# Force-Extension Graphs ## Understanding Force-Extension Graphs - Force-extension graphs display the relationship between applied force and resulting extension for different materials - Each material produces a characteristic graph shape revealing its mechanical behaviour - The graph is a powerful diagnostic tool showing how materials respond under tension ## Materials Obeying Hooke's Law - Some materials obey Hooke's Law up to a certain point - This region appears as a **straight line passing through the origin** - Force and extension are directly proportional in this region - Beyond this point, the graph begins to curve - The material no longer follows Hooke's Law - Behaviour becomes non-linear

The limit of proportionality is the point beyond which force and extension are no longer directly proportional

## Key Features on Force-Extension Graphs ### Limit of Proportionality - The point where the straight-line region ends and the graph begins to curve - Beyond this point, Hooke's Law no longer applies - Extension is no longer proportional to applied force - Identified visually as the point where the line begins to flatten or curve ### Force Constant (Spring Constant) - Represented by the symbol $k$, measured in N m$^{-1}$ - Equals the **gradient of the straight-line portion** of the graph - $k = \frac{\Delta F}{\Delta x}$ - Only valid up to the limit of proportionality - Larger $k$ values indicate stiffer materials ### Elastic Limit - The maximum force a material can withstand while still returning to its original length - Always occurs **after** the limit of proportionality - Beyond this point, permanent deformation occurs - Material will not fully recover when force is removed

The elastic limit and limit of proportionality are close together but not identical points

## Force-Extension Behaviour of Different Materials ### Metal Wires - Metal wires exhibit both elastic and plastic behaviour - **Elastic region**: - Wire obeys Hooke's Law - Loading and unloading curves follow the same path - Wire returns to original length when force removed - **Beyond elastic limit**: - Plastic deformation begins - Wire undergoes permanent extension - Unloading curve has the same gradient as loading curve - Unloading curve is parallel to loading curve but shifted horizontally - This horizontal shift represents permanent extension The force-extension graph for a typical metal wire shows a clear linear region followed by plastic deformation:

| Phase | Behaviour | Graph Feature | |-------|-----------|---------------| | Elastic | Reversible deformation | Straight line, loading = unloading | | Plastic | Permanent deformation | Curved line, unloading parallel to loading | ### Rubber Bands - Rubber is an elastic material that does **not** obey Hooke's Law - Shows significant non-linear behaviour from the start - Creates a **hysteresis loop**: - Loading curve (stretching) follows one path - Unloading curve (contracting) follows a different path below the loading curve - The enclosed area represents energy "lost" to thermal energy - Area between curves = thermal energy dissipated during the stretch-release cycle - Area $(X + Y)$ under loading curve = minimum energy required to stretch to extension $e$ - Area $X$ alone = net work done or thermal energy released The hysteresis loop for rubber shows the loading and unloading curves:

The hysteresis loop indicates energy is dissipated as heat when rubber is repeatedly stretched and released

### Polymeric Materials (Polythene) - Polymers like polythene do not obey Hooke's Law at any point - Experience plastic deformation immediately when force is applied - Easy to stretch into new shapes due to molecular chain structure - Difficult to return to original shape once deformed - Graph shows continuous curve with no linear region - Very useful for applications requiring permanent reshaping (e.g., plastic bottles, bags) ## Comparing Material Behaviours The force-extension characteristics of different materials can be compared on the same graph:

