Question

A playground carousel is rotating counterclockwise about its center on frictionless bearings. A person standing still on the ground grabs onto one of the bars on the carousel very close to its outer edge and climbs aboard. Thus, this person begins with an angular speed of zero and ends up with a nonzero angular speed, which means that he underwent a counterclockwise angular acceleration. The carousel has a radius of 1.53 m, an initial angular speed of 3.50 rad/s, and a moment of inertia of 124 kg·m2. The mass of the person is 42.3 kg. Find the final angular speed of the carousel after the person climbs aboard.

Answer 1

Answer:

1.94601 rad/s

Explanation:

$I_1$ = Moment of inertia of carousel = 124 kgm²

$\omega_1$ = Angular speed of carousel = 3.5 rad/s

$\omega_2$ = Angular speed of person

r = Radius of carousel = 1.53 m

m = Mass of person = 42.3 kg

Moment of inertia of person

$I_2=mr^2\\\Rightarrow I_2=42.3\times 1.53^2\\\Rightarrow I_2=99.02007\ kgm^2$

As the angular momentum is conserved in the system

$I_1\omega_1=(I_1+I_2)\omega_2\\\Rightarrow \omega=\frac{I_1\omega_1}{(I_1+I_2)}\\\Rightarrow \omega_2=\frac{124\times 3.5}{(124+99.02007)}\\\Rightarrow \omega_2=1.94601\ rad/s$

The angular speed of the carousel after the person climbs aboard is 1.94601 rad/s