Chapter 4.1, Problem 47E

Solution

To identify: how many teeth cross the cutting surface each second.

The teeth cross the cutting surface each second is $12571.44$

Given information:

A radial arm saw has a circular cutting blade with a diameter of 10 in. It spins at 2000 rpm. If there are 12 cutting teeth per inch on the cutting blade.

  

Explanation:

To determine how many teeth cross the cutting surface each second by a radial arm saw which has a circular cutting blade with a diameter of 10inches and it spins at 2000 revolutions per minute (rpm), if it has 12 cutting teeth per inch on the cutting blade

Recall that circumference of a circle is $2\pi r$

where r is the radius of the circle:

  $\begin{array}{l}2r=10\\ r=5\end{array}$

Therefore, circumference C of the circular cutting blade is equal to:

  $\begin{array}{c}C=2r\pi \\ =2\left(5\right)\left(3.1428\right)\\ =31.428\text{ inches}\end{array}$

Then total number of blades on the circumference of the circular cutting blade will be:

  $31.428\times 12=377.136$ .

As, there are 12cutting teeth per inch on the cutting blade.

As the circular blade spins at 2000 revolutions per minute (rpm).

Convert 2000 revolution per minute (rpm) to revolution per second (rps)by dividing it by 60 as 1 minute is equal to 60 seconds.

Thus,

  $\begin{array}{c}2000\text{rpm}=\frac{2000}{60}\\ =\frac{200}{6}\text{rpm}\end{array}$

Hence number to teeth crossing the cutting surface each second is equal to $377.136\times \frac{200}{6}=12571.44$ .