Question

A disk has a mass 3.5 kg and radius 15 cm, rotating with angular speed 15 rev/s when a second disk of 5.0 kg is dropped onto it. If second disk has diameter 18 cm and mass 5.0 kg, what is the common final angular speed of the system?

Answer 1

`9.9" rev per sec"`, rounded to one decimal place.

Explanation:

Law of Conservation of angular momentum states:
The total angular momentum of a system about an axis remains constant, when the net external torque acting on the system about the given axis is zero.

This is applicable in the given question as a stationary second disk is dropped on a rotating disk. It is assumed that second disk is also mounted on the same shaft.

Angular momentum `vecL=vecI xxvec omega`
where angular velocity of disk rotating with angular speed `f` is`omega=2pif`
Moment of inertia of disk `I=1/2m_"disk"R^2`

Initial Angular momentum `=vecL_1+vecL_2`
`=|vecL_1|+0`
`=(1/2m_1R_1^2)(2pif)`
`=(3.5xx(0.15)^2)(pixx15)`
`=1.18125pi`

If `f_c` is the final common angular velocity of the system,
Final Angular momentum`=(vec I_1+vec I_2)xxvecomega_c`

Inserting given values and equating scalar part with (1) we get, (remember to change the given diameter of second disk to its radius)
`(1/2m_1R_1^2+1/2m_2R_2^2)2pif_c=1.18125pi`
`=>(3.5xx(0.15)^2+5.0xx(0.18/2)^2)pif_c=1.18125pi`
`=>f_c=1.18125/(3.5xx(0.15)^2+5.0xx(0.18/2)^2)`
`f_c=9.9" rev per sec"`, rounded to one decimal place.