A uniform cylindrical turntable of radius 1.80 m and mass 25.2 kg rotates counte

Question

A uniform cylindrical turntable of radius 1.80 m and mass 25.2 kg rotates counterclockwise in a horizontal plane with an initial angular speed of 4pi rad/s. The fixed turntable bearing is frictionless. A lump of clay of mass 2.41 kg and negligible size is dropped onto the turntable from a small distance above it and immediately sticks to the turntable at a point 1.70 m to the east of the axis.

(a) Find the final angular speed of the clay and turntable. Magnitude in rad/s and direction (clockwise or counterclockwise)

b) Is mechanical energy of the turntable-clay system constant in this process? Yes or No ?

What, if any, is the change in internal energy? in Joules

(c) Is momentum of the system constant in this process? Yes or No?

What, if any, is the amount of impulse imparted by the bearing? magnitude in kg

Explanation / Answer

a)

In your situation the moment of inertia of the "turntable system" is changed by adding a fixed mass of clay a distance 1.90 m from the center. Since dropping a piece of clay onto the turn table does not result in a torque acting on the turntable system, angular momentum must be conserved.

Angular momentum is conserved:

I??? = I???

zeroes indicate initial and ones indicate final values!

We want to know the angular velocity of the turntable and clay system after the clay is placed on the table. Solving the above equation for ?? gives:

?? = I??? / I?

We have not been given values for either I? or I? so these must be calculated:

I? = The moment of inertia of the turntable
= The moment of inertia of a cylinder with mass m and radius r around its z axis =

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