Question

A Texas cockroach of mass 0.117 kg runs counterclockwise around the rim of a lazy Susan (a circular disk mounted on a vertical axle) that has a radius 17.3 cm, rotational inertia 5.20 x 10-3 kg·m2, and frictionless bearings. The cockroach's speed (relative to the ground) is 1.91 m/s, and the lazy Susan turns clockwise with angular velocity ω0 = 2.87 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?

Answer 1

given,

mass of cockroach = 0.117 Kg

radius = 17.3 cm

rotational inertia = 5.20 x 10⁻³ Kg.m²

speed of cockroach = 1.91 m/s

angular velocity of Susan (ω₀)= 2.87 rad/s

final speed of cockroach = 0 m/s

Initial angular velocity of Susan

L_s = I ω₀

L_s = 5.20 x 10⁻³ x 2.87

L_s=0.015 kg.m²/s

initial angular momentum of the cockroach

L_c = - mvr

L_c = - 0.117 x 1.91 x 0.173

L_c = - 0.0387 kg.m²/s

total angular momentum of Both

L = 0.015 - 0.0387

L = - 0.0237 kg.m²/s

after cockroach stop inertia becomes

I_f = I + mr^2

I_f = 5.20 x 10⁻³+ 0.117 x 0.173^2

I_f = 8.7 x 10⁻³ kg.m²/s

final angular momentum of the disk

L_f = I_f ω_f

L_f = 8.7 x 10⁻³ x ω_f

using conservation of momentum

L_i = L_f

-0.0237 =8.7 x 10⁻³ x ω_f

`\omega_f = (-0.0237)/(8.7 * 10^(-3))`

`\omega_f = -2.72\ rad/s`

angular speed of Susan is
`\omega_f = -2.72\ rad/s`

The value is negative because it is in the opposite direction of cockroach.

`|\omega_f |= 2.72\ rad/s`

b) the mechanical energy is not conserved because cockroach stopped in between.