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## Q. 3.10

Written Communication

A mechanical engineer was running some tests to validate the spring constant of a new spring (part #C134). A mass was placed on a spring, and the resulting compression displacement was measured.

## Verified Solution

Hooke’s Law (discussed more in Chapter 5) states that the force exerted on a spring is proportional to the displacement of the spring. This can be expressed by

$F = kx$

Where F is the applied force, x is the displacement, and k is the spring constant. The data was recorded in the following table in SI units.

mass           displacement
0.01                      0.0245
0.02                       0.046
0.03                       0.067
0.04                        0.091
0.05                        0.114
0.06                        0.135
0.07                        0.156
0.08                         0.1805
0.09                       0.207
0.1                         0.231

The engineer is tasked with developing a professional table and graph that communicates the data and explains the Hooke’s Law relationship for the spring.
First, the engineer needs to calculate the resulting force from the applied mass using $w$ = mg and construct a table that illustrates the force and displacement data.

Note the following best practices regarding Table 3.7.

Table 3.7
Results of Spring Test Data

 Mass (kg) Force (N) Displacement (m) 0.01 0.098 0.0245 0.02 0.196 0.0460 0.03 0.294 0.0670 0.04 0.392 0.0910 0.05 0.490 0.1140 0.06 0.588 0.1350 0.07 0.686 0.1560 0.08 0.784 0.1805 0.09 0.882 0.2070 0.10 0.980 0.2310

• The engineer has added the calculated force values
• Units for each column have been added
• Appropriate borders to separate the data have been added
• The number of significant digits in each column is now consistent
• The headings are capitalized and bolded
• The data is aligned to make each column easy to read
Second, the engineer must communicate the spring rate relationship in the data table. A scatter plot is chosen and created in Figure 3.11. This graph effectively illustrates the relationship between force and displacement and demonstrates how well the data aligns with the linear relationship predicted by Hooke’s Law.

Note the best practices regarding Figure 3.11:
• The axes are clearly labeled, including appropriate units
• A descriptive title accompanies the graph
• A trend line clearly demonstrates the linear relationship between the variables
• The number of gridlines is minimal and used only for visual aids
• The data spans the axes, eliminating large areas of empty space in the graph.

Using the table and graph, the engineer can quickly estimate and communicate the spring constant for the spring as 4 N/m and validate that against the design requirements. 