91Ó°ÊÓ

1. The mass of the cockroach,$\text{M}=0.17\text{kg}.$
2. The rim of the lazy Susan (a circular disk mounted on a vertical axle) is,$\text{R}=15\text{cm}=0.15\text{m}$
3. The rotation inertia of the lazy Susan (a circular disk mounted on a vertical axle) is,$\text{I}=5.0×{10}^{−3}{\text{kgm}}^2$.
4. The speed of the cockroach relative to the ground is,$\text{v}=2.0\text{m}/\text{s}$
5. The angular speed of the Susan is,${\text{Ó¬}}_0=2.8\text{rad}/\text{s}$.

Using the law of conservation of angular momentum, find the angular speed of the lazy Susan after the cockroach does stop.

Formula:

The law conservation of angular momentum is,

Initial angular momentum of the system = Final angular momentum of the system

(a)

According to the conservation of the angular momentum,

Initial angular momentum of the system = Final angular momentum of the system

$\begin{array}{c}⇒\text{mvR}−{\text{IӬ}}_0=−({\text{mR}}^2+\text{I}){\text{Ӭ}}_{\text{f}}\\ ⇒{\text{Ӭ}}_{\text{f}}=\frac{\text{mvR}−{\text{IӬ}}_0}{({\text{mR}}^2+\text{I})}\end{array}$ (The clockwise motion is taken as negative.)

Therefore, the angular speed of the lazy Susan after the cockroach does stop is, ${\text{Ó¬}}_{\text{f}}=4.2\mathrm{rad}/\mathrm{s}$.

(b)

According to the law of conservation of energy,

The Initial mechanical energy of the cockroach-Susan system is,

$\begin{array}{c}{\text{K}}_{\text{i}}=\text{K}.{\text{E}}_{\text{iCockroach}}+\text{K}.{\text{E}}_{\text{iSusan}}\\ =\frac{1}{2}{\text{mv}}^2+\frac{1}{2}{\text{IӬ}}_0^2\\ {\text{K}}_{\text{i}}=\frac{1}{2}\left(0.17\right){\left(2\right)}^2+\frac{1}{2}(5×{10}^{−3}){\left(2.8\right)}^2\\ {\text{K}}_{\text{i}}=0.34\text{J}\end{array}$

The final mechanical energy of the cockroach-Susan system

$\begin{array}{c}{\text{K}}_{\text{f}}=\text{K}.{\text{E}}_{\text{fCockroach}}+\text{K}.{\text{E}}_{\text{fSusan}}\\ ⇒{\text{K}}_{\text{f}}=0+\frac{1}{2}{\text{IӬ}}_{\text{f}}^2\\ ⇒{\text{K}}_{\text{f}}=\frac{1}{2}(5×{10}^{−3}){\left(4.2\right)}^2\\ ⇒{\text{K}}_{\text{f}}=19.6×{10}^{−3}=44.1×{10}^{−3}\text{J}\end{array}$

As,${\text{K}}_{\text{i}}≠{\text{K}}_{\text{f}},$ the mechanical energy of the cockroach-Susan system is not conserved