The Angular Momentum

To solve the problem, we need to consider the conservation of angular momentum, which states that the initial angular momentum of the system must be equal to the final angular momentum of the system, provided no external torques act on it.

The initial angular momentum of the ring, , can be calculated as:

where is the initial moment of inertia of the ring, calculated as:

So, the initial angular momentum is:

When the four small spheres, each with mass , are added to the ring, the new moment of inertia becomes:

The final angular momentum , assuming conservation of angular momentum, is the same as the initial angular momentum:

Thus,

Substitute the expression for :

Solve for :

The radius terms cancel out, giving:

Thus, the new angular velocity of the ring with the spheres attached is:

The areal velocity of a planet is defined as the rate at which the planet sweeps out area in its orbit. For a planet of mass and distance from the Sun, the areal velocity is given by: where is the radius vector and is the angular velocity.
Step 1: Relationship between angular momentum and areal velocity
The angular momentum of a planet is given by: where is the mass of the planet and is the angular momentum per unit mass.
Now, using the expression for areal velocity , we can solve for :
Step 2: Final angular momentum
Substitute into the expression for angular momentum: Thus, the angular momentum of the planet is , which corresponds to option (C).