Chapter 4.1, Problem 74E

Solution

To identify: The angular speed, in radians per second, of the wheel and of both sprockets when Ben reaches his peak racing speed of in high gear. Ben Shultz’s bike wheels are 28 in. in diameter and for high gear the pedal sprocket is 9 in. in diameter and the wheel sprocket is 3 in. in diameter.

  

Angular speed of the wheel: $28.286\text{rad}/\text{s}$ .

Angular speed of the sprockets: $9.429\text{rad}/\text{s}$ .

Given information:

The given is Ben Shultz’s bike wheels are 28 in. in diameter, and for high gear the

Pedal sprocket is 9 in. in diameter and the wheel sprocket is 3 in. in diameter.

Explanation:

Consider the given information.

Definition of angular speed $w$ for some angular displacement $△\theta$ and sometime interval $\Delta t$ .

So, $w=\left|\frac{△\theta }{\Delta t}\right|$

Linear speed $v$ is defined as $v=\frac{\Delta s}{\Delta t}$ for some displacement along the circumference $\Delta s$ and some interval $\Delta t$ . Take note that $\Delta s=r△\theta$ by the definition of the radian measurement $\left(\theta =\frac{s}{r}\right)$ .

Substituting these two gives an expression of linear speed.

  $w=\left|\frac{v}{r}\right|$

Substituting the resultant equation from the second step into the first one gives an expression for our angular speed $w$ into which we can plug in our given values to evaluate (as shown in the following steps).

  $w=\left|\frac{792\text{ in}/\text{s}}{28\text{ in}}\right|=28.286\text{rad}/\text{s}$

Velocity of the bike v and radius of the wheels r to get angular speed of the wheel $w$ . Take note that a revolution is a dimensionless quantity since a radian is a dimensionless quantity.

  $\left|v\right|=\left|wr\right|=\left|\left(28.286\text{rad/s}\right)\left(3\text{in}\right)\right|=84.857\text{in/s}$

Angular speed of the wheel $w$ and radius of the wheel’s gear $r$ to get speed of the chain $v$ .

  $w=\left|\frac{84.857\text{in/s}}{9\text{in}}\right|=9.429\text{rad/s}$

Speed of the chain $v$ and radius of the sprocket’s chain $v$ to get angular speed of the sprocket $w$ .

So,

Angular speed of the wheel: $28.286\text{rad/s}$

Angular speed of the sprockets: $9.429\text{rad}/\text{s}$