1.5.1 Effects of Forces
Key Concepts ⚙️
- When a force \$F\$ is applied to an elastic solid, it stretches by an amount \$x\$.
- For small extensions, the relationship is linear: \$F = kx\$ (Hooke’s Law).
- \$k\$ is the spring constant, a measure of stiffness.
- The plot of \$F\$ vs. \$x\$ is a straight line in the elastic region.
- Beyond the elastic limit, the material may deform permanently.
Analogy & Example 🎯
Think of a rubber band stretched between your fingers. The further you pull, the more force you feel. If you pull just enough, the band snaps back to its original length—this is the elastic region. Pull it too far, and it may not return—this is beyond the elastic limit.
Experimental Procedure 🧪
- Set up a vertical spring attached to a fixed support.
- Attach a mass hanger to the end of the spring.
- Measure the initial length \$L_0\$ of the spring with no load.
- Add a known mass \$m\$ to the hanger and record the new length \$L\$.
- Calculate the extension \$x = L - L_0\$.
- Compute the applied force \$F = mg\$ (using \$g \approx 9.81\,\text{m/s}^2\$).
- Repeat for different masses to get a set of \$(x, F)\$ pairs.
- Plot \$F\$ (y‑axis) against \$x\$ (x‑axis) and draw the best‑fit straight line.
- Determine the slope of the line; this is the spring constant \$k\$.
Load‑Extension Graph 📊
The graph should look like this (conceptually):
\$F\$ (N) ↗ \$x\$ (m)
Key points to note:
- The initial point (0,0) represents no load.
- The slope of the linear section equals \$k\$.
- Any curvature or deviation indicates the material is leaving the elastic region.
Sample Data Table 📚
| Mass (kg) | Extension (m) | Force \$F\$ (N) |
|---|
| 0.50 | 0.025 | 4.9 |
| 1.00 | 0.050 | 9.8 |
| 1.50 | 0.075 | 14.7 |
From the table, \$k = \frac{F}{x} \approx 196\,\text{N/m}\$.
Exam Tips 📑
- Show a clear sketch of the load‑extension graph with labels and a straight‑line fit.
- Explain how the slope gives the spring constant and why the graph is linear only in the elastic region.
- Remember to use \$g = 9.81\,\text{m/s}^2\$ when converting mass to force.
- Discuss what happens if the material is stretched beyond its elastic limit.
- Use appropriate units and check dimensional consistency.