| Material | Obeys Hooke's Law? | Elastic Region? | Plastic Region? | Key Characteristic | |----------|-------------------|-----------------|-----------------|-------------------| | Metal wire | Yes (initially) | Yes | Yes | Linear then plastic deformation | | Rubber | No | Yes | No | Large hysteresis loop | | Polythene | No | Minimal | Yes | Immediate plastic deformation | ## Investigating Force-Extension Characteristics Experimentally ### Aim - To investigate the relationship between applied force and extension for various materials - Materials tested: metal springs, rubber bands, polythene strips - Determine whether materials obey Hooke's Law - Identify elastic limits and breaking points ### Variables - **Independent variable**: Force/Load (N) - **Dependent variable**: Extension (m) - **Control variables**: - Original length of material - Cross-sectional area/thickness of material - Type of material in each test ### Required Equipment | Equipment | Purpose | |-----------|---------| | Clamp stand, clamp and boss | Secure apparatus and suspend materials | | G-clamp | Fix clamp stand to bench securely | | Metre ruler (resolution 1 mm) | Measure extension accurately | | Metal spring, rubber band, polythene strip | Materials to investigate | | Pulley | Reduce friction, allow masses to hang vertically | | Mass hanger + 100g slotted masses | Provide controlled, incremental forces | | Fiducial marker (tape) | Mark reference point for measuring extension | | Set square | Ensure ruler is perpendicular to bench | ### Experimental Method 1. **Measure and record the original length** - Clamp material vertically from stand - Secure metre ruler vertically alongside using set square to ensure perpendicularity - Mark a reference point near the bottom of the material with tape (fiducial marker) - Record initial ruler reading at the marker with no load applied 2. **Apply incremental loads** - Attach mass hanger to bottom of material - Add a 100 g mass to the hanger - Allow material to come to rest (avoid oscillations) - Record new ruler reading at the fiducial marker 3. **Repeat measurements** - Continue adding 100 g masses one at a time (at least 5-10 readings) - Record ruler reading after each addition - Continue until material shows signs of permanent deformation or approaches breaking point 4. **Calculate extensions** - Extension = Final reading - Initial reading - Record all data in a table 5. **Repeat for different materials** - Conduct identical procedure for rubber band - Conduct identical procedure for polythene strip - Compare behaviours ### Analysing Results 1. **Calculate force applied** - Force $F = mg$ where $m$ is total mass (in kg) and $g = 9.81$ N kg$^{-1}$ 2. **Plot force-extension graph** - Force (N) on y-axis - Extension (m) on x-axis - Plot points and draw line of best fit 3. **Determine spring constant** - According to Hooke's Law: $F = kx$ - Comparing with $y = mx$: gradient = spring constant $k$ - Calculate gradient using: $k = \frac{\Delta F}{\Delta x}$ from the **straight-line section only** - Only use linear region (up to limit of proportionality) 4. **Identify key points** - Limit of proportionality: where line begins to curve - Elastic limit: last point before permanent deformation (if observable) - Compare graphs for different materials ### Minimising Errors **Systematic errors:** - Use set square to ensure ruler is perpendicular to bench - Read ruler at eye level to avoid parallax error - Use fiducial marker touching ruler for consistent reading point **Random errors:** - Ensure material is stationary before taking readings - Repeat experiment several times and calculate mean extension values - Take multiple diameter measurements (if needed) at different positions ### Safety Precautions - **Wear safety goggles** - materials may snap under tension and recoil dangerously - Place cushion or landing mat below mass hanger to catch falling masses - Never stand directly beneath hanging masses - Check apparatus is securely clamped before adding masses - Gradually increase load - avoid sudden large additions

Materials under high tension store significant elastic potential energy and can snap violently

## Interpreting Graph Axes - Always carefully check which variable is plotted on which axis - Exam questions may swap axes: - Standard: Force on y-axis, Extension on x-axis → gradient = $k$ - Reversed: Extension on y-axis, Force on x-axis → gradient = $\frac{1}{k}$ - If axes are swapped, spring constant = $\frac{1}{\text{gradient}}$ ## Worked Example: Determining Spring Constant from Graph A spring is tested by hanging increasing masses from it. The graph plots length of spring (y-axis) against applied load (x-axis). At zero load, the spring measures 10 cm. When 0.36 N is applied, the length increases to 14.5 cm. **Calculate the spring constant.** **Solution:** The graph has length on the y-axis and force on x-axis, so the gradient represents extension per unit force. - Extension $\Delta L = 14.5 - 10 = 4.5$ cm $= 0.045$ m - Force change $\Delta F = 0.36 - 0 = 0.36$ N Gradient of graph $= \frac{\Delta L}{\Delta F} = \frac{0.045}{0.36} = 0.125$ m N$^{-1}$ Since $F = kx$, we need $k = \frac{F}{x}$, which is the reciprocal of the gradient when axes are swapped: $$k = \frac{1}{\text{gradient}} = \frac{1}{0.125} = 8.0 \text{ N m}^{-1}$$