A horizontal disk is rotating counter-clockwise about its axis of symmetry at `14 rps` . Its moment of inertia with respect to its axis of symmetry is `8 kg.m^2` . A second disk, of moment of inertia...

A horizontal disk is rotating counter-clockwise about its axis of symmetry at `14 rps` . Its moment of inertia with respect to its axis of symmetry is `8 kg.m^2` . A second disk, of moment of inertia equal to `2 kg.m^2` with respect to its axis of symmetry, rotating clockwise about the same axis at `7 rps` , is dropped on top of the first disk. The two disks stick together and rotate as one about their common axis of symmetry. What is the final angular velocity of the system?

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The system has some angular momentum `L_0` at its initial state. To calculate its value, we first calculate the angular momentum of each disk and then we add them together. The formula for calculating the angular momentum of a rotating body is given by (1):

`(1) L = Iw `

Where `I` is the moment of the inertia with respect to the axis of rotation, and `w` is the angular frequency of the rotation. Note that the angular momentum is a vector, pointing in the direction given by the vector cross product `v^^ r` , where `v` is the velocity of the mass element located at position `r` relative to the axis of rotation. For our disks, this direction is orthogonal to the disk. We will adopt the following: for counter-clockwise rotation this vector points "up" and for clockwise rotation the vector points "down".

Let's calculate the angular momentum `L_a` of the first disk.

`L_a = (8 kg.m^2)(14 rps) = 112 kg.m^2/s`

And `L_b` for the second disk:

`L_b = (2 kg.m^2)(-7 rps) = -14 kg.m^2/s`

Note the minus sign, since the disk is rotating in the opposite direction relative to the first disk.

Now, the initial angular momentum of the system is simply:

`L_0 = L_a + L_b = (112 - 14) kg.m^2/s = 98 kg.m^2/s`

After we drop the second disk over the first, they stick together with final angular velocity `w_f`  and moment of inertia `I_a + I_b`  (the moment of inertia of a body is the sum of the moment of inertia of its parts - relative to the correct axis of rotation). Since no external torque is being applied to the system, the total angular momentum is conserved, so:

`L_f =...

(The entire section contains 2 answers and 560 words.)

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