Forces and ElasticityAQA GCSE Physics: Revision notes

Elastic deformation occurs when an object returns to its original shape and size after the deforming force is removed. The material has not been permanently changed, and the deformation is temporary.

Inelastic deformation occurs when an object does not return to its original shape and size after the deforming force is removed. The material is permanently changed and will not recover its original form.

Most materials exhibit elastic behaviour up to a certain point (called the limit of proportionality), beyond which they begin to show inelastic behaviour.

Hooke's Law states that the force applied to a spring is directly proportional to its extension, provided the limit of proportionality is not exceeded.

This relationship is expressed by the equation:

F = ke

Where:

• F = force applied (in Newtons, N)
• k = spring constant (in N/m)
• e = extension (in metres, m)

Key points:

• The extension is the increase in length from the original length
• The spring constant k is a measure of the stiffness of the spring – a larger k value means a stiffer spring that requires more force to stretch it
• This law only applies within the elastic limit – once exceeded, the relationship is no longer linear
• The equation can also be rearranged to find k = F/e

A force–extension graph is plotted with force (N) on the vertical axis and extension (m) on the horizontal axis.

Reading the spring constant from the graph:

1. The graph shows a straight-line relationship through the origin up to the limit of proportionality
2. The gradient (slope) of the straight-line section equals the spring constant k
3. To calculate the gradient, use: k = ΔF / Δe (change in force divided by change in extension)
4. Choose two points on the straight-line section that are far apart for accuracy
5. The spring constant is measured in N/m

Beyond the limit of proportionality:

• The graph curves upwards (becomes non-linear)
• The relationship is no longer directly proportional
• Hooke's Law no longer applies
• The material is beginning to undergo inelastic deformation

Aim: To investigate the relationship between force applied to a spring and its extension, and to determine the spring constant.

Equipment:

• Spring
• Ruler (or measuring scale)
• Masses and mass hanger
• Clamp and stand (to hold the spring vertically)
• Pointer or fixed reference marker

Method:

1. Clamp the spring vertically and attach a pointer at its bottom to mark the initial length
2. Record the initial length of the spring (e = 0)
3. Add a mass to the hanger and allow the system to reach equilibrium
4. Measure the new length of the spring and calculate the extension (new length – original length)
5. Record the force applied (F = mg, where g = 10 m/s²)
6. Repeat by adding further masses in equal increments (e.g., 50 g each time)
7. Record extension and corresponding force for each mass addition
8. Continue until the spring reaches its limit of proportionality or deforms visibly
9. Plot a force–extension graph with force on the vertical axis and extension on the horizontal axis
10. Identify the straight-line region and calculate the spring constant from the gradient

Safety and accuracy considerations:

• Ensure the spring is not overloaded beyond its elastic limit
• Use a reference point (pointer) to ensure accurate extension measurements
• Allow time for equilibrium after each mass addition
• Repeat measurements for reliability

The limit of proportionality is a crucial point on a force–extension graph where the linear relationship between force and extension breaks down.

Significance:

• Below the limit of proportionality: The material obeys Hooke's Law; force and extension are directly proportional; the graph is a straight line through the origin; the material will fully recover its original shape when the force is removed (elastic behaviour)

• At the limit of proportionality: This is the last point where F = ke applies accurately; beyond this point, the relationship is no longer linear

• Beyond the limit of proportionality: The graph curves upwards; Hooke's Law no longer applies; for the same increase in force, the extension increases by a larger amount; the material begins to undergo permanent (inelastic) deformation; when the force is removed, the material will not fully return to its original length

Key distinction:

• The limit of proportionality is where the straight-line section ends
• The elastic limit is slightly beyond this point – the maximum stress beyond which permanent deformation occurs
• In GCSE-level questions, these terms are often used interchangeably, but the limit of proportionality is the point where the graph deviates from linearity

When a spring is stretched or compressed, it stores elastic potential energy. This energy is released when the spring returns to its original length.

Elastic Potential Energy Equation:

Ee = ½ke²

Where:

• Ee = elastic potential energy (in Joules, J)
• k = spring constant (in N/m)
• e = extension (in metres, m)

Key points:

• This equation is only valid within the elastic limit where Hooke's Law applies
• The energy stored is proportional to the square of the extension – doubling the extension stores four times the energy
• The energy is stored as elastic potential energy in the bonds between atoms in the material
• When the spring is released, this potential energy is converted to kinetic energy (if the spring is free to move) or dissipated as heat and sound (if the spring is held fixed)

Alternative form (using Hooke's Law): Since F = ke, we can also express elastic potential energy as:

Ee = ½Fe (where F is the force at maximum extension)

This shows that energy depends on both the force applied and the extension achieved.