A large disk of mass M and radius R is spinning like a CD of merry-go-round - that is, about an axis perpendicular to the plane of the disk. It is rotating at an angular velocity ? (initial). At some instant, a sphere of mass M/4 , which is initially not rotating, is dropped onto the disk at a distance of 3R/ 4 from the center. The sphere sticks to the disk and begins rotating with it. Find the final velocity of the disk-sphere combination.

^^^ the ? mark is the lowercase "omega"

Add the two I, using the parallel axis theorem.

To find the final velocity of the disk-sphere combination, we need to use the principle of conservation of angular momentum.

Step 1: Calculate the initial angular momentum of the system.
The initial angular momentum of the spinning disk is given by L1 = I1 * ω1, where I1 is the moment of inertia of the disk and ω1 is the initial angular velocity of the disk.

Step 2: Calculate the initial angular momentum of the sphere.
Since the sphere is initially not rotating, its initial angular momentum is zero.

Step 3: Calculate the final angular momentum of the system.
After the sphere sticks to the disk, their combined system will have a new moment of inertia and angular velocity. The final moment of inertia of the combined system is given by the sum of the moment of inertia of the disk and the moment of inertia of the sphere. Let's denote the final moment of inertia as I2.

I2 = I_disk + I_sphere

To calculate I_disk, we use the formula for the moment of inertia of a solid disk rotating about its central axis, which is given by:

I_disk = (1/2) * M * R^2

To calculate I_sphere, we use the formula for the moment of inertia of a sphere rotating about an axis passing through its center, which is given by:

I_sphere = (2/5) * (M/4) * (3R/4)^2

Step 4: Apply the conservation of angular momentum.
According to the conservation of angular momentum, the initial angular momentum of the system is equal to the final angular momentum of the system.

L1 = L2

I1 * ω1 = I2 * ω2

Substituting the values for I1, ω1, I2, and solving for ω2 will give us the final angular velocity of the disk-sphere combination.

Step 5: Calculate the final linear velocity of the disk-sphere combination.
The final linear velocity of the disk-sphere combination can be calculated using the formula:

v_final = R * ω2

Substitute the value of ω2 obtained from step 4, along with the given values for M and R, to find the final linear velocity.

Note: In the calculation, make sure to use consistent units for all variables (e.g., kilograms for mass, meters for radius) to obtain accurate results.