Young’s Modulus (last edited February 15, 2011)

Dr. Larry Bortner

 Purpose

To investigate the elasticity of materials by showing that the stress is proportional to the strain. To find Young’s modulus for two materials.

 Background

In talking about objects in an introductory physics course, we first assume a point particle when describing its motion. As we get into more complicated motions and what causes the motion, we take into account that all objects have a volume and a shape, with the assumption that they are perfectly rigid. In the real world this is not the case and how a force is applied determines how the shape of an object changes.

 

This distortion is quantified by the strain, a measure of how much the shape changes compared to its original shape. Information of the cause of the distortion such as the angle of the force and the area of the object where it was applied is contained in the stress. The distorting force can be applied normal (perpendicular) to a surface (either in or out), tangential to the surface, or uniformly perpendicular across the entire surface of the object. The corresponding stresses are tension or compression, shear, and hydraulic. In each case, under certain conditions the resulting strain is proportional to the stress.

 

In this experiment, we investigate the change in length of a wire under a varying tension. The constant of proportionality between the tensile stress and strain is called Young’s modulus, also known as the elastic modulus. (Modulus is just another word for constant.)

 Molecular Model of Stress and Strain

To see how this linear stress-strain relationship comes about, let’s look at a solid metal wire on the atomic scale. The metal atoms of the wire are bound together by electrical forces. The exact nature of these forces is complicated but we can model the solid approximately as points (the atoms) in a three-dimensional lattice, interconnected with springs. If we anchor one end of the wire and stretch it out with a known force at the other end, each little spring aligned along the length of the wire will stretch by the same amount since the same applied force is transmitted uniformly to each. Although the stretch of an individual spring may be microscopic, there are a lot of them and the sum of all of these teeny changes can produce a macroscopic change in the length of the wire, that is, one that is measurable in a lab.

 

Just like a spring, for a certain range of applied forces, the wire obeys Hooke’s law. This means that the change in length is proportional to the applied force. Also like a spring, when the applied force is removed, the wire returns to its original length. If we go beyond that certain range, if we exceed some maximum applied force, the wire or spring is permanently deformed or it breaks. The only difference between a wire and a spring is that the magnitude of the stretch.

 

This stretching behavior is summarized in a stress-strain diagram such as the one shown in Fig. 1.

Figure 1.jpg

Figure 1 Typical tensile stress-strain curve for a metal. Of interest in this experiment is the linear, or elastic, region.

 

As the stress is increased between points a and b on the graph, the stress-strain relationship for the wire is linear and elastic. Between points b and c the behavior is still elastic but is no longer linear. After point c, called the yield strength, the material enters the plastic deformation region, which means that the stretch of the wire is permanent. (For example, if the wire is stressed to point d on the graph and the stress is slowly decreased, the stress-strain curve follows the dotted line instead of the original curve and there is a leftover strain when all stress is removed.) At point e the wire reaches its breaking point. Differences in the shape and limits of the stress-strain diagram determine whether a material is considered ductile or brittle, elastic or plastic.

 Macroscopic Behavior

We assume that the material put under stress is homogeneous and isotropic. Homogeneous means that the elastic properties are the same throughout the bulk of the sample. Isotropic means that the properties do not depend on direction. Since the direction of the force is along the axis of the wire in this experiment, this means that the orientation of the material within the wire does not affect its behavior.

 

You are probably aware from observation of the stretching of rubber bands or the making of balloon animals that when the length of an object is stretched, its diameter decreases. This definitely happens on a smaller scale in the case of a stretched wire, but we do not take this into account in our analysis of this experiment.

 

The illustration in Fig. 2 indicates the relevant physical quantities for defining Young’s modulus:

·         the applied force F

·         the cross sectional area A

·         the initial, unstressed length L0

·         the change in length due to the stress, ΔL

Figure 2.jpg

Figure 2 Quantities of interest for a cylindrical object under tensile stress.

 

We now define the stress and strain:


 

The stress S is the force per unit area:

 

(1)                    

 

The strain e is the fractional change in the length:

 

(2)                    

 

Within the linear, elastic region of the stress- strain diagram (from a to b on Fig. 1), Young’s modulus E is defined as the ratio of stress to strain:

 

(3)                     

 

Equivalently, Young’s modulus is the constant of proportionality between the stress and the resulting strain which it produces, or the slope of the line in the plot of stress vs. strain:

 

(4)                  

 

Since the strain is a ratio and therefore has no unit, Young’s modulus has the same units as the stress, which is Newtons per square meter (N/m2), or Pascals (Pa). In practice, it is usually stated in units of MPa (106 Pa) or GPa (109 Pa).

