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In understanding the movement of a door struck by mud, we start with the concept of moment of inertia. It's like mass in linear motion but for rotating objects. The moment of inertia \( I \) translates to how difficult it is to change the rotational speed. For us to calculate it, the shape and axis of rotation matter.
For a rectangular door hinged at one side, use the equation:
- \( I = \frac{1}{3} mL^2 \) - where \( m \) is the mass of the door, and \( L \) is its width.In our exercise, a door weighing 40 kg, 1 meter wide, gives:
- \( I = \frac{1}{3} \times 40.0 \times (1.00)^2 = 13.33 \ \text{kg} \ \text{m}^2 \)This formula shows us that width contributes more than height to rotational resistance. Also, check other objects like rods or cylinders. They each have unique formulas for inertia.

When an object rotates, its angular momentum \( L \) remains constant if no external torque acts. This is the conservation of angular momentum, a key principle in physics. In simpler terms, it means the door and mud must conserve momentum when they interact.
- We start with the door at rest, so its initial angular momentum \( L_i \) is 0.
- The mud, traveling at 12 m/s, transfers its linear momentum to the door as angular momentum:
- \( L_f = m_m \cdot v \cdot \frac{L}{2} \)
- With 0.5 kg of mud, - \( L_f = 0.5 \times 12.0 \times \frac{1.00}{2} = 3.0 \ \text{kg} \ \text{m}^2/\text{s} \)By conservation, the initial \( L_i \)and final \( L_f \) angular momenta must equal. This is how impact velocity and position create rotation.

After understanding momentum conservation, we calculate angular speed \( \omega \). This speed tells how fast the door spins post-impact.
The relation of angular momentum \( L \) and angular velocity is:- \( I \cdot \omega = L \)Here, \( I \) stands for moment of inertia and \( \omega \) for angular speed in radians per second.
Plugging known values in, from the conservation equation:- \( 13.33 \cdot \omega = 3.0 \)- Solving, \( \omega = \frac{3.0}{13.33} = 0.225 \) radians/second.This result tells the door's new spinning rate due to the mud.Such calculated outcomes are crucial in designing mechanisms where precise rotational control is necessary. Remember, everything affects speed—mass, distance from the rotation point, and impact strength.