Elastic Strain Energy (AQA A Level Physics): Revision Note

Elastic strain energy

• Work has to be done to stretch a material

• Before a material reaches its elastic limit (whilst it obeys Hooke's Law), all the work done is stored as elastic strain energy

• The work done, or the elastic strain energy is the area under the force-extension graph

• This is true for whether the material obeys Hooke’s law or not

  • For the region where the material obeys Hooke’s law, the work done is the area of a right-angled triangle under the graph

  • For the region where the material doesn’t obey Hooke’s law, the area is the full region under the graph

    • To calculate this area, split the graph into separate segments and add up the individual areas of each

• When a material does obey Hooke's law, the elastic strain energy, E can be calculated using:

$E=\frac{1}{2}F∆L$

• Where:

  • $E$ = elastic strain energy (or work done) (J)

  • $F$ = average force (N)

  • $∆L$ = extension (m)

• This equation is equivalent to the area of a right-angled triangle under the force-extension graph

• Using Hooke's law $F=k∆L$, the elastic strain energy can also be written as:

$E=\frac{1}{2}k{\left(∆L\right)}^2$

• Where:

  • $k$ = spring constant (N m^–1)

Breaking stress

• As greater force is applied on a material, the stress on it increases

• The breaking stress is the maximum stress a material can stand before it fractures (breaks)

  • A material with high breaking stress is considered ductile

  • This means it can stretch a long way before breaking

  • A common example of this is copper, as well as being a good electrical conductor, copper is ductile so it is a suitable material for making wires

• The ultimate tensile stress (UTS) is sometimes also marked on a stress-strain graph

  • This is the maximum stress that the material can withstand

• The UTS and breaking stress can depend on the condition of the material such as its temperature

  • This is very important for engineers when considering materials for a particular structure

  • The material might need to stand extreme temperatures and loads which are taken into consideration

Spring energy

• When a vertical spring is extended and contracted, its energy is converted into other forms

• Although the total energy of the spring will remain constant, it will have changing amounts of:

  • elastic potential energy (EPE)

  • kinetic energy (KE)

  • gravitational potential energy (GPE)

• When a vertical mass is hanging on a spring and it moves up and down, its energy will convert between the three in various amounts

• At position A:

  • The spring has some EPE since it is slightly compressed

  • Its KE is 0 since it is stationary

  • Its GPE is at a maximum because the mass is at its highest point

• At position B:

  • The spring has some EPE since it is slightly stretched

  • Its KE is at a maximum as it passes through the equilibrium position at its maximum speed

  • It has some GPE since the mass is still above the ground

• At position C:

  • The spring has its maximum EPE because it is at its maximum extension

  • Its KE is 0 since it is stationary

  • Its GPE is at a minimum because it is at its lowest point above the Earth's surface

• For a horizontal mass on a spring system, there is no gravitational potential energy to consider

  • The spring only converts between kinetic and elastic potential energy