Consider a turntable to be a circular disk of moment of inertia It rotating at a constant angular velocity ωi around an axis through the center and perpendicular to the plane of the disk (the disk's "primary axis of symmetry"). The axis of the disk is vertical and the disk is supported by frictionless bearings. The motor of the turntable is off, so there is no external torque being applied to the axis. Another disk (a record) is dropped onto the first such that it lands coaxially (the axes coincide). The moment of inertia of the record is Ir. The initial angular velocity of the second disk is zero. There is friction between the two disks. After this "rotational collision," the disks will eventually rotate with the same angular velocity.

Question

Quinn Patton

Physics

30 July, 04:12

Consider a turntable to be a circular disk of moment of inertia It rotating at a constant angular velocity ωi around an axis through the center and perpendicular to the plane of the disk (the disk's "primary axis of symmetry"). The axis of the disk is vertical and the disk is supported by frictionless bearings. The motor of the turntable is off, so there is no external torque being applied to the axis. Another disk (a record) is dropped onto the first such that it lands coaxially (the axes coincide). The moment of inertia of the record is Ir. The initial angular velocity of the second disk is zero. There is friction between the two disks. After this "rotational collision," the disks will eventually rotate with the same angular velocity.

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Ansley Arellano · Answer 1

Note: Angular momentum is always conserved in a collision.

The initial angular momentum of the system is

L = (It) (ωi)

where It = moment of inertia of the rotating circular disc,

ωi = angular velocity of the rotating circular disc

The final angular momentum is

L = (It + Ir) (ωf)

where ωf is the final angular velocity of the system.

Since the two angular momenta are equal, we see that

(It) (ωi) = (It + Ir) (ωf)

so making ωf the subject of the formula

ωf = [ (It) / (It + Ir) ] ωi