# PM-1 YOUNG'S MODULUS

1. PURPOSE

To measure the value of Young's Modulus for a copper wire; to investigate the effects of exceeding the elastic limit; and to determine the approximate breaking point.

2. APPARATUS

Vertical stand to support the wire, optical lever, copper wire, set of kilogram weights, weight hanger, scale and telescope (or scale and lamp).

3. THEORY

The elastic modulus, or Young's modulus of a wire, Y, is defined:

 [1]
```
F/A    F  L
Y ≡ ———— = —— —
Δx/L   Δx A
```

where F is the force stretching the wire, L the length of the wire, A the cross sectional area of the wire, Δx the elongation, and Y is Young's modulus of the wire.

4. PROCEDURE

Cut a suitable length of copper wire and fasten one end securely in the chuck at the top of the apparatus. The smaller chuck is slipped on the other end of the wire, and fastened a few inches from the end. This chuck is positioned loosely in the hole in the optical lever support table, and should be adjusted so it is not too low in the hole, to allow for downward movement of the chuck during stretch. The optical lever is a mirror on a three legged mount. Two legs fit in a groove in the support table, the third leg rests on top of the movable chuck.

Do not touch the mirror surface"its soft metallic mirror surface is on the front, unprotected. It scratches and tarnishes easily and cannot be cleaned without damage.

It can be shown that if a mirror rotates through an angle, an image seen in the mirror rotates through twice that angle. Look up the law of reflection in a text, and verify this. The wire stretch causes one leg of the optical lever to lower, tilting the mirror back. The angle of tilt may be related to the amount of stretch and the spacing of the lever legs. This is a straightforward geometric calculation.

The angle of tilt of the mirror is observed indirectly in either one of two ways:

(1) Telescope and scale. The telescope and scale are supported on a separate stand, the telescope being on a level with the mirror. The scale should lie in a vertical plane. If a curved scale is used, the scale should be placed so that the mirror is exactly at the center of its arc (most of the scales have a 50 cm radius). If the scale is straight, it may be placed at any distance from the mirror, but then the distance must be measured; 1/2 meter to 2 meters should be convenient.

The telescope is used to observe an image of the scale reflected in the mirror. Focus the telescope at the distance of the mirror so you can see the mirror clearly in the field of view. The telescope is now properly aimed at the mirror, but the scale image is completely out of focus. Refocus the telescope for a greater distance, so that objects twice as far away as the mirror are in focus. The field of view is narrow, so the scope still may not be correctly aimed to see the scale. Move your hands about until you see an image of your hand. This will show where the telescope is aimed. Further adjustments of the components should bring the scale into view. Finally, make a more precise focus adjustment by eliminating parallax between the telescope cross hairs and the scale image. Parallax is the effect of motion between the cross hairs and scale as the eye is moved slightly from side to side past the eyepiece. This shifting motion would introduce error into all readings, and must be eliminated.

(2) Lamp and scale. A lamp and lens system produces a beam of light which is reflected by the mirror onto the scale. A cross wire behind the lens may be brought into focus on the scale, serving as an indicator of position. As the mirror tilts, the beam moves up the scale.

This method requires some explanation. The lens and cross wire are essentially in the same plane at the front end of the sliding tube of the light source. This lens does not focus the cross wire at all, it's function is to concentrate the light onto the mirror. When optimally adjusted it produces an image of the lamp filament on the mirror. To focus the cross wire image onto the scale requires another lens, fastened to the front of the mirror, and of an appropriate focal length for the distances you have chosen. If the lamp and scale are the same distance from the mirror, then this lens should have a focal length equal to this distance. The lens may be chosen from an optician's lens set. These lenses are rated in diopters, the diopter rating being the reciprocal of the lens focal length in meters.

Reminder: You must measure:

a) the original length and diameter of the wire (before stretching) b) the mirror to scale distance c) the optical lever leg spacing

Make a dry run to determine reasonable amounts of weights to add, and to find the approximate elastic limit. Check whether the wire responds immediately to added weight, or takes some time to stretch fully. Such a time lag is called creep.

Fig. 3 shows (schematically) what may happen. For small applied forces the stretch is linearly related to force, the wire obeying Hooke's law. But when the elastic limit is reached the slope of the curve increases. If weights are removed from the wire, reducing the applied force, the data falls on a different line, displaced from the other one. When all weight is removed there will be a permanent stretch, S, in the wire.

This failure of the values of one variable retrace the same values when the direction of change of the other variable is reversed is called hysteresis.

With a new wire, add weights in increments and take data on scale reading vs. total weight on the wire. When you are about 3/4 of the way to the elastic limit, remove weights in increments and see if this data retraces the same curve. If it does not, any one of several things could be the cause:

1) The elastic limit may have already been exceeded. 2) Not enough time was allowed for creep. 3) The wire may have slipped in the upper chuck. 4) Scale readings were imprecise, or scale may have been bumped and moved from its initial position.

Now add weights in increments and continue well past the elastic limit. Remove weights in increments all the way to zero and plot the new curve. Have you permanently stretched the wire? If so, how much? Have you changed the Young's Modulus? If so, how much? Do not trust Fig. 1 to answer these questions, use your data and graph.

Plot your data on a single graph.

5. QUESTIONS:

(1) Suppose the material stretches by keeping its volume constant, that is, stretch is compensated for by decrease in its cross sectional area. This is a simple model which we might test against the data. If the elastic limit is not exceeded, the material returns to its original length and cross sectional area, so we suppose that no permanent rearrangement of its crystalline structure occurred.

But if the elastic limit is exceeded, and a permanent stretch occurs, there certainly has been an internal restructuring. Keeping our hypothesis that the volume is still the same (and therefore the density is also), would we expect that the permanently stretched wire has the same Young's modulus as it did before stretching? If Y is the same, show this conclusively. If it changes, does it increase, or decrease due to stretching?

Text and line drawings © 1991, 2004 by Donald E. Simanek.