PAG: Investigating Force & Extension

Force ($F$): In this context, force is the load applied to the spring, typically provided by the weight of suspended masses, measured in Newtons (N). It is calculated using $W = mg$, where $m$ is mass in kg and $g$ is the gravitational field strength.

Extension ($x$ or $\Delta L$): This is the increase in length of the spring from its original, unstretched position. It is calculated by subtracting the original length ($L_0$) from the new length ($L_n$) after a load is applied: $x = L_n - L_0$.

Spring Constant ($k$): A measure of the stiffness of the spring, representing the force required per unit of extension. A higher spring constant indicates a stiffer spring that requires more force to stretch.

Hooke's Law: The principle stating that the extension of a spring is directly proportional to the force applied, provided the limit of proportionality is not exceeded. This is expressed mathematically as $F = kx$.

Length vs. Extension: It is critical to distinguish between the total length of the spring and the extension. Hooke's Law relates force to the change in length, not the absolute length.

Spring Constant vs. Gradient: If Force is plotted on the y-axis and Extension on the x-axis, the gradient of the linear section equals the spring constant ($k$). If the axes are swapped, the gradient is $1/k$.

Gradient Calculation: When asked to find the spring constant from a graph, always use the largest possible triangle on the linear section of the line. Ensure you use the formula $k = \frac{\Delta F}{\Delta x}$.

Unit Consistency: Check that extension is converted to meters (m) if the spring constant is required in N/m. Many practical measurements are taken in mm or cm, which must be scaled by $10^{-3}$ or $10^{-2}$ respectively.

Origin Check: A Force-Extension graph for a spring obeying Hooke's Law must pass through the origin $(0,0)$. If it does not, it may indicate a systematic error in measuring the original length.

Sanity Check: If a calculated spring constant is extremely high (e.g., millions) for a small laboratory spring, verify that you haven't confused mass (g) with force (N) or failed to convert cm to m.

Parallax Error: This occurs when the measurement on the ruler is read from an angle rather than at eye level. To avoid this, always ensure your line of sight is perpendicular to the ruler and the fiducial marker.

Mass vs. Weight: Students often plot mass in grams on the y-axis. You must convert mass to weight ($W = mg$) to get the force in Newtons, using $g \approx 9.81 \text{ N/kg}$.

Exceeding the Limit: If the spring is loaded too heavily, it will 'overstretch.' This is visible on a graph when the line begins to curve towards the extension axis, indicating that the spring constant is no longer constant.