Energy in deformations

Hooke's Law

Forces can cause objects to deform (i.e. change their shape). The way in which an object deforms depends on its dimensions, the material it is made of, the size of the force and direction of the force.

If you measure how a spring stretches (extends its length) as you apply increasing force and plot extension (e) against force (F);

the graph will be a straight line.

Note: Because the force acting on the spring (or any object), causes stretching; it is sometimes called tension or tensile force.

This shows that Force is proportional to extension. This is Hooke's law. It can be written as:

F = ke

Where:.

F = tension acting on the spring.

e is extension = (l-l_o); l is the stretched length and l_o is original length, and.

k is the gradient of the graph above. It is known as the spring constant.

The above equation can be rearranged as

Spring constant = Applied force/extension

The spring constant k is measured in Nm^-1 because it is the force per unit extension.

The value of k does not change unless you change the shape of the spring or the material that the spring is made of.

A stiffer spring has a greater value for the spring constant

We can apply the concept of spring constant to any object obeying Hooke's law. Such an object is called (linearly) elastic.

1.       An elastic object will return to its original form if the force acting on it is removed.

2.       Deformation in an elastic object increases linearly with the force.

In fact, a vast majority of materials obey Hooke's law for at least a part of the range of their deformation behaviour. (e.g. glass rods, metal wires).

In the diagram above, if you extend the spring beyond point P, and then unload it completely; it won't return to its original shape. It has been permanently deformed. We call this point theelastic limit - the limit of elastic behaviour.

If a material returns to its original size and shape when you remove the forces stretching or deforming it (reversible deformation), we say that the material is demonstrating elastic behaviour.

If deformation remains (irreversible deformation) after the forces are removed then it is a sign of plastic behaviour.

Hooke's Law Back to Deformation of Solids

Energy in deformations

Whenever we apply force to an object, it will cause deformation. If the deformation caused is within the elastic limit, the work done in deforming the object is stored within it as potential energy. We call this (elastic) 'strain energy'. It can be released from the object by removing the applied force. The strain energy then performs work in un-deforming the object and returns to its original state. We can calculate the energy stored in a deformed object, from the force- extension graph. As a example let us consider the case of a metal wire. The diagram is a force-extension graph of the wire within its elastic limit. It is a straight-line graph. If we apply a tensile force (T) of 10N to this wire it will extend to 0.02M.

The work (W) done by the wire is the shaded triangular area under the straight line.

In this example W is:.

W = ½ force x extension = ½ x 10 x 0.02 = 0.1J

The metal wire in the above example, stores energy perfectly as it releases all the energy stored in the form of extensional strain without any loss of energy.

Some materials cannot store energy perfectly. For example, rubber is not a good material for storing energy even though popularly it is called 'elastic'.

Actually, it does not obey Hooke's law. We can see this from its force-extension graph. We get two different curves, one for applying increasing force (loading curve) and a different one when we decrease the force (unloading curve). On unloading, rubber gives up less energy (area under the unloading line) than the energy it takes up to deform (area under the loading line). The difference between these two energies (area enclosed by the loop) is the energy lost. This energy loss is absorbed by the molecules of rubber and is eventually dissipated as heat. Consequently the rubber gets noticeably hot if we stretch it and un-stretch it repeatedly. The graph of the above type, forming a loop is called hysteresis curve.

Area of the hysteresis loop represents the energy lost.

Molecules and atoms (considered the same)

While the kinetic theory of matter considers the motion of molecules to be free and random, there are forces between molecules.

The force acting between molecules is an electrostatic force. If we consider two molecules, the force is:

repulsive between opposite electron clouds
repulsive between opposite nuclei
attractive between one electron cloud and the opposing nucleus

As molecules have forces between them they must have potential energy.
It is convenient to think of two molecules as having zero P.E. at infinite distance, where the attractive force between them is also zero.
Work is done when two attracting molecules are separated. So their P.E. must increase when they are parted.
However, since molecules at infinite distance have zero P.E., those at intermediary distances have negative P.E. relative to this level.
Conversely, repelling molecules have positive P.E. . Repulsive forces do work to move molecules apart.

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Relation between force (F) and P.E. change (δE - delta E)

Consider two attracting molecules(X & Y). The force of attractionF of Y on X moves X a small distance δr (delta 'r') towards Y.
The distance δr is so small that the force F may be considered constant.

The work done W in moving the force F a distance δr is given by: (remember work = force x distance force moves)

If the change in the P.E. of X is δE (delta E), then:

Note the minus sign signifies the decrease in P.E. of X as it is attracted towards Y.

Elimenating δW from the first two equations:

in the limit, as δEand δr tend to zero,

This means that the negative of the gradient of an E-r graph equals the force F acting.

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Curve of molecular P.E. (E) vs molecular separation (r)

The value of the equilibrium separation r_o depends on the solid and temperature. An approximate value is 3 x 10^-10 m.

The point of inflexion is where the gradient is maximum on the right hand side of the graph.

P.E. and K.E. changes between molecules

If two molecules in a solid are at absolute zero, they have no kinetic energy and their separation is r_o . Now consider the two molecules at a higher temperature, with a shared kinetic energyε . This energy creates an imbalance between the attractive and repulsive forces.

B         r maximum    molecules attracted     PE max, KE zero

BC       PE converted into KE

C        r = r_o equilibrium position     PE min     KE max

CA      KE converted into PE

A        r minimum    molecules repelled     PE max, KE zero

AC      PE converted into KE

C       r = r_o equilibrium position     PE min     KE max

CB      KE converted into PE

In solids molecules can only vibrate small distances about fixed positions. The reason is that the kinetic energy ε is much smaller than the potential energy ε_o (approx. 10%) . Hence a solid has a fixed shape and volume.

Effect of temperature on equilibrium position

C represents the equilibrium position r_o at absolute zero. In this state the molecules have no kinetic energy and consequently do not oscillate.

Consider a low temperature P.E. level at AB. The mid-point of ABmoves r_o to the right.

At a higher P.E. level A^'B^' corresponding to a higher temperature. The mid-point A^'B^' of moves r_o even further to the right.

So as the temperature rises, r_o increases. The molecules move apart and the solid is observed to expand.

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Curve of intermolecular force (F) vs molecular separation (r)

Force is the negative of the gradient of the E-r graph.

When F is positive, the force is repulsive.
When F is negative, the force is attractive.

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