Chapter 16, Problem 35A

To determine

(a)

The depth of cut for the given piece of round stock.

Answer to Problem 35A

Depth of cut for the given piece of round stock should be $9.02$ mm.

Explanation of Solution

Given information:

The formula for calculating the depth of round stock piece is given by -  $C=\frac{D}{2}-0.5\times \sqrt{4\times {\left(\frac{D}{2}\right)}^2-F^2}$

Where, $C=\text{depth of cut, }D=\text{diameter, }F=\text{length of flat}$

  

Also, given that - $D=34.80\text{ mm, }F=30.50\text{ mm}$.

Calculation:

As per the problem statement, the formula for calculating the depth of round stock piece is given by -  $C=\frac{D}{2}-0.5\times \sqrt{4\times {\left(\frac{D}{2}\right)}^2-F^2}$

Here, we have been given $D=34.80\text{ mm, }F=30.50\text{ mm}$.

Putting above values in given formula for depth of cut and following order of operation as below.

  $\begin{array}{l}C=\frac{D}{2}-0.5\times \sqrt{4\times {\left(\frac{D}{2}\right)}^2-F^2}\\ \Rightarrow C=\frac{34.80}{2}-0.5\times \sqrt{4\times {\left(\frac{34.80}{2}\right)}^2-{30.50}^2}\text{ {Divide within parentheses}}\\ \Rightarrow C=\frac{34.80}{2}-0.5\times \sqrt{4\times {\left(17.4\right)}^2-{30.50}^2}\text{ {Square under radical sign}}\\ \Rightarrow C=\frac{34.80}{2}-0.5\times \sqrt{4\times 302.76-930.25}\text{ {Multiply within radical sign} }\\ \Rightarrow C=\frac{34.80}{2}-0.5\times \sqrt{1211.04-930.25}\text{ {Subtract within radical sign}}\\ \Rightarrow C=\frac{34.80}{2}-0.5\times \sqrt{280.79}\text{ {Square root}}\\ \Rightarrow C=\frac{34.80}{2}-0.5\times 16.75679\text{ {Divide}}\\ \Rightarrow C=17.4-0.5\times 16.75679\text{ {Multiply}}\\ \Rightarrow C=17.4-8.37839\text{ {Subtract}}\\ \Rightarrow C=9.02160\\ \Rightarrow C=9.02\text{ {Rounding off to 2 decimal places}}\end{array}$

Hence, depth of cut for the given piece of round stock should be $9.02$ mm.

To determine

(b)

To evaluate required depth of cut for the given piece of round stock.

Answer to Problem 35A

Depth of cut for the given piece of round stock should be $8.74$ mm.

Explanation of Solution

Given information:

The formula for calculating the depth of round stock piece is given by -  $C=\frac{D}{2}-0.5\times \sqrt{4\times {\left(\frac{D}{2}\right)}^2-F^2}$

Where, $C=\text{depth of cut, }D=\text{diameter, }F=\text{length of flat}$

  

Also, given that - $D=55.90\text{ mm, }F=40.60\text{ mm}$.

Calculation:

As per the problem statement, the formula for calculating the depth of round stock piece is given by -  $C=\frac{D}{2}-0.5\times \sqrt{4\times {\left(\frac{D}{2}\right)}^2-F^2}$

Here, we have been given $D=55.90\text{ mm, }F=40.60\text{ mm}$.

Putting above values in given formula for depth of cut and following order of operation as below.

  $\begin{array}{l}C=\frac{D}{2}-0.5\times \sqrt{4\times {\left(\frac{D}{2}\right)}^2-F^2}\\ \Rightarrow C=\frac{55.90}{2}-0.5\times \sqrt{4\times {\left(\frac{55.90}{2}\right)}^2-{40.60}^2}\text{ {Divide within parentheses}}\\ \Rightarrow C=\frac{34.80}{2}-0.5\times \sqrt{4\times {\left(27.95\right)}^2-{40.60}^2}\text{ {Square under radical sign}}\\ \Rightarrow C=\frac{34.80}{2}-0.5\times \sqrt{4\times 781.2025-1648.36}\text{ {Multiply within radical sign} }\\ \Rightarrow C=\frac{34.80}{2}-0.5\times \sqrt{3124.81-1648.36}\text{ {Subtract within radical sign}}\\ \Rightarrow C=\frac{34.80}{2}-0.5\times \sqrt{1476.45}\text{ {Square root}}\\ \Rightarrow C=\frac{34.80}{2}-0.5\times 16.75679\text{ {Divide}}\\ \Rightarrow C=27.95-0.5\times 16.75679\text{ {Multiply}}\\ \Rightarrow C=27.95-19.2123\text{ {Subtract}}\\ \Rightarrow C=8.7377\\ \Rightarrow C=8.74\text{ {Rounding off to 2 decimal places}}\end{array}$

Hence, depth of cut for the given piece of round stock should be $8.74$ mm.

To determine

(c)

To find out the required depth of cut for the given piece of round stock.

Answer to Problem 35A

Depth of cut for the given piece of round stock should be $5.48$ mm.

Explanation of Solution

Given information:

The formula for calculating the depth of round stock piece is given by -  $C=\frac{D}{2}-0.5\times \sqrt{4\times {\left(\frac{D}{2}\right)}^2-F^2}$

Where, $C=\text{depth of cut, }D=\text{diameter, }F=\text{length of flat}$

  

Also, given that - $D=91.40\text{ mm, }F=43.40\text{ mm}$.

Calculation:

As per the problem statement, the formula for calculating the depth of round stock piece is given by -  $C=\frac{D}{2}-0.5\times \sqrt{4\times {\left(\frac{D}{2}\right)}^2-F^2}$

Here, we have been given $D=91.40\text{ mm, }F=43.40\text{ mm}$.

Putting above values in given formula for depth of cut and following order of operation as below.

  $\begin{array}{l}C=\frac{D}{2}-0.5\times \sqrt{4\times {\left(\frac{D}{2}\right)}^2-F^2}\\ \Rightarrow C=\frac{91.40}{2}-0.5\times \sqrt{4\times {\left(\frac{91.40}{2}\right)}^2-{43.40}^2}\text{ {Divide within parentheses}}\\ \Rightarrow C=\frac{91.40}{2}-0.5\times \sqrt{4\times {\left(45.7\right)}^2-{43.40}^2}\text{ {Square under radical sign}}\\ \Rightarrow C=\frac{91.40}{2}-0.5\times \sqrt{4\times 2088.49-1883.56}\text{ {Multiply within radical sign} }\\ \Rightarrow C=\frac{91.40}{2}-0.5\times \sqrt{8353.96-1883.56}\text{ {Subtract within radical sign}}\\ \Rightarrow C=\frac{91.40}{2}-0.5\times \sqrt{6470.4}\text{ {Square root}}\\ \Rightarrow C=\frac{91.40}{2}-0.5\times 80.4388\text{ {Divide}}\\ \Rightarrow C=45.7-0.5\times 80.4388\text{ {Multiply}}\\ \Rightarrow C=45.7-40.2194\text{ {Subtract}}\\ \Rightarrow C=5.48060\\ \Rightarrow C=5.48\text{ {Rounding off to 2 decimal places}}\end{array}$

Hence, depth of cut for the given piece of round stock should be $5.48$ mm.