Physics 9702/52 — February/March 2016

Background Concept

The suggested model equates the energy transferred from gravitational potential energy (GPE) to elastic strain energy when the object reaches its lowest point.

• Loss in GPE over a fall distance $h$ is $mgh$, where $m$ is the object’s mass and $g$ is the acceleration of free fall.
• Elastic strain energy stored in a stretched cord (obeying Hooke’s law) is

$$E_e = \frac{1}{2}kx^2$$

where $k$ is the spring constant and $x$ is the extension beyond its natural (unstretched) length.

In this question the cord has natural length $L$, and at the lowest point the cord length is $h$, so extension is $x = h - L$. The suggested relationship is therefore

$$\frac{1}{2}k(h - L)^2 = mgh.$$

To test a relationship experimentally you usually (i) decide what to vary and measure, (ii) control the other important variables, then (iii) rearrange to a straight-line form so that a graph provides a clear test and lets you extract constants such as $g$.

Understanding the Question

You are asked to design a laboratory experiment using cords of the same unstretched length $L = 50.0\ \text{cm}$ but different spring constants $k$.

For each cord you must determine:

• $k$ (a property of that cord), and
• $h$ (the maximum distance the object falls).

Then you must explain how to use your results to:

• calculate values of $\dfrac{(h-L)^2}{h}$,
• plot a graph with that expression on the $y$-axis, and
• obtain $g$ from the graph.

The question also explicitly asks for: procedure, measurements, control of variables, analysis, and safety, plus a diagram of the arrangement.

Approach

A robust plan is:

1. Use a static calibration to find $k$ for each cord: apply known forces and measure extension, using Hooke’s law $F = kx$.
2. Use a consistent “bungee-style” release to measure the maximum fall distance $h$ for the same object mass $m$ for each cord.
3. Rearrange the given equation so that the required $y$-axis quantity is proportional to $1/k$. This suggests plotting $\dfrac{(h-L)^2}{h}$ against $\dfrac{1}{k}$.
4. The gradient gives $2mg$, so with known $m$ you can determine $g$.

Step-by-Step Reasoning

1) Measuring the spring constant $k$

Hooke’s law for a spring/cord in its linear region is:

$$F = kx.$$

So for each cord:

• Clamp the cord vertically.
• Measure its natural length $L$ (given as $50.0\ \text{cm}$; still good practice to verify).
• Hang a known mass $M$ so the applied force is $F = Mg$.
• Measure the new length and compute extension $x = (\text{stretched length}) - L$.
• Repeat for several masses so you can plot $F$ vs $x$.

A straight line indicates Hooke’s law applies; the gradient of the line is $k$.

Why do several loads? Because it reduces random error and lets you see if the cord stays in the elastic region (non-linearity suggests you are stretching too far).

2) Measuring the maximum distance fallen $h$

You need the maximum drop distance from the release point to the lowest point.

A practical method:

• Fix a metre rule or tape measure vertically next to the motion.
• Put a clear pointer on the mass (or a thin horizontal marker) to read position.
• Release the mass from the same height each time (important control variable).
• Use video (phone on a tripod) to capture the motion; the lowest point is where the mass is momentarily at rest and changes direction.

Alternative marking method (if allowed): place a light card/marker that just gets touched at the lowest point; adjust until it just touches, then read $h$. Video is usually more reliable.

Repeat 3+ times and average $h$ for each cord because the turning point can vary slightly.

3) Linearising and graph choice

Starting with

$$\frac{1}{2}k(h - L)^2 = mgh,$$

divide both sides by $h$ and rearrange:

$$\frac{(h-L)^2}{h} = \frac{2mg}{k}.$$

This is of the form:

$$y = (2mg)x$$

if you choose

$$y = \frac{(h-L)^2}{h}, \qquad x = \frac{1}{k}.$$

So:

• compute $y$ for each cord using your measured $h$ and known $L$,
• compute $x = 1/k$ for each cord from the calibration,
• plot $y$ against $x$.

If the model is correct, the plot is a straight line through the origin. The gradient is:

$$\text{gradient} = 2mg.$$

Therefore, using the known mass $m$:

$$g = \frac{\text{gradient}}{2m}.$$

(Units check: $y$ has units of metres; $x$ has units of $\text{N}^{-1}\text{ m}$ because $k$ is $\text{N m}^{-1}$. So gradient has units $\text{N}$, matching $2mg$.)

4) Control of variables

To make it a fair test of how $h$ depends on $k$:

• Keep $m$ the same object each time.
• Use cords with the same $L$ and measure $L$ consistently.
• Use the same release height and method (release from rest, no push).
• Keep the measuring scale aligned and fixed (avoid parallax).
• Keep the cord vertical and the mass not swinging (release carefully; repeat trials).

5) Safety

Main hazards: falling mass, cord snapping, and stand tipping.

• Clamp and weight the stand; check all clamps are tight.
• Keep people clear below the mass; use a soft landing tray/mat.
• Do not overstretch; test cords for damage first; wear eye protection.

Key Takeaways

• Find $k$ experimentally using Hooke’s law ($F$ vs $x$).
• Measure a turning-point distance ($h$) reliably, ideally with video.
• Linearise a relationship to a straight-line graph; choose axes so the gradient contains the constant you want.
• Use the gradient of the graph to determine $g$ via $g = \text{gradient}/(2m)$.

Common Mistakes

• Plotting $\dfrac{(h-L)^2}{h}$ against $k$ instead of against $1/k$ (would give an inverse curve, not a straight line).
• Not controlling the release height; changing the start point changes $h$.
• Measuring extension from the wrong reference (confusing $h$ with $h-L$).
• Using only one load to find $k$ (large uncertainty and no check of Hooke’s law).
• Stretching beyond the elastic limit so $k$ is not constant.

Things to Be Careful About

• Use consistent units (convert cm to m before calculations of $y$ so the graph units are correct).
• Define $h$ clearly as the distance fallen from the release point to the lowest point.
• Avoid parallax when reading rulers; keep the scale close to the motion.
• Ensure the mass does not swing; sideways motion makes $h$ difficult to measure accurately.
• When drawing the best-fit line, use a large triangle to calculate the gradient for reduced percentage uncertainty.