Question

a uniform disk with a radius r and mass m is rotating about its center with an angular velocity of !0. a second disk with half the radius and the same mass is dropped from rest onto the rotating disk so that their centers are at the same position. after a short time they rotate together. what is the new angular velocity of the combined system?

Answer 1

The new angular velocity of the combined system, after a smaller disk is dropped onto a larger rotating disk, is
`\((1)/(5)\)` times the initial angular velocity of the larger disk.

To find the new angular velocity of the combined system after the second disk is dropped onto the rotating disk, we can use the principle of conservation of angular momentum.

The angular momentum
`\(L\)`of an object is given by the product of its moment of inertia
`\(I\)`and angular velocity
`\(\omega\):`

`\[L = I \omega\]`

The moment of inertia of a uniform disk rotating about its center is given by the expression
`\(I = (1)/(2) m r^2\),` where
`\(m\)` is the mass of the disk and
`\(r\)` is the radius.

The conservation of angular momentum states that the total angular momentum of an isolated system remains constant unless acted upon by an external torque.

Initially, the first disk is rotating with an angular velocity
`\(\omega_0\)` and has an angular momentum
`\(L_1\):`

`\[L_1 = I_1 \omega_0 = (1)/(2) m r^2 \omega_0\]`

When the second disk is dropped onto the first one, the moment of inertia of the system becomes the sum of the individual moments of inertia:

`\[I_{\text{total}} = I_1 + I_2 = (1)/(2) m r^2 + (1)/(2) m \left((r)/(2)\right)^2 = (5)/(8) m r^2\]`

Let
`\(\omega_{\text{final}}\)`be the final angular velocity of the combined system. The final angular momentum
`\(L_{\text{final}}\)` is given by:

`\[L_{\text{final}} = I_{\text{total}} \omega_{\text{final}}\]`

Since angular momentum is conserved,
`\(L_{\text{final}}\)` must be equal to
`\(L_1\)`initially:

`\[(1)/(2) m r^2 \omega_0 = (5)/(8) m r^2 \omega_{\text{final}}\]`

Solving for
`\(\omega_{\text{final}}\),` we get:

`\[\omega_{\text{final}} = (1)/(5) \omega_0\]`

So, the new angular velocity of the combined system is
`\((1)/(5)\)`times the initial angular velocity of the first disk.