Overview
This lecture covers the two main equations related to elasticity, explains the spring constant and elastic potential energy, and demonstrates their use with examples.
Elasticity Equations
- The equation ( F = k e ) links force (F) to extension (e), with k as the spring constant.
- The equation ( \text{Elastic Potential Energy} = \frac{1}{2} k e^2 ) calculates energy stored in a stretched spring.
- Only the extension (e) is squared in the elastic potential energy equation, not the entire term.
Spring Constant
- The spring constant (k) measures how stiff or elastic a spring is.
- A low k value means the spring is more elastic (easier to stretch); a high k means it is stiffer.
- To find k: measure the extension (new length - natural length), then use ( k = F / e ).
Elastic Potential Energy Example
- Using the equations: a spring stretched from 0.6m to 0.8m by a 14N force has an extension of 0.2m.
- The spring constant is ( k = 14/0.2 = 70 ) N/m.
- The elastic potential energy becomes ( 0.5 \times 70 \times (0.2)^2 = 1.4 ) joules.
Interpreting Force-Extension Graphs
- The gradient (slope) of the straight section of a force-extension graph equals the spring constant.
- The area under the curve represents the energy stored in the spring (elastic potential energy).
- The elastic limit (limit of proportionality) is the point where the spring stops obeying Hooke’s Law.
Key Terms & Definitions
- Spring Constant (k) — a measure of how much force is needed to stretch or compress a spring.
- Extension (e) — the change in length from a spring’s natural length when force is applied.
- Elastic Potential Energy — energy stored in an object when it is stretched or compressed.
- Elastic Limit/Limit of Proportionality — the maximum extension where Hooke’s Law is still valid.
Action Items / Next Steps
- Practice solving problems using ( F = k e ) and ( \text{Elastic Potential Energy} = \frac{1}{2} k e^2 ).
- Review the concepts of force-extension graphs and identify the spring constant and elastic limit.