A solid wood door 1.00 m wide and 2.00 m high is hinged along one side and has a total mass of 40.0 kg. Initially open and at rest, the door is struck at its center by a handful of sticky mud with mass 0.500 kg, traveling perpendicular to the door at 12.0 m/s just before impact. Find the final angular speed of the door. Does the mud make a significant contribution to the moment of inertia?

Verified step by step guidance

1. Calculate the moment of inertia (I) of the door about the hinge axis. The door can be approximated as a rectangular slab, and the moment of inertia for a rectangle rotating about an axis along one edge is given by the formula: I = \( \frac{1}{3} m L^2 \), where m is the mass of the door and L is the width of the door.
2. Calculate the angular momentum (L) imparted by the mud. Angular momentum can be calculated using the formula: L = r \times p, where r is the distance from the hinge to the point of impact (half the width of the door, since the mud hits the center), and p is the linear momentum of the mud, given by p = m \times v, where m is the mass of the mud and v is its velocity.
3. Determine the final angular speed (\( \omega_f \)) of the door using the conservation of angular momentum. The initial angular momentum of the door is zero (since it starts at rest), so the final angular momentum of the door is equal to the angular momentum imparted by the mud. Use the formula: \( \omega_f = \frac{L}{I} \), where L is the angular momentum calculated in step 2 and I is the moment of inertia calculated in step 1.
4. Assess the contribution of the mud to the moment of inertia of the system. Calculate the moment of inertia of the mud about the hinge axis using the formula: I_mud = m \times r^2, where m is the mass of the mud and r is the distance from the hinge to the point of impact.
5. Compare the moment of inertia of the mud (I_mud) with the moment of inertia of the door (I) to determine if the mud makes a significant contribution to the total moment of inertia of the system.