Chapter 31, Problem 26A

The hole locations of the block in Figure 31-11 are checked by placing the block on a surface plate and indicating the bottom of each hole using a height gage with an indicator attachment. Determine the height gage settings from the bottom of the part to the bottom of the holes. Assume that the actual hole diameters and locations are the same as the given dimensions. The setting for Hole Number 1 is given.

To determine

(1)

The main scale setting.

The vernier scale setting.

Answer to Problem 26A

The main scale setting is $0.450^{″}-0.475^{″}$.

The vernier scale setting is $2$.

Explanation of Solution

Given information:

The diameter of the hole $1$ is $0.376^{″}$ and location of center of hole is $0.640^{″}$.

The figure below shows the hole locations from surface plate.

Figure-(1)

Write the expression for the height gauge settings.

  $\text{HGS}=\text{A}-\frac{d_1}{2}$ ...... (I)

Here, the distance between the center of the holes is $\text{A}$, the diameter of hole $1$ is $d_1$.

Write the expression for the relationship between vernier scale setting and main scale setting.

  $\text{HGS}=\text{MSS}+\left(\text{VSS}\times \text{LC}\right)$ ...... (II)

Here, the height gauge setting is $\text{HGS}$, the main scale setting is $\text{MSS}$, the vernier scale setting is $\text{VSS}$, and the least count is $\text{LC}$

Calculation:

Substitute $0.640^{″}$ for $\text{A}$ and $0.376^{″}$ for $d_1$ in Equation (I).

  $\begin{array}{c}\text{HGS}=0.640^{″}-\frac{0.376^{″}}{2}\\ =0.640^{″}-0.188^{″}\\ =0.452^{″}\end{array}$

The $\text{HGS}$ value $0.452^{″}$ is lie on main scale in range of $0.450^{″}-0.475^{″}$.

Hence, the main scale setting is $0.450^{″}-0.475^{″}$.

As the lower value is required for the measurement, taking $0.4850"$ for $\text{MSS}$.

Substitute $0.452^{″}$ for $\text{HGS}$, $0.001^{″}$ for $\text{LC}$, and $0.452^{″}$ for $\text{MSS}$ in Equation (II).

  $\begin{array}{l}\text{0}\text{.45<msup><mo>2</mo><mo>″</mo></msup>}=0.450^{″}+\left(\text{VSS}\times 0.001^{″}\right)\\ 0.002^{″}=\text{VSS}\times 0.001^{″}\\ \text{VSS}=2\end{array}$

Hence, the vernier scale setting is $2$.

Conclusion:

The main scale setting is $0.450^{″}-0.475^{″}$.

The vernier scale setting is $2$.

To determine

(2)

The main scale setting.

The vernier scale setting.

Answer to Problem 26A

The main scale setting is $1.500^{″}-1.525^{″}$.

The vernier scale setting is $19$.

Explanation of Solution

Given information:

The diameter of the hole $2$ is $0.258^{″}$ and distance $\text{B}$ is $1.008^{″}$.

Write the expression for the height gauge settings.

  $\text{HGS}=\text{A}+\text{B}-\frac{d_2}{2}$ ...... (III)

Here, the distance between the centers of hole $1$ and hole $2$ is $\text{B}$ and the diameter of hole $2$ is $d_2$.

Calculation:

Substitute $0.640^{″}$ for $\text{A}$, $1.008^{″}$ for $\text{B}$, and $0.258^{″}$ for $d_2$ in Equation (III).

  $\begin{array}{c}\text{HGS}=0.640^{″}+1.008^{″}-\frac{0.258^{″}}{2}\\ =0.640^{″}+1.008^{″}-0.129^{″}\\ =1.519^{″}\end{array}$

The $\text{HGS}$ value $1.519^{″}$ is lie on main scale in range of $1.500^{″}-1.525^{″}$.

Hence, the main scale setting is $1.500^{″}-1.525^{″}$.

As the lower value is required for the measurement, taking $1.500^{″}$ for $\text{MSS}$.

Substitute $1.519^{″}$ for $\text{HGS}$, $0.001^{″}$ for $\text{LC}$, and $1.500^{″}$ for $\text{MSS}$ in Equation (II).

  $\begin{array}{l}1.519^{″}=1.500^{″}+\left(\text{VSS}\times 0.001^{″}\right)\\ 0.019^{″}=\text{VSS}\times 0.001^{″}\\ \text{VSS}=19\end{array}$

Hence, the vernier scale setting is $19$.

Conclusion:

The main scale setting is $1.500^{″}-1.525^{″}$.

The vernier scale setting is $19$.

To determine

(3)

The main scale setting.

The vernier scale setting.

Answer to Problem 26A

The main scale setting is $1.025^{″}-1.050^{″}$.

The vernier scale setting is $15$.

Explanation of Solution

Given information:

The diameter of the hole $3$ is $0.188^{″}$ and distance between centers of hole $2$ and hole $3$ is $0.514^{″}$.

Write the expression for the height gauge settings.

  $\text{HGS}=\text{A}+\text{B}-\text{C}-\frac{d_3}{2}$ ...... (IV)

Here, the distance between the centers of hole $2$ and hole $3$ is $\text{C}$, and the diameter of hole $3$ is $d_3$.

Calculation:

Substitute $0.640^{″}$ for $\text{A}$, $1.008^{″}$ for $\text{B}$, $0.514^{″}$ for $\text{C}$, and $0.188^{″}$ for $d_3$ in Equation (IV).

