91Ó°ÊÓ

Angular velocity is a measure of how quickly an object rotates or revolves around a particular point or axis. In our example, it's how fast the rod with the basket, containing the puck, spins after the collision. The unit for angular velocity is radian per second (rad/s).
To calculate angular velocity (\( \omega \)), you need to know the time period (\( T \)) taken for one complete revolution. Here, the rod takes 0.736 seconds for one revolution. You can use the formula \( \omega = \frac{2\pi}{T} \) to find angular velocity.
This formula states that if you know how long it takes for one complete spin, you can find how fast the object is rotating by dividing the full angle of rotation (which is \( 2\pi \) radians) by the time it takes to complete that rotation. In our case, substituting \( T \) as 0.736 seconds gives us \( \omega \approx 8.54 \, \text{rad/s} \), indicating a rather swift rotation.

The moment of inertia is a measure of an object's resistance to changes in its rotation. Imagine it as the rotational equivalent of mass in linear motion. It depends on how the mass is distributed relative to the rotation axis.
For a rod pivoted at one end, its moment of inertia is calculated using the formula \( I_{\text{rod}} = \frac{1}{3} m_r L^2 \) where \( m_r \) is the mass of the rod, and \( L \) is its length.
For the puck, because it's at a distance from the pivot point, its moment of inertia is \( I_{\text{puck}} = m_p L^2 \) where \( m_p \) is the puck's mass. By adding the two, you get the total moment of inertia for the system: \( I = I_{\text{rod}} + I_{\text{puck}} \).
In our exercise, this gives us \( I \approx 1.499 \, \text{kg} \cdot \text{m}^2 \). This means it takes a certain amount of torque, or rotational force, to change the spinning speed of the rod with the puck.

Collision dynamics deals with how objects interact during impacts and how their momentums are conserved. Here, the key principle is the conservation of angular momentum.
Before the puck hits the basket attached to the rod, it has linear momentum, which then converts into angular momentum after it is caught. The law of conservation of angular momentum states that in the absence of external torques, the initial angular momentum (\( L_i \)) is equal to the final angular momentum (\( L_f \)).
For the puck-rod system, \( L_i = m_p \cdot v_p \cdot L \) and \( L_f = I \cdot \omega \), where \( v_p \) is the speed of the puck before the collision. By equating these, \( m_p \cdot v_p \cdot L = I \cdot \omega \), you can solve for the puck's initial speed:\( v_p = \frac{I \cdot \omega}{m_p \cdot L} \).
Substituting the values from our example results in \( v_p \approx 39.21 \, \text{m/s} \). This shows how conservation laws help determine the speed of the puck right before the collision.