A Texas cockroach of mass 0.180 kg runs counterclockwise around the rim of a laz

Question

A Texas cockroach of mass 0.180 kg runs counterclockwise around the rim of a lazy Susan (a circular disk mounted on a vertical axle) that has a radius 13.5 cm, rotational inertia 3.17 x 10-3 kg·m2, and frictionless bearings. The cockroach's speed (relative to the ground) is 2.27 m/s, and the lazy Susan turns clockwise with angular velocity 0 = 2.53 rad/s. The cockroach finds a bread crumb on the rim and, of course, stops. (a) What is the angular speed of the lazy Susan after the cockroach stops? (b) Is mechanical energy conserved as it stops?

Explanation / Answer

given:

m=0.180 kg

r=13.5 cm=0.135m

I=3.17*10^-3 kg.m^2

v=2.27m/s

0 = 2.53 rad/s.

(a) What is the angular speed of the lazy Susan after the cockroach stops? this is a problem about conservation of rotational momentum.

rotational momentum is given by: L = I

Initially, the disk's momentum is:
Lo = Io = 3.17*10^-3*2.53 = 0.008kgm^2/s

Similarly, the momentum of the cockroach in the opposite direction is

Lc = -I = -mr^2*v/r = -mvr = -0.180*2.27*0.165 = -0.0674 kgm^2/s

After the cockroach stops, the total inertia of the now only spinning disk is I + mr^2, therefore its final angular momentum is

Lf = (I + mr^2)f
= (3.17*10^-3 + 0.180*0.165^2)f
= 0.0080705f

We know momentum is conserved i.e.:

Lc + Lo = Lf
Or
-0.0674 + 0.008 = 0.0080705f

f = -7.36rad/s

b) mechanical energy conserved as it stops.

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