  $\begin{array}{c}\text{HGS}=0.640^{″}+1.008^{″}-0.514^{″}-\frac{0.188^{″}}{2}\\ =0.640^{″}+1.008^{″}-0.514^{″}-0.094^{″}\\ =1.648^{″}-0.608^{″}\\ =1.040^{″}\end{array}$

The $\text{HGS}$ value $1.040^{″}$ is lie on main scale in range of $1.025^{″}-1.050^{″}$.

Hence, the main scale setting is $1.025^{″}-1.050^{″}$.

As the lower value is required for the measurement, taking $1.025^{″}$ for $\text{MSS}$.

Substitute $1.040^{″}$ for $\text{HGS}$, $0.001^{″}$ for $\text{LC}$, and $1.025^{″}$ for $\text{MSS}$ in Equation (II).

  $\begin{array}{l}1.040^{″}=1.025^{″}+\left(\text{VSS}\times 0.001^{″}\right)\\ 0.015^{″}=\text{VSS}\times 0.001^{″}\\ \text{VSS}=15\end{array}$

Hence, the vernier scale setting is $15$.

Conclusion:

The main scale setting is $1.025^{″}-1.050^{″}$.

The vernier scale setting is $15$.

To determine

(4)

The main scale setting.

The vernier scale setting.

Answer to Problem 26A

The main scale setting is $0.700^{″}-0.725^{″}$.

The vernier scale setting is $4$.

Explanation of Solution

Given information:

The diameter of the hole $4$ is $0.496^{″}$ and distance between centers of hole $1$ and hole $4$ is $0.312^{″}$.

Write the expression for the height gauge settings.

  $\text{HGS}=\text{A}+\text{D}-\frac{d_4}{2}$ ...... (V)

Here, the distance between hole $1$ and hole $4$ is $\text{D}$, and the diameter of hole $4$ is $d_4$.

Calculation:

Substitute $0.640^{″}$ for $\text{A}$, $0.312^{″}$ for $\text{D}$, and $0.496^{″}$ for $d_4$ in Equation (V).

  $\begin{array}{c}\text{HGS}=0.640^{″}+0.312^{″}-\frac{0.496^{″}}{2}\\ =0.640^{″}+0.312^{″}-0.248^{″}\\ =0.952^{″}-0.248^{″}\\ =0.704^{″}\end{array}$

The $\text{HGS}$ value $0.704^{″}$ is lie on main scale in range of $0.700^{″}-0.725^{″}$.

Hence, the main scale setting is $0.700^{″}-0.725^{″}$.

As the lower value is required for the measurement, taking $0.700^{″}$ for $\text{MSS}$.

Substitute $0.704^{″}$ for $\text{HGS}$, $0.001^{″}$ for $\text{LC}$, and $0.700^{″}$ for $\text{MSS}$ in Equation (II).

  $\begin{array}{l}0.704^{″}=0.700^{″}+\left(\text{VSS}\times 0.001^{″}\right)\\ 0.004^{″}=\text{VSS}\times 0.001^{″}\\ \text{VSS}=4\end{array}$

Hence, the vernier scale setting is $4$.

Conclusion:

The main scale setting is $0.700^{″}-0.725^{″}$.

The vernier scale setting is $4$.

To determine

(5)

The main scale setting.

The vernier scale setting.

Answer to Problem 26A

The main scale setting is $1.500^{″}-1525^{″}$.

The vernier scale setting is $14$.

Explanation of Solution

Given information:

The diameter of the hole $5$ is $0.127^{″}$ and the distance between centers of hole $4$ and hole $5$ is $0.810^{″}$.

Write the expression for the height gauge settings.

  $\text{HGS}=\text{A}+\text{D}+\text{E}-\frac{d_5}{2}$ ...... (VI)

Here, the distance between hole $1$ and hole $4$ is $\text{D}$, the distance between centers of hole $4$ and hole $5$ is $\text{E}$, and the diameter of hole $5$ is $d_5$.

Calculation:

Substitute $0.640^{″}$ for $\text{A}$, $0.312^{″}$ for $\text{D}$, $0.810^{″}$ for $\text{E}$, and $0.127^{″}$ for $d_5$ in Equation (VI).

  $\begin{array}{c}\text{HGS}=0.640^{″}+0.312^{″}+0.810^{″}-\frac{0.127^{″}}{2}\\ =0.640^{″}+0.312^{″}+0.810^{″}-0.0635^{″}\\ =1.762^{″}-0.248^{″}\\ =1.514^{″}\end{array}$

The $\text{HGS}$ value $1.514^{″}$ is lie on main scale in range of $1.500^{″}-1525^{″}$.

Hence, the main scale setting is $1.500^{″}-1525^{″}$.

As the lower value is required for the measurement, taking $1.500^{″}$ for $\text{MSS}$.

Substitute $1.514^{″}$ for $\text{HGS}$, $0.001^{″}$ for $\text{LC}$, and $1.500^{″}$ for $\text{MSS}$ in Equation (II).

  $\begin{array}{l}1.514^{″}=1.500^{″}+\left(\text{VSS}\times 0.001^{″}\right)\\ 0.014^{″}=\text{VSS}\times 0.001^{″}\\ \text{VSS}=14\end{array}$

Hence, the vernier scale setting is $14$.

Conclusion:

The main scale setting is $1.500^{″}-1525^{″}$.

The vernier scale setting is $14$.