YOUNG S MODULUS. Introduction

Transcription

1 Introduction In this experiment, we study the elasticity of some solid materials by measuring their Young s modulus. A highly useful optical method is also introduced for the measurement of a very small displacement. Theory LB1B_YM.DOC 1.007, May 13, 2002 Ken Cheney [SECTION]-1

2 Figure 1 An important aspect of the physical properties of a solid material is its elasticity. If one stretches or compresses a material with a force F, the corresponding deformation x for most materials is observed to be proportional to F. Furthermore, the geometric shape of the material also plays a role in determining x. For simplicity let us consider a simple case in which the material being studied is cut into a piece of length L and uniform cross-sectional area A. If A is doubled, then for the same force F applied the resultant deformation is only half as much, since the same force F applied on the material is now being shared by the two identical portions, each with the same cross-sectional area A. This means. Also, if the length L of the material to be stretched or compressed is doubled, then so would be x for the same force F applied. This is because each of the two segments of the material with length L is subject to the same force F and has a deformation x, making the total amount of deformation 2x. Thus x 1/A. Combining these considerations, we get x FL/A, or x = 1 Y FL A [SECTION]-2

3 Here the constant of proportionality is 1/Y, where Y is defined as the Young s modulus of the material. Note that the formula above can also be re-written as Y = F/A... (1) X/L where F/A, the force applied per unit area, is defined as the stress while x/l, the amount of deformation per unit length, is defined as the strain. The Young s modulus provides a quantitative assessment for the elasticity of a certain material. The greater the value of Y, the more the force F that would be needed to deform the material by a given amount. In this experiment, we shall measure the value of Y for some of the following common materials: cast iron, steel and aluminum. Equipment (1) iron stand (2) tripod (3) Helium-Neon (He-Ne) laser (4) steel, aluminum and cast iron wires (5) vernier caliper, meter stick, etc. Procedure Refer to the experimental setup in Figure 2. A segment of the wire with length L is being clamped between points A and B on the iron stand. The lower end of the wire (at point B) is attached to a group of weights which provide the stretching force: F=Mg. To find the Young s modulus, we need to measure all the other quantities in Eq.(1). [SECTION]-3

4 Figure 2 (1) Measure the length L of the wire with a meter stick and the diameter d of the wire with a vernier caliper. The cross-sectional area A of the wire can be found from A = 1 4 πd 2 (2) Now to measure x, the deformation (stretch) of the wire. Since x is very small (perhaps less than a millimeter), we cannot expect to measure it accurately with a meter stick. To this end we employ a very useful technique illustrated in Figures 2 and 3. This method is based on the original idea by the British physicist Lord Cavendish at Cambridge University in 1798, when he made the first successful measurement of the universal gravitational constant. (Since there was no laser available then, he had to use the sunlight.) [SECTION]-4

5 The mechanism for the measurement of x is depicted in Figure 3 below. Figure 3 As shown in figure 3, the lower end of the wire (point B) is clamped onto a cylinder. Also resting on the cylinder is the back leg of a tripod, whose front surface is a mirror. When the wire has no stretch (x=0), the back leg of the tripod is in such a position that the mirror is in a vertical plane. A horizontal laser beam incident on the mirror will then be reflected right back, hitting point 1 on the screen. As weights are added to stretch the wire, however, the cylinder begins to descend by an amount x, causing the tripod to tilt (see Figure 3). The normal line of the mirror surface of the tripod (which is perpendicular to the mirror surface) now makes an angle θ with respect to the horizontal. As a result, the reflected laser beam makes an angle 2θ with respect to the horizontal. hitting point 2 on the screen. Since the separation between the mirror surface of the tripod and the screen can be quite large, a small value of x can cause a relatively large displacement D of the laser spot on the screen. From the figure we see that x/a= tan(θ) and D/b= tan(2θ). Since x<<a and D<<b, the angle is very small. Thus x/a= tan(θ) θ and D/b= tan(2θ) 2θ. Combining these results, we obtain x a 2b D Measure a and b, both fixed quantities, with a meter stick. Stretch the wire with several different weight M. For each value of M, locate the position of the reflected laser spot on the screen and find the corresponding value of D, then calculate x from the formula above. (3) Repeat the measurement for wires made of different materials, as required by the instructor. [SECTION]-5

6 Analysis From the experimental data obtained, plot the deformation x of the wire versus the mass M attached to stretch it. From the formula x = 1 Y FL A = 1 Y we know that the plot should yield a straight line whose slope is given by k = gl YA Find the slope from the plot (by using the standard "rise over run" method) and solve for the experimental value of the Young s modulus from the formula above: Y exp = gl ka MgL A = gl YA M km Find the standard value of the Young s modulus for each of the wires measured from your 2A textbook or the appendix in this manual, and calculate the per cent error: e = Y st Y exp Y st 100% Special Remark: For the dual purpose of making the approximation tan(θ) θ accurate and making D large enough for easy measurement, please make b, the separation between the tripod and the screen, as large as possible. Placing the screen near the far edge of the table can increase b to about 4 meters. C:\KEN\WP\HANDOUTS\LB1B\LB1B_YM 7/20/96 [SECTION]-6