## Episode 228: The Young modulus

Lesson for 16-19

- Activity time 170 minutes
- Level Advanced

The Young modulus is often regarded as the quintessential material property, and students can learn to measure it. It is a measure of the stiffness of a material; however, in practice, other properties of materials, scientists and engineers are often interested in, such as yield stress, have more influence on the selection of materials for a particular purpose.

Lesson Summary

- Discussion: Defining the Young modulus (20 minutes)
- Student activity: Studying data (20 minutes)
- Student experiment: Measuring the Young modulus (60 minutes)
- Student experiment: An alternative approach using a cantilever (30 minutes)
- Discussion: Comparing experimental approaches (10 minutes)
- Student questions: Involving the Young modulus (30 minutes)

#### Discussion: Defining the Young modulus

A typical value of *k* might be 60 N m^{-1}.

What does this mean? (60 N will stretch the sample 1 m.) What would happen in practice if you did stretch a sample by 1 m? (It will probably snap!)

A measure of stiffness that is *independent* of the particular sample of a substance is the Young modulus *E*.

Recall other examples you have already met of sample independent

properties that only depend upon the substance itself:

- density = massvolume
- electrical resistivity = resistance × arealength
- specific heating capacity = energy transferredmass × temperature change
- thermal conductivity = power × lengtharea × temperature difference

We need to correct

*k* for sample shape and size (i.e. length and surface area).

Episode 228-1: The Young modulus (Word, 53 KB)

Note the definitions, symbols and units used:

Quantity | Definition | Symbol | Units |
---|---|---|---|

Stress | tensionarea=FA | σ (sigma) | N m^{-2}={Pa |

Strain | extension per original length= Δ xx | ε (epsilon) | No units (because it’s a ratio of two lengths) |

Young modulus | stressstrain | E | N m^{-2} = Pa |

Strains can be quoted in several ways: as a %, or decimal. E.g. a 5% strain is 0.05.

Episode 228-2: Hooke's law and the Young modulus (Word, 75 KB)

#### Student activity: Studying data

It is helpful if students can learn to find their way around tables of material properties. Give your students a table and ask them to find values of the Young modulus. Note that values are often given in GPa (1 × 10^{9} Pa).

Some interesting values of *E* :

- DNA ~ 10
^{8}Pa - spaghetti (dry) ~ 10
^{9}Pa - cotton thread ~ 10
^{10}Pa - plant cell walls ~ 10
^{11}Pa - carbon fullerene nanotubes ~ 10
^{12}Pa

Episode 228-3: Materials database (Word, 115 KB)

#### Student experiment: Measuring the Young modulus

You can make measuring the Young modulus *E* a more interesting lab exercise than one which simply follows a recipe. Ask students to identify the quantities to be measured, how they might be measured, and so on. At the end, you could show the standard version of this experiment (with Vernier scale etc.) and point out how the problems have been minimized.

What needs to be measured? Look at the definition: we need to measure load (easy), cross-sectional area *A* , original length *x*_{0} (so make it reasonably long), and extension Δ *x* .

Problems? Original length – what does this correspond to for a particular experimental set up? Cross-sectional area: introduce the use of micrometer and/or vernier callipers. Is the sample uniform? If sample gets longer, won’t it get thinner? Extension – won’t it be quite small?

Should the sample be arranged vertically or horizontally?

Divide the class up into pairs and brainstorm possible methods of measuring the quantities above, including the pros and cons of their methods.

Some possibilities for measuring Δ *x* :

Attach a pointers to the wire

- Pro: measures Δ
*x*directly - Con: may affect the sample; only moves a small distance

Attach a pointer to the load

- Pro: measures Δ
*x*directly, does not effect the sample - Con: only moves a small distance

Attach a pulley wheel

- Pro:
amplifies

the Δ*x* - Con: need to convert angular measure to linear measure, introduces friction

Attach a pointer to the pulley wheel

- Pro:
amplifies

the Δ*x*even more - Con: need to convert angular measure to linear measure, introduces friction

Exploit an optical level

- Pro: a
frictionless

pointer,amplifies

the Δ*x*even more - Con: need to convert angular measure to linear measure, more tricky to setup?

Illuminate the pointer etc to produce a magnified shadow of the movement

- Pro: easy to see movement
- Con: need to calculate magnification, can be knocked out of place

use a lever system to amplify or diminish the load and provide a pointer

- Pro: useful for more delicate or stiff samples; can use smaller loads
- Con: fixing the sample so it doesn’t
slip

, need to convert angular measure to linear measure

Different groups could try the different ideas they come up with. Depending upon the time available, it may be worth having some of the ideas already set up.

Give different groups different materials, cut to different sizes, for example: metal wires (copper, manganin, constantan etc), nylon (fishing line), human hair (attach in a loop using Sellotape), rubber. Note that in the set up above, the sample is at an angle to the ruler – a source of systematic error.

#### Safety

Students should wear eye protection, provide safe landing for the load should sample break, e.g. a box containing old cloth. For the horizontal set up: bridges

over the sample to trap the flying ends, should the sample snap.

Good experimental practice: measure extension when adding to the load and when unloading, to check for any plastic behaviour.

Episode 228-4: Measuring the stiffness of a material (Word, 59 KB)

Episode 228-5: Stress–strain graph for mild steel (Word, 68 KB)

Information about the use of precision instruments (micrometer screw gauge, Vernier callipers and Vernier microscope).

Episode 228-6: Measure for measure (Word, 82 KB)

#### Student experiment: An alternative approach using a cantilever

An alternative approach to measuring the Young modulus is to bend a cantilever. (Potential engineering students will benefit greatly from this.)

For samples too stiff to extend easily (e.g. wooden or plastic rulers, spaghetti, glass fibres) the deflection *y* of a cantilever is often quite easy to measure and is directly related to its Young modulus *E* .

If the weight of the cantilever itself is *m**g*, and the added load is *M**g* and *L* is the length of the cantilever (the distance from where the cantilever is supported *to* where the load is applied):

#### For a rectangular cross section, dimension in the direction of the load is *d* , other dimension is *b*

- y = 4 (Mg + 5mg/16)
*L*^{ 3} - E b
*d*^{ 3}

(for square cross-section *d* = *b* )

#### For a circular cross-section radius *r*

- y = 4 (Mg + 5mg/16)
*L*^{ 3} - 3 π
*r*^{ 4}E

#### Discussion: Comparing experimental approaches

Finish with a short plenary session to compare the pros and cons of the different experimental approaches.

#### Student questions: Involving the Young modulus

Questions involving stress, strain and the Young modulus, including data-handling.

Episode 228-7: Calculations on stress, strain and the Young modulus (Word, 59 KB)

Episode 228-8: Stress, strain and the Young modulus (Word, 26 KB)