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·m^{2}, 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 ω_{0} =
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.**

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