Hooke's law is a concept that can be applied to objects that can deform, when acted on by a force, and return to their original state when alleviated of that force. This includes objects such as springs, which can be both compressed or stretched, as well as elastics. But what exactly does Hooke's law tell us about these objects? Hooke's law can be defined with the following formula:
The formula states that the force required to compress or stretch an object that is elastic in its properties is equal to, and directly proportional to the product of its spring constant (k), and displacement of length (x) from an initial neutral position known as equilibrium. The relationship is simple: to compress or stretch an elastic objects greater distances requires more force, and also requires more force the greater the spring constant is.
Finding the Spring Constant
The spring constant, k, is a unique value, measured in newtons per meter (N/m), that is different for different types of springs/elastics. Since it is proportional to the force applied to displace a spring/elastic, we can deduce that rigid objects such as large compression springs used in industrial machinery will have a large spring constant while an easily stretchable household elastic will have a low spring constant.
In order to find the spring constant, we must isolate k. We can do this by simulating a situation in which the spring force is proportional to another force. In order for this to occur, the net force must equal 0, meaning that acceleration must equal 0.
Method
In order to find the spring constant, we must isolate k. We can do this by simulating a situation in which the spring force is proportional to another force. In order for this to occur, the net force must equal 0, meaning that acceleration must equal 0.
Method
By hanging a mass from an elastic band, we can create a situation in which spring force is equal to another type of force - in this case, gravitational force. By attaching an elastic band to a peg in the wall and attaching different masses to it, we can find the spring constant k by measuring the displacement of the elastic that is a result of the gravitational force exerted by each of the masses hung on the elastic. This is shown in the below equation:
Figure 3.1: Method used to find the displacement of an elastic under different gravitational forces.
Masses that the elastic could sustain such as this water bottle weighing 515g were hung from the elastic in a bucket (mass = 102g).
Raw Data
Figure 3.2: Mass and Length of Elastic Band
Mass (g)
|
Length (cm)
|
0
|
6.5
|
102
|
7.7
|
249
|
8.5
|
457
|
9.5
|
604
|
10.1
|
764
|
12.5
|
972
|
15.6
|
Processed Data
We first must convert mass and length into standard units before calculating gravitational force and displacement. Our initial length for displacement is the length of the elastic band when it is holding 0 g.
Sample Calculations
Fg = mg
= (0.457kg)(9.8m/s)
= 4.48 N
d = Final Length - Initial Length
= 0.085m - 0.0065m
= 0.030 m
Figure 3.3: Gravitational Force of Mass and Displacement of Elastic Band
Gravitational Force (N)
|
Displacement (m)
|
0
|
0
|
1.00
|
0.012
|
2.44
|
0.020
|
4.48
|
0.030
|
5.92
|
0.036
|
7.49
|
0.060
|
9.53
|
0.091
|
Finding the Slope
While it is possible to algebraically find the spring constant, k, by dividing the gravitational force by the displacement for each case and finding the average k value, we can also do this graphically by finding the slope using a linear regression analysis. Since k = mg/x, the resulting slope should equal the value of k the same way the slope of a distance vs time graph will give velocity.
*Note that Fg must be on the Y axis and d must be on the X axis in order for the value of the slope to reflect that of the spring constant.
While it is possible to algebraically find the spring constant, k, by dividing the gravitational force by the displacement for each case and finding the average k value, we can also do this graphically by finding the slope using a linear regression analysis. Since k = mg/x, the resulting slope should equal the value of k the same way the slope of a distance vs time graph will give velocity.
*Note that Fg must be on the Y axis and d must be on the X axis in order for the value of the slope to reflect that of the spring constant.
Figure 3.4: Linear Regression to Find the Spring Constant
According to the regression analysis, the collected data shows a strong correlation in which the R-squared value is 0.9357. The resulting slope of the best fit line is 109 and therefore, we can say that the spring constant for the elastic band is 109 N/m.
We can confirm this value manually by picking two points on the best fit line and and finding the divisor of change in gravitational force and change in displacement. In this case, we pick the lowest and highest points on the best fit line.
Evaluation of the Method
The R-squared value of the collected data does not show an entirely perfect correlation; while it is not weak, it's not the strongest either. While fault in the employment of the procedure may have been the resistance of the wall. Masses were hung on the elastic by placing them in a bucket and its large size caused it to have contact with the wall. On top of this, the elastic band was hooked from a peg in the wall that was located in an inside corner, increasing the contact with the wall. Often, the elastic had to be manually stretched before allowed to naturally hang so the friction of the wall wouldn't hold up the stretching. This may explain inconsistencies in the data such as the 4th and 5th data points.
