Answer
(a) $\frac{\omega_R}{\omega_F} = \frac{N_F}{N_R}$
(b) $\frac{\omega_R}{\omega_F} = 4.0$
(c) $\frac{\omega_R}{\omega_F} = 1.5$
Work Step by Step
(a) Let the linear speed be N teeth/s.
Note that the linear speed is the same for both sprockets. Therefore,
$\omega_R = (\frac{N~teeth/s}{N_R~teeth/rev})(2\pi ~rad/rev)$
$\omega_R = \frac{(2\pi)~N}{N_R}~rad/s$
Similarly, $\omega_F = \frac{(2\pi)~N}{N_F}~rad/s$
We can find the ratio of $\omega_R$ to $\omega_F$.
$\frac{\omega_R}{\omega_F} = \frac{N_F}{N_R}$
(b) $\frac{\omega_R}{\omega_F} = \frac{N_F}{N_R} = \frac{52}{13} = 4.0$
(c) $\frac{\omega_R}{\omega_F} = \frac{N_F}{N_R} = \frac{42}{28} = 1.5$