Two objects are moving in the x, y plane as shown. The magnitude of their total angular
momentum (about the origin O) is:

To determine the total angular momentum of two objects moving in the x, y plane about the origin O, we need to consider the individual angular momenta of each object and then combine them. Angular momentum is a vector quantity that depends on both the position and the momentum of the objects. Let's break this down step by step.

Understanding Angular Momentum

Angular momentum (L) for a single object can be calculated using the formula:

• L = r × p

Where:

• r is the position vector from the origin to the object.
• p is the linear momentum of the object, given by p = mv, where m is mass and v is velocity.

Calculating Individual Angular Momenta

For each object, we need to find the position vector and the momentum. Let's denote the two objects as Object 1 and Object 2.

Assume:

• Object 1 has mass m1 and is located at position (x1, y1) with velocity (vx1, vy1).
• Object 2 has mass m2 and is located at position (x2, y2) with velocity (vx2, vy2).

The angular momentum for Object 1 can be calculated as:

• L1 = r1 × p1 = (x1, y1) × (m1 * vx1, m1 * vy1)

Using the determinant method for the cross product in two dimensions, we find:

• L1 = m1 * (x1 * vy1 - y1 * vx1)

Similarly, for Object 2:

• L2 = r2 × p2 = (x2, y2) × (m2 * vx2, m2 * vy2)

Thus, we have:

• L2 = m2 * (x2 * vy2 - y2 * vx2)

Finding Total Angular Momentum

The total angular momentum about the origin O is simply the vector sum of the angular momenta of both objects:

• L_total = L1 + L2

Substituting the expressions we derived:

• L_total = m1 * (x1 * vy1 - y1 * vx1) + m2 * (x2 * vy2 - y2 * vx2)

Example Calculation

Let’s say:

• Object 1 has m1 = 2 kg, (x1, y1) = (3, 4), and (vx1, vy1) = (1, 2).
• Object 2 has m2 = 3 kg, (x2, y2) = (1, 2), and (vx2, vy2) = (2, 1).

Calculating for Object 1:

• L1 = 2 * (3 * 2 - 4 * 1) = 2 * (6 - 4) = 2 * 2 = 4 kg·m²/s

Calculating for Object 2:

• L2 = 3 * (1 * 1 - 2 * 2) = 3 * (1 - 4) = 3 * (-3) = -9 kg·m²/s

Now, summing them gives:

• L_total = 4 - 9 = -5 kg·m²/s

This negative value indicates the direction of the angular momentum vector, which is important in understanding the motion of the system.

Final Thoughts

By following these steps, you can calculate the total angular momentum of any number of objects in the x, y plane. Remember, the key is to accurately determine the position and momentum of each object and apply the cross product correctly. This approach not only helps in solving problems in mechanics but also deepens your understanding of rotational dynamics.