Physics 211 MAD 04/00

Experiment 8
Young's Modulus

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Introduction

Young's Modulus is one of the fundamental properties of a material. It is similar to the spring constant of a coiled spring, but is a slightly more general concept. It relates the stress (force per area) placed on a wire to the strain (change in length divided by length) for the same wire. This experiment makes use of an optical lever to ``magnify'' the elongation in the wire. The optical lever uses the rotation of a small mirror to translate the small elongation of the wire into a large change in reading on a meterstick.

Theory

Stress:

Stress is defined as the force per unit area of the wire. The area is $\pi r^2$ (r being the radius of the wire) and the force is simply the weight hanging below on the wire (M g). So:

$$Stress = \frac{M g}{\pi r^2}.$$ (1)

Strain:

Strain is the elongation (e) divided by the length of the wire (L).

$$Strain = \frac{e}{L}.$$ (2)

Lever:

See the figures below for some information. A small rotation of the mirror ($\theta$) leads to the light ray from the scale to the telescope to reflect through an angle of $2 \theta$. From the first figure, it can be seen that

$$\tan 2 \theta = \frac{D}{R} \rightarrow \theta \sim \frac{D}{2R}.$$ (3)

On the other hand, the elongation is related to the angle $\theta$ and the radius of the cylinder d to which the mirror is attached:

$$e = d \theta.$$ (4)

Combining these expressions,

$$e = \frac{dD}{2R}.$$ (5)

Final:

The final result can be found by combining eqns. 1, 2 and 5:

$$Y = \frac{2 M g L R}{\pi r^2 d D},$$ (6)

where all the variables are defined above or in the figures.

Apparatus

You need an assortment of weights, the Young's Modulus apparatus and the optical lever apparatus.

Procedure

1)

Hang several 100 grams to straighten the wire as best you can. Measure the radius of the cylinder (d; the wire is wrapped around this and the mirror is attached to it), the radius of the wire (r), the unstressed length (L) and the current mass (m₀). Even though there is some stress applied, we'll consider only additional stress beyond what's there now.

Record the values below (10 points):

2)

Focus the telescope at infinity (point it out the window and focus). Make sure you can see the cross hairs, too! There are two adjustments that you can make, one affects both the cross hairs and the focus, the other affects only the focus. Once you have the focus adjusted properly, please be careful not to change it.

3)

Adjust the telescope, mirror and scale so that you can see a reflection of the scale in the mirror, while looking through the telescope. This will require many fine adjustments and lots of patience; your instructor will be around with helpful advice. Make sure that you are originally looking at the top of the scale, since stretching the wire will tilt the mirror downwards.

Once you have the apparatus adjusted, have your instructor verify that it is OK (10 points):

4)

Once all the adjustments have been made, measure the distance between the mirror and the scale (R). Once you've made this measurement, you do not want to move the telescope or the attached scale. Also, at this time you can read the scale through the telescope; this will give the unstretched scale reading (D₀).

Record the values below (10 points):

5)

Now comes the easy part. Add some mass to the hanger and read the new scale reading D₁. The distance D in figure 1 is the distance between the unstretched scale reading D₀ and the stretched scale reading D₁. Make a table in Excel showing the mass M, unstretched scale reading D₀ (this should be the same for all trials), D₁ and D.

6)

Add more mass and read the new D₁. Record in your table. Repeat. Try to have 7-10 different masses; increase mass by 100 g each time.

7)

Remove all the added masses, leaving the original m₀ only. Repeat steps 5) and 6) using the same masses you used before. Go through this process 4 times, so that you have 4 values of D for each mass M. Each run should be a seperate column in Excel.

8)

Average the 4 D's for each M. Print and attach the Excel spreadsheet containing all your data. (20 points)

9)

Plot the average D (y-axis) vs. M (x-axis) the slope is related to Young's Modulus via eqn. 6.

$$D = \frac{2 g L R}{\pi r^2 d} \frac{1}{Y} M.$$ (7)

The first fraction contains all the fixed numbers; only Y (Young's Modulus) is unknown. From the slope of the best fit straight line, you can determine Y. If your data doesn't look particularly linear, chances are that the first data point or two are at fault: you may still be pulling the bends out of the wire. If so, ignore the first point or two (this is why you want at least 7 data points).

Calculate the coefficient $\frac{2 g L R}{\pi r^2 d}$ remembering to convert all diameters to radii, and all lengths to meters. As usual, if you don't keep the units straight in your calculations, the numerical value you get at the end is meaningless. Record the value here (including the units!): (10 points)

Print and attach the graph, showing the equation of the line with the slope and intercept. (20 points)

Determine Y from the slope of your graph and eqn. 6. Record the value here: (10 points)

You can find values of Young's Modulus in Table 13-1 of the textbook. Given the value of Y determined above, what do you think the wire is made of? (10 points)