Question

Short answer

The ratio of the angular velocities is\(\frac{{{\omega _1}}}{{{\omega _2}}} = \frac{{{R_2}}}{{{R_1}}}\).

Step-by-step solution

Given data

The radius of the turntable is\({R_1}\).

The radius of the roller is\({R_2}\).

Understanding angular velocities

When there is no slipping between the turntable and the rubber roller, the tangential speed of the table will be equivalent to the tangential speed of the roller.

Determine the ratio of the angular velocities

Since the speeds of the roller and table are similar, the relation to calculate the ratio of the angular velocities is given as follows:

\(\begin{aligned}{c}{v_1} &= {v_2}\\{\omega _1}{R_1} &= {\omega _2}{R_2}\\\frac{{{\omega _1}}}{{{\omega _2}}} &= \frac{{{R_2}}}{{{R_1}}}\end{aligned}\)

Thus, \(\frac{{{\omega _1}}}{{{\omega _2}}} = \frac{{{R_2}}}{{{R_1}}}\) is the required ratio.