# PHY C9: Deformation & Springs

Here are the topics for this post:

• What is deformation?
• Types of deformation
• Behaviour of Springs
• Tensile tests & Force-extension graphs
• Hooke’s Law
• The Spring Constant
• Spring Systems

Here we go!

What is Deformation?
Change in shape or size of an object caused by the application of a force.

Types of Deformation:

• Tensile deformation (extension of an object)
• Compressive deformation (compression of an object)
• Bending (we won’t be covering this here)

What is a Tensile Test?
A test to study an object’s BEHAVIOUR under a constant load (force), by applying a load on the object (usually a long thin wire).

In this test, you can take measurements of:

• F, Force (weight of load)
• e, Extension (change in length of object)

With this information, a GRAPH can be plotted*.
The graph looks like this: It is a LINEAR graph, until the load exceeds the proportionality limit (point X).

*In practical experiments,
it would be wiser to measure the LENGTHs (l) instead of extension (e).
Because the extension may be small, so the chance of error is high.
If, instead, you measured the initial length (lo) & the extended length (l),
the chance of error is lower.

It’s also practical to use a strain gauge to measure small extensions, instead of a metre rule.

Thus, the equation is:

F = k(l – lo)

If the graph is plotted as MASS against extension, then the GRADIENT is k/g.
Because:
F = kx
mg = kx
m = (k/g)x

What is Hooke’s Law?
“Extension of a body is proportional to the applied load until the proportionality limit is exceeded.”

Since the graph is linear (before exceeding the proportionality limit),
the y-value & x-value must be related by a CONSTANT.

The equation is:
Force ∝ extension
Force = constant x extension
F = ke

That constant of proportionality is the SPRING CONSTANT.

What is a Spring Constant?
Force per unit extension of an object.

Identical objects have identical spring constants,
but different objects of the same material can have different spring constants, since other factors such as length & thickness affect it.

However, different objects of the same material have identical YOUNG MODULI.

Composite Spring Systems

 Arrangement Behaviour Extensions, e Spring Constants, k Springs in Series Force extends each spring normally The total extension is the sum of each spring’s regular extension etotal = e1 + e2 1/ktotal = 1/k1 + 1/k2   Because: F = ktotaletotal F/ktotal = e F/ktotal = e1 + e2 F/ktotal = F/k1 + F/k2 1/ktotal = 1/k1 + 1/k2 Springs in Parallel Force is distributed (shared) across the springs Extension of each spring is equal etotal = e1 = e2 ktotal = k1 + k2   Because: Ftotal = ktotaletotal Ftotal = F1 + F2 Ftotal = k1e + k2e ktotale = k1e + k2e ktotal = k1 + k2