Chapter 8, Problem 13P

To determine

The ratio of angular velocities ${\text{ω}}_{\text{1}}{\text{/ω}}_{\text{2}}$

Answer to Problem 13P

The ratio of angular velocities ${\text{ω}}_{\text{1}}{\text{/ω}}_{\text{2}}$ is equal to ${\text{R}}_{\text{2}}{\text{/R}}_{\text{1}}$

Explanation of Solution

Given:

Turntable of radius R₁ is in contact with the rubber roller of radius R₂ at their outer edges, so at their outer edge tangential speed of turntable must be equal to tangential speed of the roller.

  $\text{So, v (turntable) = v (roller)}$

Formula used:

The relation between the radius, angular velocity and tangential speed is

  $\text{v = Rω}$

Calculation:

Substitute the given values,

  $\begin{array}{l}\text{v (turntable) = v (roller)}\\ {\text{R}}_{\text{1}}\times {\text{ω}}_{\text{1}}{\text{ = R}}_{\text{2 }}\times {\text{ω}}_{\text{2}}\\ \text{So, }\frac{{\text{ω}}_{\text{1}}}{{\text{ω}}_{\text{2}}}\text{ = }\frac{{\text{R}}_{\text{2}}}{{\text{R}}_{\text{1}}}\end{array}$

Conclusion:

Thus, the ratio of angular velocities ${\text{ω}}_{\text{1}}{\text{/ω}}_{\text{2}}$ is equal to ${\text{R}}_{\text{2}}{\text{/R}}_{\text{1}}$