To solve this problem, we can use the principle of conservation of mechanical energy.
(a) If we treat the arm as an ideal pendulum, with all of its mass concentrated as a point mass at the free end, we can use the conservation of mechanical energy to find the maximum angle.
The projectile has an initial kinetic energy when it hits the pendulum. After the collision, the system of the pendulum and the projectile will have a combination of potential and kinetic energy.
Let's denote the maximum angle by θ.
1. Initial kinetic energy of the projectile:
The initial kinetic energy is equal to (1/2) * M * V^2, where M is the mass of the projectile and V is its velocity.
2. Maximum potential energy at the highest point:
The potential energy at the highest point is maximum and equal to M * g * L * (1 - cos(θ)), where g is the acceleration due to gravity and L is the length of the pendulum arm.
3. Maximum kinetic energy at the lowest point:
The maximum kinetic energy at the lowest point is zero since the pendulum momentarily comes to a rest.
Applying the conservation of mechanical energy, we equate the initial kinetic energy to the sum of the maximum potential energy and maximum kinetic energy:
(1/2) * M * V^2 = M * g * L * (1 - cos(θ))
From this equation, we can solve for θ.
(b) If we treat the arm as a thin rigid rod with its mass evenly distributed along its length, the situation can be modeled as a compound pendulum.
In this case, we consider both the rotational kinetic energy of the arm and the potential energy of the center of mass of the arm.
The rotational kinetic energy is given by (1/2) * I * ω^2, where I is the moment of inertia of the arm and ω is the angular velocity.
The potential energy of the center of mass is M * g * h, where h is the vertical height of the center of mass.
Applying the conservation of mechanical energy, we equate the initial kinetic energy to the sum of the rotational kinetic energy and potential energy:
(1/2) * M * V^2 = (1/2) * I * ω^2 + M * g * h
To solve this equation, we need to find the moment of inertia of the arm and the vertical height of the center of mass.
For a thin rod rotating about its end, the moment of inertia is given by (1/3) * M * L^2.
The vertical height of the center of mass can be found using trigonometry. Since the arm swings in a vertical plane, the height is given by h = L - L * cos(θ).
Substituting these values into the equation, we can solve for θ.
Remember to convert all angles to radians.
I hope this explanation helps! Please let me know if you have any further questions.