Question

A Texas cockroach of mass 0.165 kg runs counterclockwise around the rim of a lazy Susan (a circular disk mounted on a vertical axle) that has a radius 18.3 cm, rotational inertia 5.16 x 10^-3 kg·m², and frictionless bearings. The cockroach's speed (relative to the ground) is 2.61 m/s, and the lazy Susan turns clockwise with angular velocity ω₀ = 3.07 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? with S.I units

Answer 1

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 = (0.165 x 2.61 x 0.183- 5.16 x 10^-3 x 3.07)/[5.16 x 10^-3 + 0.165 x 0.183^2] = 5.8927 rad/s

Hence, wf = 5.8927 rad/s

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