A uniform cylindrical turntable of radius 1.90 m and mass 30 kg rotates counterc

Question

A uniform cylindrical turntable of radius 1.90 m and mass 30 kg rotates counterclockwise in a horizontal plane with an initial angular speed of 4 rad/s. The fixed turntable bearing is frictionless. A lump of clay of mass 2.25 kg and negligible size is dropped onto the turntable from a small distance above it and immediately sticks to the turntable at a point 1.80 m to the east of the axis.

a.) Find the resulting angular speed of the clay and the turntable.

b.)Calculate the initial and final kinetic energy of the system.

Explanation / Answer

let

R = 1.9 m
M = 30 kg
wi = 4*pi rad/s
m = 2.25 kg
r = 1.8 m

a) Apply conservation of angular momentum

final angular momentum = initial angular momentum

If*wf = Ii*wi

(0.5*M*R^2 + m*r^2)*wf = (0.5*M*R^2)*wi

==> wf = (0.5*M*R^2)*wi/(0.5*M*R^2 + m*r^2)

= (0.5*30*1.9^2)*4*pi/(0.5*30*1.9^2 + 2.25*1.8^2)

= 11.1 rad/s

b) KEi = 0.5*Ii*wi^2

= 0.5*(0.5*M*R^2)*wi^2

= 0.5*(0.5*30*1.9^2)*(4*pi)^2

= 4276 J

KEf = 0.5*If*wf^2

= 0.5*(0.5*M*R^2 + m*r^2)*wf^2

= 0.5*(0.5*30*1.9^2 + 2.25*1.8^2)*(11.1)^2

= 3785 J

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