tex]m_r = 1.2 kg[/tex]
The initial speed of the raven is [tex]u_r = 5.0m/s[/tex]
The final speed of the raven is [tex]v_r = 1.5 m/s[/tex]
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 [tex]m_r * u_r * \frac{L}{2}[/tex]
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 [tex]m_r * v_r * \frac{L}{2}[/tex]
The final angular velocity of the Gate is [tex]\frac{1}{3} m_g L^2 w[/tex]
Substituting this formula
[tex]m_r * u_r * \frac{L}{2} = \frac{1}{3} m_g L^2 w + m_r * v_r * \frac{L}{2}[/tex]
[tex]\frac{1}{3} m_g L^2 w = m_r * v_r * \frac{L}{2} - m_r * u_r * \frac{L}{2}[/tex]
[tex]\frac{1}{3} m_g L^2 w = m_r * \frac{L}{2} * [u_r - v_r][/tex]
Where [tex]w[/tex] is the angular velocity
Substituting value
[tex]\frac{1}{3} (4.5)(2)^2 w = 1.2 * \frac{2}{2} * [5 - 1.5][/tex]
[tex]6w = 4.2[/tex]
[tex]w = \frac{6}{4.2}[/tex]
[tex]w = 1.43\ rad/sec[/tex]
