PAG: Investigating Force & Extension

The mathematical expression for this relationship is $F = kx$, where $F$ is the force in Newtons (N), $k$ is the spring constant in Newtons per meter (N/m), and $x$ is the extension in meters (m).

Elastic Potential Energy is stored in the material when it is stretched; this energy is equal to the work done on the material, calculated as the area under a Force-Extension graph.

The formula for stored energy within the linear region is $E_e = \frac{1}{2}Fx$ or $E_e = \frac{1}{2}kx^2$.

Linearity: As long as the material remains within its limit of proportionality, the graph of Force vs. Extension will be a straight line passing through the origin.

Initial Measurement: Secure the spring to a clamp stand and measure its natural length using a meter ruler, ensuring the ruler is vertical and level with the top of the spring.

Incremental Loading: Add masses (e.g., 100g increments) to the spring and record the new total length after each addition, allowing the spring to come to rest to avoid oscillations.

Calculating Extension: For each load, subtract the original length from the new length ($x = L_{new} - L_{original}$).

Data Verification: It is best practice to remove the masses one by one and record the lengths during unloading to check if the spring returns to its original length, indicating elastic behavior.

Graph Interpretation: When plotting Force (y-axis) against Extension (x-axis), the gradient of the linear section is equal to the spring constant $k$.

Unit Consistency: Always convert measurements from centimeters or millimeters to meters before calculating $k$ or energy to ensure the result is in standard SI units (N/m or Joules).

Origin Check: A Force-Extension graph must pass through $(0,0)$ because zero force must result in zero extension; if it doesn't, check for systematic errors in length measurement.

Safety: Wear eye protection when stretching springs or wires, as they store significant energy and can snap or recoil violently.

Parallax Error: Reading the ruler from an angle rather than at eye level leads to inaccurate length measurements; use a set square or a fiducial marker to improve precision.

Length vs. Extension: A common mistake is plotting the total length of the spring on the graph instead of the extension; this results in a graph that does not pass through the origin.

Exceeding the Limit: Adding too much weight can permanently damage the spring, making subsequent readings invalid for determining the original spring constant.