Lab 14 Measurement of Youngrsquo;s Modulus of Metallic Wire by Static Tension Method

Background and application

Purpose

This experiment covers a basic concept of Youngrsquo;s modulus. Our purposes of this experiment are:

1. to determine the Youngrsquo;s modulus of a steel wire and

2. to learn how to measure a tiny deformation;

3. to handle the experiment data using a graphic method

Theory

The shape of solid can be changed by the action of external force，which is called“Deformation”. When the external force within certain limit and after the external force action stoping，the deformation completely disappears，which is called“Elastic deformation”.When the external force is too high，the deformation can not disappear completely and surplus deformation remains，which is called“Plastic deformation”，i.e. that deformation still exists after the external force is removed，which is an irreversible process. Increasing the external force gradually till surplus deformation appears can achieve the object elastic limit.

Youngs modulus, also known as the tensile modulus or elastic modulus, is a measure of the stiffness of an elastic material and is a quantity used to characterize materials. In solid mechanics, the slope of the stress-strain curve at any point is called the tangent modulus. The tangent modulus of the initial, linear portion of a stress-strain curve is called Youngs modulus. It can be experimentally determined from the slope of a stress-strain curve created during tensile tests conducted on a sample of the material.

In Figure 1, the length of a tested wire is , and F is the magnitude of the force applied perpendicularly to the area A of the wire. The strain, or the unit deformation, is then represented by the dimensionless quantity , here represents the initial length of the wire and the stretch produced by the force F.

Figure 1: A stress on a wire F/S leading to a strain/l

The stress is proportional to the strain before a permanent deformation for the specimen occurs, and the constant of the proportionality is called the modulus of elasticity, Youngrsquo;s modulus represented by the symbol Y. Hence, we have the equation as below:

(1)

Once the stress, F/A, and strain,l/l, are measured by the equipment in our lab the Youngrsquo;s modulus, Y, can be determined for a specific sample in the lab. If mg represents the weight of a mass m applied to the wire, and d is its average diameter , then

(2)

Now the problem is focused on how to determine the value , a very tine value after the sample is stretched because of the elasticity.

There is an optical lever device to be used for measuring the tine change of the wire.

This device is seen as a detail on the left side of Figure 2. The optical lever with a mirror rides on a small platform. Two legs b and c of it stay in the groove of the platform while one leg rides on the chuck moved freely through the yoke. The chuck clamped to the lower end of the tested wire is just through the hole of it. Once the tested wire is stretched by a weight the mirror is consequently tilted up also, as a result, the reading of the ruler viewed from the telescope is changed as follows. The changes of the reading is proportional to the strain of the stretched wire.

Figure 2: Measuring by an optical lever

(a, b and c are three legs of the optical level; a is on the chuck while b and c ride in the groove of the small platform)

From Figure 2, it is seen that the elongationl is given by

(3)

where D is the distance from the mirror to the scale and N is the difference between readings on the scale produced by the applied weight. By the motion of the mirror through the angle, the reflected beam has therefore been turned through an angle 2. As a result, we get

(4)

where K is the perpendicular distance (8 cm) of the mirror to the point on the chuck. Hence, the L is obtained by

(5)

Equation (1) becomes

(6)

Measure d, K and D, L,and N. We can measured the Youngrsquo;s modulus.

Apparatus

1. A steel wire or rod for testing，upper end of which is clamped by a chuck fixed on top of tripod

2. A heavy tripod with 3 leveling screws and with a yoke，a chuck clamped to t he lower end of the tested wire is just through the hole of it

3. Weights as the load for the wire hanging on the lower end of the tested wire

4. Optical lever with a mirror

5. A telescope

6. Measuring scale

Procedure

1. Aligning the system

a. Place the optical lever on the small platform of the tripod with the mirror setting vertically. Set the telescope and the scale at least 1 meter away from the mirror and at about the same height.

b. Adjust the telescope (elevation, height and focus), the tripod (direction) and

剩余内容已隐藏，支付完成后下载完整资料

---

用静态拉伸法测量金属丝的杨氏模量

实验简介

本实验的目的是:1.学习如何测定和计算钢丝(或其它金属丝)的杨氏模量; 2.学习如何利用光学杠杆测量被测材料细微的形变.光杠杆是本实验的关键部件，学生要理解为什么它能把细微的长度变化的测量变得可能。理解什么是应变和什么是应力。它们之间的比例系数就是杨氏模量。学生通过本实验除了测量原理外应该学会如何使测量变得精确，例如对负重的处理（轻托轻放），望远镜中如何观察和读出标尺。最后如何画图，如何计算斜率以及最后如何给出正确的杨氏模量测量值的表达式。我们也希望学生能估算他们的测量误差以及探讨误差的来源。

目标

本实验涵盖了杨氏模量的基本概念。本实验的目的是：

1．确定钢丝的杨氏模量；

2．学习如何测量微小形变；

3．用图解法处理实验数据

理论

固体的形状可以由外力的作用来改变，这就是所谓的“形变”。当外力在一定限度内停止作用后，形变完全消失，称为弹性形变，当外力过大，形变不能完全消失，产生剩余形变，被称为“塑性形变”，即形变撤去外力后仍然存在，这是一个不可逆的过程。通过逐渐增大外力作用直到产生剩余形变，能够达到材料的弹性极限。

