A Texas cockroach of mass 0.17 kg runs counterclockwise around the rim of a lazy

Question

A Texas cockroach of mass 0.17 kg runs counterclockwise around the rim of a lazy Susan (a circular disk mounted on a vertical axle) that has radius 15 cm, rotational inertia 4.6 × 10-3 kg·m2, and frictionless bearings. The cockroach's speed (relative to the ground) is 2.0 m/s, and the lazy Susan turns clockwise with angular speed 0-4.0 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? rad/s (b) Is mechanical energy conserved as it stops? O Yes No

Explanation / Answer

Given,

m = 0.17kg ; r = 15 cm = 0.15 m ; I = 4.6 x 10^-3 kg-m^2 ; vc = 2 m/s ; w0 = 4 rad/s

a)The angular momentum of the cockroach is:

Lic = m v r

that of Susan is:

Ls = I w

Total intial angular momentum of the system

L = Lic + Ls

L = m v r + I w

after change in the system, the rotational inertia becomes:

If = I + m r^2

Lf = If wf => wf = Lf/If

wf = (mvr + Iw)/(I + mr^2)

wf = (0.17 x 2 x 0.15 - 4.6 x 10^-3 x 4)/[4.6 x 10^-3 + 0.17 x 0.15^2] = 8.24 rad/s

Hence, wf = 3.87 rad/s

b)No, the mechanical energy is not conserved as it stops.

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