Question

Jacob and Sophia are playing with a merry-go-round on a playground.
The merry-go-round can be modeled as a flat disk with mass M = 223.0
kg and radius R = 3.3 m that freely rotates horizontally without friction
about its center axis. When the merry-go-round is already spinning at
23 rotations per minute, Jacob exerts a constant force FJ = 12.3 N
tangent to the outer edge of the merry go round in the direction of the
merry-go-round’s rotation. At the same time Sophia puts her foot at
a distance r = 3.1 m from the center, which exerts a constant force of
friction FS = 21.2 N. How long will it take for the merry-go-round to
come to a stop?

Answer 1

In this problem two opposite torque are acting

Torque due to Jacob

$$=R\times F=3.3\times12.3=40.59$$

Torque due to Sophia

$$=21.2\times 3.1=65.72$$

Effective torque

$$=65.72-40.59=25.13\ N\ m$$

Now 23 rotation per minute

$$=\frac{23\times 2\pi}{60}=2.408\ rad\ per\ sec=\omega_i$$

Using equation of motion for torque-

$$\omega_f=\omega_i+\alpha t$$

Now

$$\omega_f=0$$

$$\alpha=\frac{torque}{I}$$

where,

$$I=moment\ of\ inertia=\frac{1}{2}MR^2=1214.235$$

So,

$$\alpha=\frac{25.13}{1214.235}=0.0207$$

So,

$$t=|\frac{\omega_i}{\alpha}|=\frac{2.408}{0.0207}=116.33\ sec$$