Let the radius of the rear sprocket is\({r_r}\)
Tangential speed,
\(v = r\omega \)
Where\(\omega \)is angular velocity or speed and r is radius.
Since rear wheel and rear shock put are rotating together, so their angular speeds are equal,
Therefore, angular speed of rear sprocket can be determined by using the formula.
\(\begin{aligned}{}{v_{rw}} &= {R_{rw}}{\omega _r}\\5.00\,\,{m \mathord{\left/{\vphantom {m s}} \right.} s} &= \left( {0.330\,\,m} \right){\omega _r}\\{\omega _r} &= 15.15\,\,{{rad} \mathord{\left/{\vphantom {{rad} s}} \right.} s}\end{aligned}\)
So angular speed of the rear sprocket is \({\omega _r} = 15.15\,\,{{rad} \mathord{\left/{\vphantom {{rad} s}} \right.} s}\)
Since the front sprocket and rear sprocket are connected, therefore they will have same tangential speed,
\(\begin{aligned}{}{v_r} &= {v_f}\\{r_r}{\omega _r} &= {r_f}{\omega _f}\\{r_r}\left( {15.15\,\,{{rad} \mathord{\left/{\vphantom {{rad} s}} \right.} s}} \right) &= \left( {12\,\,cm} \right)\left( {3.768\,\,\,{{rad} \mathord{\left/{\vphantom {{rad} s}} \right.} s}\,} \right)\\{r_r} &= 2.98\,\,cm\end{aligned}\)
Therefore, the radius of the rear sprocket is 2.98 cm .