Forces & Matter
Stretching, Bending, and Compressing
When forces act on an object, they can change its shape. An object can be stretched, bent, or compressed. In every case, more than one force is required to change the shape — a single force would simply cause the object to accelerate rather than deform. For example, to stretch a spring you must pull both ends in opposite directions, or fix one end and pull the other.
Stretching, bending, or compressing an object always requires more than one force. A single force would cause acceleration, not deformation.
Elastic and Inelastic Distortion
When a force deforms an object, the deformation can be either elastic distortion or inelastic distortion.
In elastic distortion, the object returns to its original shape and size once the force is removed. A stretched rubber band that springs back to its original length is an example of elastic behaviour. In inelastic distortion, the object is permanently deformed — it does not return to its original shape. Bending a paperclip is an example of inelastic distortion.
A spring behaves elastically as long as it is not stretched beyond its elastic limit. Beyond this point, the spring undergoes inelastic distortion and will not return to its original length when the force is removed.
Hooke’s Law
Hooke’s law states that the extension of a spring is directly proportional to the force applied, as long as the elastic limit is not exceeded. The equation is:
$$F = k \times x$$
Where \(F\) is the force applied in newtons (N), \(k\) is the spring constant in newtons per metre (N/m), and \(x\) is the extension in metres (m).
A stiffer spring has a larger spring constant — it requires more force to produce the same extension. The spring constant can be found by rearranging the equation: \(k = \frac{F}{x}\).
A spring has a spring constant of 40 N/m. Calculate the force needed to extend the spring by 0.15 m.
\(F = k \times x = 40 \times 0.15 = 6\) N
The force needed is 6 N.
Hooke’s law: \(F = k \times x\). The extension is directly proportional to the force, but only up to the elastic limit. Beyond this point, the relationship is no longer linear and the spring does not return to its original length.
Force-Extension Graphs
A force-extension graph shows the relationship between force and extension for a material. For a spring obeying Hooke’s law, the graph is a straight line through the origin — this is a linear relationship. The gradient of the straight-line section equals the spring constant \(k\).
If the spring is stretched beyond its elastic limit, the graph curves — the extension increases more quickly for each additional unit of force. This is a non-linear relationship. The point where the line starts to curve marks the elastic limit.
Different materials produce different shaped graphs. A rubber band, for example, does not obey Hooke’s law — its force-extension graph is curved from the start. It still returns to its original shape (elastic), but the relationship between force and extension is non-linear.
Energy Stored in a Stretched Spring
When a spring is stretched (or compressed) within its elastic limit, work is done on the spring, and energy is stored as elastic potential energy. The amount of energy stored can be calculated using the equation:
$$E = \frac{1}{2} \times k \times x^2$$
Where \(E\) is the energy transferred in stretching (joules, J), \(k\) is the spring constant (N/m), and \(x\) is the extension (m). This equation only applies when the spring obeys Hooke’s law — within the elastic limit.
A spring with a spring constant of 80 N/m is stretched by 0.10 m. Calculate the elastic potential energy stored in the spring.
\(E = \frac{1}{2} \times k \times x^2 = \frac{1}{2} \times 80 \times 0.10^2 = \frac{1}{2} \times 80 \times 0.01 = 0.4\) J
The elastic potential energy stored is 0.4 J.
Elastic potential energy: \(E = \frac{1}{2} k x^2\). This equation applies only when the spring obeys Hooke’s law (within the elastic limit). Notice that energy depends on extension squared — doubling the extension quadruples the energy stored.
Core Practical: Investigating Springs
In this core practical, you investigate the relationship between force and extension for a spring and calculate the work done.
- Method: Clamp a spring vertically. Measure its natural (unstretched) length with a ruler. Add masses one at a time, recording the new length each time. Calculate the extension (new length minus original length) for each mass.
- Variables: Independent variable = force (weight of masses). Dependent variable = extension. Control variables = same spring, same starting position, same measuring equipment.
- Analysis: Plot a force-extension graph. The straight-line region confirms Hooke’s law. The gradient equals the spring constant. If the line curves, the elastic limit has been exceeded.
- Accuracy: Read the ruler at eye level to avoid parallax error. Wait for the spring to stop oscillating before reading. Repeat readings and calculate a mean to improve reliability.