Introduction to Stretching and Squashing

In our previous chapters, we looked at how forces make objects move. But forces can do something else too: they can change the shape of an object. Whether you are jumping on a trampoline, stretching a hair tie, or sitting on a sofa, you are seeing elastic behaviour in action. In this chapter, we will explore Hooke’s Law and how different materials respond when we pull on them.

1. Changing Shape: Tension and Compression

When you apply a force to an object, it can change shape in two main ways:

  • Tension: This happens when forces pull on an object, making it longer (stretching).
  • Compression: This happens when forces push on an object, making it shorter (squashing).

In this chapter, we focus mostly on extension. Extension is simply how much longer an object gets when you stretch it. To calculate it, use this simple idea:

\(Extension = Total \ length - Original \ length\)

2. Hooke’s Law

Robert Hooke, a 17th-century scientist, discovered a special relationship for certain materials like metal springs. He found that if you double the force, the extension also doubles. This is called direct proportionality.

Hooke's Law Definition: The extension of an object is directly proportional to the force applied to it, provided the limit of proportionality is not exceeded.

In mathematical terms, this is often written as:

\(F = k \times x\)

Where:
\(F\) = Force applied (measured in Newtons, \(N\))
\(x\) = Extension (measured in metres, \(m\))
\(k\) = The spring constant (a measure of how "stiff" the spring is, in \(N/m\))

Key Takeaway:

If a material follows Hooke’s Law, its force-extension graph will be a straight line passing through the origin (0,0).

3. Elastic vs. Plastic Behaviour

Not all materials return to their original shape after being stretched. We categorise behaviour into two types:

Elastic Behaviour

An object shows elastic behaviour if it returns to its original shape and size once the force is removed. Example: A standard exercise resistance band or a metal spring (within its limits).

Plastic Behaviour

An object shows plastic behaviour if it remains permanently deformed (stretched) even after the force is removed. It does not return to its original shape. Example: Stretching a piece of chewing gum or soft clay.

The Elastic Limit: This is the "point of no return." If you stretch a spring too far, it passes its elastic limit and will stay permanently bent or long. It has changed from elastic to plastic behaviour.

4. Force-Extension Graphs

The best way to see how a material behaves is to plot a graph of Force (\(F\)) on the y-axis against Extension (\(x\)) on the x-axis.

A. Metal Springs and Wires

For a helical spring or a metal wire, the graph starts as a straight line through the origin. This shows that they obey Hooke's Law. Eventually, the graph starts to curve. This curving point is called the limit of proportionality.

B. Rubber Bands

Rubber bands are different! They do not obey Hooke's Law. If you plot their graph, it is a curve from the very beginning. However, rubber bands are still very elastic because they return to their original shape when you let go.

Quick Review:
Straight line through origin = Obeying Hooke's Law.
Curved line = Not obeying Hooke's Law.

5. Investigating Extension (Prescribed Practical 1.22)

You need to know how to investigate how extension varies with force for helical springs, metal wires, and rubber bands. This is a common exam topic!

The Method:

  1. Setup: Hang a spring from a secure clamp and stand. Attach a ruler vertically next to the spring.
  2. Original Length: Record the initial length of the spring (with no weights attached) at eye level to avoid parallax error.
  3. Add Force: Add a known mass (e.g., \(100g\), which is approximately \(1N\) of force).
  4. Measure: Record the new length of the spring.
  5. Calculate Extension: Subtract the original length from the new length.
  6. Repeat: Continue adding weights one by one, recording the extension each time.
  7. Safety: Wear safety goggles in case the spring or wire snaps!

Comparing Materials:

  • Helical Springs: Give a beautiful straight-line graph initially.
  • Metal Wires: Only stretch a tiny amount before snapping, so you need very heavy weights and a long wire to measure it accurately.
  • Rubber Bands: Give a curved "S" shaped graph.

Common Mistakes to Avoid

  • Length vs. Extension: Many students forget that extension is the increase in length, not the total length. Always subtract the original length!
  • Units: Ensure your force is in Newtons (\(N\)). If a question gives you mass in grams (\(g\)), convert it to kilograms (\(kg\)) and then multiply by \(g\) (\(10 \ N/kg\)) to get the weight.
  • Origin: A Hooke's Law graph must go through \((0,0)\). If there is no force, there should be no extension.
Final Summary Key Takeaways:
  • Hooke's Law states extension is directly proportional to force.
  • Elastic objects return to their original shape; Plastic objects do not.
  • The limit of proportionality is where the straight-line relationship ends.
  • Springs and wires obey Hooke's Law initially; rubber bands do not.