 Experiment summary

 

·         You are given a piece of wire hanging vertically in a sturdy frame, a two kilogram mass hanger attached to the end of the wire and a stack of kilogram masses.

·         The wire is connected to a bubble level and balancing micrometer, which allows you to determine small changes in the length of the wire.

·         You record the initial length and initial diameter of the wire.

·         You then place different amounts of mass on the mass hanger and for each amount you determine the cumulative change in length of the wire.

·         In the analysis you calculate and plot stress vs. strain.

·         From this graph you determine whether the stress on the wire remains in the linear region.

·         You find a numerical value of Young’s modulus for the wire, and you determine the material that composes the wire by comparing this experimental value to a table of accepted values.

 Leveling the Apparatus

The equipment should be set up as shown in Fig. 3. The wire is firmly held at its top end by a collet. The two kilogram mass hanger is attached to the other end, initially with no additional mass. A second collet, clamped to the wire near its lower end, connects the wire to the balancing micrometer.

Figure 3.jpg

Figure 3 The Young's modulus apparatus.

                                   

There are two separate leveling procedures in this experiment. First, the entire stand must be leveled using the adjustable feet on the tripod stand. The mass holder has to be centered between the two vertical rods that support the wire. If the mass holder is making contact with either rod, adjust the feet on the tripod until the mass holder is centered.

 

The second critical procedure comes in leveling the balancing micrometer in order to determine the change in length of the wire. Observe the bubble level that is attached to the balancing micrometer. Turn the dial to level the micrometer, placing the bubble in an easily repeatable position. Note that the length changes are on the order of hundredths of a millimeter. Having the micrometer perfectly level, with the bubble exactly centered, is not as important as having the bubble located at a repeatable reference mark, something that is clear and easy to judge. Make sure that you always use the same vantage point when observing the bubble. Have your head and eyes in the same location every time you adjust the micrometer. Throughout the experiment, you determine the change in length of the wire by adjusting the micrometer to bring the bubble back to this original position.

 

The point of the micrometer screw rests in the slot of the head of a screw. This screw point must rest at the same point in the slot each time you level the micrometer.

 

A final thing to consider when balancing is that the micrometer assembly has some torsional freedom, which affects the bubble position. Keep the assembly twisted clockwise as far as it will go when you level the micrometer.

 Reading the micrometers

Figure 4 is an illustration of the Starrett handheld micrometer that you use to measure the thickness of the wire. Rotate the knurled end of the barrel to open the jaws then gently close them onto the object that you are measuring.

Figure 4.jpg

Figure 4 Starrett micrometer. Record two numbers: the nearest, lowest half millimeter from the fixed barrel (dF= 5.5 in this case) and the nearest hundredth mm from the rotating barrel (dR=22).

Over-tightening the jaws damages the micrometer. When the micrometer jaws make contact with the object that you are measuring, and you feel resistance, do not force the micrometer to tighten further.

 

To record the micrometer measurement, follow this process:

 

1.    Using the edge of the rotating barrel as your reference mark, read the scale on the fixed barrel (up to the edge of the moving barrel) to obtain the measurement to the nearest half millimeter. Record this number as d F.

2.    Using the center line inscribed on the fixed barrel as your reference mark, take the reading from the scale inscribed around the moving barrel. Record this number as d R.

3.    In the Analysis (not on your data sheet) use this formula to get your measurement d in mm: d = d F + d R/100.

a.     One example: if d F=1.5 and d R=7 the result is d = 1.57 mm.

b.     Another example: if d F=1.5 and d R=35 the result is d = 1.85 mm.

 

Figure 5 is an illustration of the dial on the balancing micrometer. The micrometer measures the height of the dial above the zero reference mark to the nearest 0.01 mm. The change in height of the dial is the change in length of the wire.

 

Figure 5.jpg

Figure 5 Top view and side view of the balancing micrometer. The reading from this setting is 11.28 mm.

 

Read the balancing micrometer as follows:


 

1.    Look at the vertical scale beside the dial.

a.   Reading from the bottom upward, measure up to but not past the top of the dial.

b.   You may have to unbalance the bubble by setting the dial at zero.

i.      Note the vertical scale reading (it may be ambiguous).

ii.     Increase (turn counterclockwise) or decrease (clockwise) the height by making one complete turn to the next zero.

iii.    Note the scale reading.

iv.    Rebalance the bubble.

