Elasticity Of Solids
Some solids can be stretched or compressed more easily than others. A piece of copper wire, for example, can be stretched, although a similar length of rubber band can be stretched much more. However, there is a difference in the stretchiness of these two substances that goes beyond mere extent. If you let go of a rubber band after stretching it, it will spring back to its original length, but if you let go of a copper wire, it will stay stretched.
The elasticity of a substance is the extent of its ability to return to its original dimensions after a sample of it has been stretched or compressed. According to this definition, rubber has high elasticity, and copper has low elasticity. Note that elasticity, defined in this way, is qualitative (it says something about how a substance behaves) but is not truly quantitative (we aren’t assigning specific numbers to it). Scientists can and sometimes do define elasticity according to a numerical scheme, but we won’t worry about that here. It is worth mentioning that there is no such thing as a perfectly elastic or perfectly inelastic material in the real world. Both these extremes are theoretical ideals.
This being said, suppose that there does exist a perfectly elastic substance. Such a material will obey a law concerning the extent to which it can be stretched or compressed when an external force is applied. This is called Hooke’s law: The extent of stretching or compression of a sample of any substance is directly proportional to the applied force. Mathematically, if F is the magnitude of the applied force in newtons and s is the amount of stretching or compression in meters, then
s = kF
where k is a constant that depends on the substance. This can be written in vector form as
s = k F
to indicate that the stretching or compression takes place in the same direction as the applied force.
Perfectly elastic stuff can’t be found in the real world, but there are plenty of materials that come close enough so that Hooke’s law can be considered valid in a practical sense, provided that the applied force is not so great that a test sample of the material breaks or is crushed.
Elasticity Of Solids Practice Problems
Problem
Suppose that an elastic bungee cord has near perfect elasticity as long as the applied stretching force does not exceed 5.00 N. When no force is applied to the cord, it is 1.00 m long. When the applied force is 5.00 N, the band stretches to a length of 2.00 m. How long will the cord be if a stretching force of 2.00 N is applied?
Solution
Applying 5.00 N of force causes the cord to become 1.00 m longer than its length when there is no force. We are assured that the cord is “perfectly elastic” as long as the force does not exceed 5.00 N. Therefore, we can calculate the value of the constant k , called the spring constant , in meters per newton (m/N) by rearranging the preceding formula:
s = kF
k = s/F
k = (1.00 m)/(5.00 N) = 0.200 m/N
provided that F ≤ 5.00 N. Therefore, the formula for displacement as a function of force becomes
s = 0.200 F
If F = 2.00 N, then
s = 0.200 m/N × 2.00 N = 0.400 m
This is the additional length by which the cord will “grow” when the force of 2.00 N is applied. Because the original length, with no applied force, is 1.00 m, the length with the force applied is 1.00 m + 0.400 m = 1.400 m. Theoretically, we ought to round this off to 1.40 m.
The behavior of this bungee cord, for stretching forces between 0 and 5.00 N, can be illustrated graphically as shown in Fig. 103 . This is a linear function; it appears as a straight line when graphed in standard rectangular coordinates. If the force exceeds 5.00 N, according to the specifications for this particular bungee cord, we have no assurance that the function of displacement versus force will remain linear. In the extreme, if the magnitude of the stretching force F is great enough, the cord will snap, and the displacement s will skyrocket to indeterminate values.
Practice problems of these concepts can be found at: Basic States Of Matter Practice Test
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