Chapter 8, Problem 73GP

(a)

To determine

The angular speed of the rear wheel ( ${\omega }_R$ ) related to $\left({\omega }_F\right)$ .

Explanation of Solution

Introduction:

To relate the angular speed of the rear wheel of the bicycle to that of pedals and front sprocket, compare the angular velocity of the rear wheel to that of the front wheel.

The linear speed of the front wheel is given as,

  $v_F={\omega }_F.R_F$

Similarly, for the rear wheel,

  $v_R={\omega }_R.R_R$

The linear speed of the front wheel is equal to the linear speed of the rear wheel so, equating both the equations,

  ${\omega }_R.R_R={\omega }_F.R_F$

Separating the similar terms,

  $\frac{R_R}{R_R}=\frac{{\omega }_F}{{\omega }_R}$-------------------1

And given that the $N_F$ and $N_R$ are the number of teeth on the front wheel to the number of teeth on the rear wheel.

The diameter of the sprocket is d and the circumference of the sprocket

  $\begin{array}{l}2\pi R_F=N_{F}d\\ 2\pi R_R=N_{R}d\end{array}$

Deriving the above equations

  $\frac{R_R}{R_F}=\frac{N_R}{N_F}$--------------2

Solving both 1 and 2 equations,

  $\frac{{\omega }_R}{{\omega }_F}=\frac{N_F}{N_R}$

Conclusion: 

The ratio of the angular velocity of the rear wheel to the front wheel is $\frac{{\omega }_R}{{\omega }_F}=\frac{N_F}{N_R}$ .

(b)

To determine

To Evaluate: The ratio of $\frac{{\omega }_R}{{\omega }_F}$ when front and rear sprockets have 52 and 13 teeth

Answer to Problem 73GP

  $\frac{{\omega }_R}{{\omega }_F}=4$

Explanation of Solution

Given data:

The teeth on rear sprocket ( $N_R$ )= 13

The teeth on front sprocket ( $N_F$ )= 52

Formula used:

The relation between the ratios of the angular velocity of the rear wheel to the front wheel and the number of teeth on the rear and front is given as,

  $\frac{{\omega }_R}{{\omega }_F}=\frac{N_F}{N_R}$

Calculation:

Substitute the values of teeth on the front $N_F$ _, and rear $N_R$ ,

  $\begin{array}{l}\frac{{\omega }_R}{{\omega }_F}=\frac{52}{13}\\ \frac{{\omega }_R}{{\omega }_F}=4\end{array}$

Conclusion:

The ratio of the angular velocity of the rear sprocket to the front sprocket is 4.

(c)

To determine

To Evaluate: The ratio of $\frac{{\omega }_R}{{\omega }_F}$ when front and rear sprockets have 42 and 28 teeth.

Answer to Problem 73GP

  $\frac{{\omega }_R}{{\omega }_F}=1.5$

Explanation of Solution

Given data:

The teeth on r0ear sprocket ( $N_R$ )

= 28

The teeth on front sprocket ( $N_F$ )

= 42

Formula used:

  $\frac{{\omega }_R}{{\omega }_F}=\frac{N_F}{N_R}$

Calculation:

The relation between the ratios of the angular velocity of the rear wheel to the front wheel and the number of teeth on the rear and front is given as,

  $\frac{{\omega }_R}{{\omega }_F}=\frac{N_F}{N_R}$

Substitute the values of teeth on front $N_F$ and rear $N_R$ ,

  $\begin{array}{l}\frac{{\omega }_R}{{\omega }_F}=\frac{42}{28}\\ \frac{{\omega }_R}{{\omega }_F}=1.5\end{array}$

Conclusion: 

The ratio of the angular velocity of the rear sprocket to the front sprocket is 1.5.