2.    Record this measurement on the vertical scale as the number of millimeters.

3.    Look down at the top of the dial and record the number that coincides with the front edge of the vertical scale as the hundredths of a millimeter (i.e. if the dial is at 5, append 0.05 mm to the number of millimeters from Step 2); if the dial is at 63, append 0.63 mm).

 Procedure

You need the following items:


*        Young’s Modulus apparatus (Figure 3)

*        two-kilogram mass holder

*         eight one-kilogram weights, u{m}=0.001 kg

*           kilogram weight, u{m}=0.001 kg

*        meter stick with caliper attachments

*        Starrett micrometer

 

 

1.     Level the stand so that the mass holder hangs freely.

 Initial wire parameters

2.     Once the apparatus is level, use the meter stick and the vernier accessories to measure L0, the initial length of the wire between the two collets (the length between points A and B in Fig. 3).

3.     Use the Starrett micrometer to record the two numbers dF and dR that give the diameter d of the wire. Refer to Fig. 4.

4.     Record as x the initial reading of the balancing micrometer with no added masses (m=0) on the mass holder. Refer to Fig. 5 and the discussion concerning the reading of the balancing micrometer.

 Add/remove mass, measure the change in height/length 

Caution: Handle the masses carefully. If a kilogram mass is dropped, it could cause broken toes or a chipped tooth or some other injury.

 

5.     Place a 1 kg mass on the mass holder. The wire stretches and the balancing micrometer is no longer level.

6.     Turn the dial of the balancing micrometer until it is level again. (The bubble is back at the reference position. Again, be consistent in viewing this.)

7.     Record the total mass m added to the holder (do not include the mass of the holder itself).

8.     Record the new position x of the micrometer.

9.     Continue this process (steps 5 8) for the following values of m, in this order: 1, 2, 3, 4, 5, 6, 7, 8, 8.5, 7.5, 6.5, 5.5, 4.5, 3.5, 2.5, 1.5, 0.5, and 0. That is,

a.     set m,

b.     level the micrometer,

c.     record m,

d.     record x.

 Second wire

10.  Repeat steps 1−9 of the Procedure for the second wire.

 Analysis

1.     Open the Young’s modulus template and enter the values of g, L0, d F, d R for the first wire, and the uncertainties in L, d, m, and x. (The only thing in the template, besides the functions, is the table of Young’s moduli. You have to label everything else and do the plots.)

2.     Calculate d.

3.     Enter the data for m and x into two appropriately labeled columns, including the reading for m=0.

4.     Calculate the strain (Eq. 1) and the stress in GPa (Eq. 2) in adjoining columns.

5.     Propagate the error for e and S.

6.     Calculate u{e} and u{S} for each data point. (There is one value of u{e} for all data points.)

7.     Graph stress vs strain, including error bars and a trendline.

8.     Calculate Young’s modulus E and u{E} in GPa using LINEST.

9.     Compare your result to each of the values in the accompanying table of Young’s moduli for different materials to identify the wire material. Assume that the error for the accepted value is the least (smallest) significant digit. If nothing matches, or you have more than one match, what is your best guess?

Metal

E (GPa)

lead

15.7

 

magnesium

41.8

 

aluminum-nickel alloy

64.7

 

cadmium

69.3

 

silver (hard drawn)

77.5

 

antimony

78.0

 

gold

78.5

 

brass

90.2

 

zinc

104.7

 

titanium

120.2

 

copper

129.7

 

platinum

166.7

 

nichrome

186

 

steel (hard drawn)

192.2

 

steel (annealed)

200.1

 

nickel

207.0

 

tungsten (drawn)

355.0

 

Table 1 Young's moduli of various metals.

 

10.  Repeat Steps 1 9 of the Analysis for the second wire.

 Questions

1.     Propagate the error for e and show that it is independent of the value of ΔL. (Hint: One of the relative errors in the initial expression for u{e} is much smaller than the other. Make the approximation that this is zero and simplify the resulting expression.)

2.     One of our assumptions is that the wire diameter is constant. Show that when the actual change in area is taken into account, the fractional increase in the stress is equal to the strain (to first order). Assume that the volume of the wire is constant.