杨氏模量，也称为拉伸模量或弹性模量，是衡量弹性材料的刚度的一种手段，是用来定量地描述材料的参数。在固体力学中，任一点的应力-应变曲线的斜率称为切线模量。应力-应变曲线初始的线性部分的切线模量，被称为杨氏模量。由对一个材料样品进行拉伸试验所得到的应力-应变曲线的斜率，就可以得到杨氏模量的实验确定值。

在图1中，一段待测材料的长度为，F是垂直地施加在材料截面S的力的大小。应变，或单位变形，可由无量纲值表示，这里表示导线的初始长度、表示力F所产生的形变长度。

图 1在导线上的应力F/S，导致应变/l

在发生永久变形之前，应力和应变是成正比的，并且这个比值就是弹性模量，杨氏模量由字母Y表示。因此，我们有以下公式：

（1）

一旦应力F/A和应变l/l在实验室中被测量出来，则这个杨氏模量Y可以作为实验室里的一个特定的样品。若用mg代表我们所施加于导线上质量为m的物体的重量，且d表示导线的平均直径，则有

（2）

现在的问题是如何确定这个由于材料受到拉力而产生的微小形变的值：。

有一种光学杠杆装置用于测量线材微小的形变。

如图2左图所示，这个光杠杆装置是一个镜子放在基座上，两条腿b和c置于平台的槽沟里，同时一条腿能在轭圈中自由地移动。导线的一端固定在它的孔上。一旦导线被一个物体所拉伸，反射镜将相应地发生倾斜，因此，从望远镜中观察到的标尺的读数也随之改变。读数的改变量亦与导线所受的拉力成正比。

图 2由光学杠杆装置测量

（a、b、c为光杠杆的三条腿；a连在该盘上的，而b和c置于平台的槽中）

从图2可得到伸长量可由下式得到：

(3)

D是从镜子到刻度尺的距离、N是由所施加的重量产生的刻度的读数之间的差值。当镜子转动了角度，反射光束则转过2。因此，我们得到了

(4)

K是镜子到导线连接点的垂直距离（8厘米），因此，得到L

（5）

公式（1）变为:

（6）

这样通过测量d, K 和D，我们便可以测量出杨氏模量。

实验仪器

1. 用于测试的钢丝，其上端固定于三角架顶端的固定架上
2. 一个带有3个水平调节螺丝和1个轭圈的三脚架
3. 作为负载的砝码挂在测试导线的下端
4. 光学杠杆装置
5. 望远镜
6. 测量标尺

实验步骤

1. 系统校准
2. 将光杠杆放在三角架小平台上，并保持镜子垂直。设置望远镜和刻度尺与镜子保持至少1米的距离，并保持二者高度大约相同。
3. 调整望远镜（仰角、高度和焦距）、三脚架（方向）和反射镜（仰角），这样你就可以看到标尺测量刻度的反射像。
4. 倾斜的镜子或重置刻度尺，直到尺子刻度的中点与望远镜里的十字基准线重合。
5. 确定杨氏模量
6. 在托架上小心地放置2公斤的砝码，每当增加或减少砝码时都将一只手放在托架下，以确保图像不受不必要的冲击而产生振动。记住这2公斤是保持导线的初始质量，不要把它算为第一个质量。
7. 通过望远镜记录刻度尺上的初试读数D₀，并稍等一会，观察这个读数是否已停止变化。如果不是，再等一分钟，看看是否仍在改变。这样你便获得了一个正确的阅读，以避免错误的初始读数引起导线的弹性滞后。
8. 往托架上增加1公斤砝码，重复上述步骤，记录下新重量下D的读数。重复此过程直到托架上有8公斤的砝码了。
9. 三次试验。将所有数据填写到下面表格中。算出这三组数据的平均值，并计算每个值的偏差D。
10. 用米尺，来测量导线两头间的长度。
11. 用千分尺，在两不同点互成直角地测两次导线的直径，共测得四组数据。这些测量数值用厘米记录。
12. R的距离是从镜子量到刻度。

数据记录与处理

表1：实验中的变量

d	D	L	K

表2：数据记录

频率	拉伸质量/kg	增长量 (10-2m)	减少量 (10-2m)	均值 (10-2m)	Successive different value (10-2m)
0	2.000			n0=	N1= n4－n0=
1	3.000			n1=
2	4.000			n2=	N2= n5－n1=
3	5.000			n3=
4	6.000			n4=	N3= n6－n2=
5	7.000			n5=
6	8.000			n6=	N3= n7－n3=
7	9.000			n7=
均值					=

结果与讨论

在你的实验室报告中，有必要讨论以下问题：

1. 为什么距离R要这么大？
2. 为什么伸长率的测量是如此缓慢？
3. 如何减少你的不确定性？

剩余内容已隐藏，支付完成后下载完整资料

---

资料编号：[31897]，资料为PDF文档或Word文档，PDF文档可免费转换为Word