Two cars moving parallelly to each other with the velocities shown. Initial separation between cars is \( r = 10 \, \text{m} \). Find angular momentum of car A w.r.t. car B.
Show Hint
The angular momentum of a moving object with respect to a point is calculated by \( L = r \times p \), where \( r \) is the perpendicular distance and \( p \) is the linear momentum.
Step 1: Angular momentum formula.
The angular momentum \( L \) of an object with respect to a point is given by:
\[
L = r \times p
\]
where \( r \) is the perpendicular distance between the point and the line of motion of the object, and \( p \) is the linear momentum of the object.
The linear momentum \( p \) is given by:
\[
p = m \cdot v
\]
where \( m \) is the mass of the object and \( v \) is its velocity.
Step 2: Relative velocities.
- The velocity of car A is given as \( 36 \, \text{km/h} \).
- The velocity of car B is given as \( 72 \, \text{km/h} \).
- The relative velocity of car A with respect to car B is:
\[
v_{\text{rel}} = v_A - v_B = 36 \, \text{km/h} - 72 \, \text{km/h} = -36 \, \text{km/h}
\]
This negative sign indicates that car A is moving in the opposite direction of car B.
Step 3: Convert velocities to SI units.
We convert the velocities from \( \text{km/h} \) to \( \text{m/s} \) using the conversion factor \( 1 \, \text{km/h} = \frac{1}{3.6} \, \text{m/s} \):
\[
v_{\text{rel}} = -36 \, \text{km/h} = \frac{-36}{3.6} = -10 \, \text{m/s}
\]
Step 4: Calculate the angular momentum of car A w.r.t. car B.
The mass of car A is \( m_1 = m \), and the mass of car B is \( m_2 = \frac{3}{2} m \).
Now, the angular momentum of car A w.r.t. car B is:
\[
L = r \cdot m_1 \cdot v_{\text{rel}} = 10 \cdot m \cdot (-10) = -100 m
\]
The magnitude of angular momentum is:
\[
L = 100 \, m
\]
Thus, the angular momentum of car A w.r.t. car B is \( 100 \, \text{m} \).
