3.3.3 Experimental principles
Young’s elastic modulus is an important physical quantity to describe the elastic deformation properties of solid materials.Especially,it is a commonly used and important parameter for selecting mechanical component materials in engineering.To accurately determine Young’s modulus,many different methods,such as static measurement method,dynamic measurement method and wave propagation method,have been proposed.Among them,static measurement methods,including tension method,torsional method and bending method,are suitable for measuring metal samples at room temperature.For a steel wire(rod object),it is very difficult to measure the elongation by general methods because it is very small within its elastic limit.In this experiment,micro elongations are accurately measured through optical lever magnification,which is a common method to measure small length.
Usually,the shape of the solid can be changed under the action of external force,which is called as“deformation”.The deformation can be divided into two categories:elastic deformation and plastic deformation.Here,“elastic deformation”represents the object can completely recover its original state once removing the external force,while“plastic deformation”refers to the deformation that cannot disappear completely when the external force is too large.Therefore,Young’s modulus is an important physical quantity to characterize the elastic deformation of solid materials within the elastic limit.In fact,it reflects the relationship between material elastic deformation and internal stress,which is only related to the intrinsic properties of the material and is independent of the geometric size of materials and external force.This experiment focuses on studying the elastic deformation of steel wire.Thus,it is a mandatory requirement to control the strength of external force loaded on the wire within its elastic deformation limitation.
Longitudinal strain is one of the simplest elastic deformations.A uniform steel wire with the length L and sectional area S is elongated byΔL under the effect of external force F.According to Hook’s law,vertical acting force on unit area F/S(normal stress)is proportional to the specific elongationΔL/L(linear strain)of the steel wire within the elastic limit.
The proportionality coefficient E is Young’s modulus with the international unit of N/m2.Assuming the diameter of the steel wire used in this experiment is d,substitute
into Equation(3.3.1),and it can be expressed as:
Based on Equation(3.3.2),the higher Young’s modulus is,the smaller the micro-variation will be in the steel wire under the condition with constant length L,diameter d and external force F.Therefore,Young’s modulus reflects the ability of solid materials against tension(or compression)deformation induced by external force.
Usually,Young’s modulus of a steel wire is measured by Young’s modulus apparatus.The structural scheme of Young’s modulus measurement apparatus,including an optical lever and telescope with a scale ruler,is shown in Figure 3.3.1.It should be emphasized that the micro elongationΔL is too small to measure accurately by general methods.Therefore,the optical lever as well as the telescope are used to measureΔL through optical lever magnification method.As displayed in Figure 3.3.1,the optical lever G is made up of one small plane mirror and three metal screws which form one isosceles triangle.The two screws on the bottom side of mirror are called as front foot tips,and the screw on the apex is named as rear foot tip.The distance from the rear foot tip to the line between two front foot tips is defined as the optical lever constant b,which can be adjusted if necessary.When measuring,two front foot tips are firstly placed on the fixed platform of Young’s modulus measurement equipment,and the rear foot tip is on the level pallet fixed and connected with the bottom end grip holder of the metallic wire.It is necessary to keep the three screws on the same horizontal level by adjusting the position of the level pallet.
Figure 3.3.1 Scheme of Young’s Modulus measuring apparatus
Figure 3.3.2 Scheme of optical lever magnifying principle
Now,let’s introduce in detail the principle of measuring micro elongation in the experiment.The schematic diagram of optical lever magnification principle is shown in Figure 3.3.2.Supposing that the optical lever constant is b,the distance from adjusted reflecting plane mirror to the measuring scale(on the side of optical lever reflector)is D,and the variation in length of steel wire caused by adding of weight m isΔL,and the reading of the ruler in the telescope before and after adding weights are n1 and n2,respectively.When the steel wire is stressed due to the external force induced by the weights,the rear foot tip makes micro motion with the pallet.Therefore,the optical lever will change a micro angle around the front foot tip to drive optical lever reflector to turn a relative micro angle.As a result,we can observe an obvious change in the measuring scale because the micro angle displacement is magnified into larger linear displacement.Therefore,the difference between two readings isΔn=n2-n1,which is proportional to theΔL.In accordance to the geometric relation of each physical quantity in Figure 3.3.2,it can be expressed as:(https://www.daowen.com)
In addition,according to the refection law of light,the angle between the incident light and reflected light is 2θ because the normal of the mirror passes through the angle θ,which can be expressed as:
BecauseΔL is a small quantity andΔL≪b,
Then,
Namely,
In the experiment,b is 4～8 cm,D is 1～2 m,so the magnification for the optical lever,which is defined asΔn/ΔL=2(D/b),can be 25～100.Therefore,the micro-elongation of the steel wireΔL can be accurately calculated by magnifying into three easily measured quantities(b,D andΔn)using the optical lever.
Finally,substitute F=mg and Equation(3.3.8)into Equation(3.3.2),Young’s Modulus of steel wire can be written as:
