Question 11.7: The Merry-Go-Round A horizontal platform in the shape of a c...
The Merry-Go-Round
A horizontal platform in the shape of a circular disk rotates freely in a horizontal plane about a frictionless, vertical axle (Fig. 11.10). The platform has a mass M = 100 kg and a radius R = 2.0 m. A student whose mass is m = 60 kg walks slowly from the rim of the disk toward its center. If the angular speed of the system is 2.0 rad/s when the student is at the rim, what is the angular speed when she reaches a point r = 0.50 m from the center?
Conceptualize The speed change here is similar to those of the spinning skater and the neutron star in preceding discussions. This problem is different because part of the moment of inertia of the system changes (that of the student) while part remains fixed (that of the platform).
Categorize Because the platform rotates on a frictionless axle, we identify the system of the student and the platform as an isolated system in terms of angular momentum.
Analyze Let us denote the moment of inertia of the platform as I_p and that of the student as I_s. We model the student as a particle.
Write Equation 11.21 for the system:
I_i \omega_i=I_f \omega_f=\text{constant} (11.21)
I_i \omega_i=I_f \omega_fSubstitute the moments of inertia, using r < R for the final position of the student:
\left(\frac{1}{2} M R^2+m R^2\right) \omega_i=\left(\frac{1}{2} M R^2+m r^2\right) \omega_fSolve for the final angular speed:
\omega_f=\left(\frac{\frac{1}{2} M R^2+m R^2}{\frac{1}{2} M R^2+m r^2}\right) \omega_iSubstitute numerical values:
\omega_f=\left[\frac{\frac{1}{2}(100 kg)(2.0 m)^2+(60 kg)(2.0 m)^2}{\frac{1}{2}(100 kg)(2.0 m)^2+(60 kg)(0.50 m)^2}\right](2.0 \text{ rad} / s)=\left[\frac{440 kg \cdot m^2}{215 kg \cdot m^2}\right](2.0 \text{ rad} /s)=4.1 \text{ rad} /sFinalize As expected, the angular speed increases. The fastest that this system could spin would be when the student moves to the center of the platform. Do this calculation to show that this maximum angular speed is 4.4 rad/s. Notice that the activity described in this problem is dangerous as discussed with regard to the Coriolis force in Section 6.3.