Hookes Law Laboratory - Lab Report Example

A weight, W = mg, is hung from one end of an ordinary spring, causing it to stretch a distance x, then an equal and opposite force, F, is created in the spring which opposes the pull of the weight. If W is not so large as to permanently distort the spring, then this force, F, will restore the spring to its original length after the load is removed. The magnitude of this restoring force is directly proportional to the stretch in the relation below.

F = -kx

The constant k is called the spring constant. To emphasize that x refers to the change in length of the spring we write the relation as follows.

F = mg = - k ∆ l (1)

From the relation, it is evident that if a plot of F as a function of ∆ l has a linear proportion. This confirms that the spring conforms to Hooke's Law and enables us to find k mathematically. (Sears, 1981)

Thus we can assume an equation k = y = mx + c………equation 1

Objective

The objective of this experiment is to study the behavior of ordinary springs in static and dynamic situations. We will determine the spring constant, k, (K which is the stiffness of the spring), for an individual spring using both Hooke's Law and the properties of an oscillating spring system.

Apparatus

• 2 Extension Springs (long, short)
• Compression spring
• Load to be used for long springs
• 2 Long Screws
• 2 Short Screws

Experimental Setup

Procedure
1. Load a spring with different weights (5 different weights) and measure the displacement associated with each weight as shown in figure 1.
2. Repeat for different springs.
3. Document your readings and use them to derive a conclusion

Results
Table 1
Long Spring(5N - 9N)
F (N) 5 6 7 8 9
dX (mm) 4 6 13 20 25
dX (m) 0.004 0.006 0.013 0.02 0.025

Table 2
Short Spring (2.5N - 10N)
F (N) 2.5 3.5 5.5 7.5 10
dX (mm) 3 14 35 56 82
dX (m) 0.003 0.014 0.035 0.056 0.082

Table 3
Compression (0 -5N)
F (N) 1 2 3 4 5
dX (mm) 2 5 11 16 23
dX (m) 0.002 0.005 0.011 0.016 0.023

Figure 2 indicates that for forces greater than about 4.5N (notice intercept of best fit), there is a linear relation between force and extension. For small loads, such a relationship fails since the fit curve does not intercept the y-axis at zero. It is assumed that this is caused by an initial "set" in the spring which requires some initial load to overcome. This is apparent if one stretches the spring manually and then releases it. It seems to snap shut at the last moment.

For this reason, 0 and 4.5N were ignored and the rest of the data were treated by a least squares analysis to determine the coefficients of first-degree polynomial best fit.
These were used to plot the line on the graph. The slope of the line, ignoring loads of less than 4.5N, was found to be 147.36 N/m. From Equation 1, we see that we need to multiply this quantity by g to calculate a value for the spring constant of k = 217.4 ± 1.8 N/m.

Using the equation k=y=mx + c where m is the slope of the graph and c is the y-intercept. We find that;

k = 147.36N/m

A graph of force versus the magnitude of displacement resulted in the expected straight line in the range of forces examined and is consistent with Hooke’s law. The slope of this line, 147.36 N/m, is the spring constant, which agrees with the value found by taking the average of the calculated spring constant. The intercept for the best-fit straight line intersects close to the origin, which is also consistent with Hooke’s law.

The location was measured relative to the base of the mass hanger. The measurements are viewed directly; however, due to parallax, it is measured at a slight angle. However, this sighting was required for each measurement, and the displacement was the difference between the location and the reference. Thus, this systematic error from parallax should be minimized. Consequently, a motion sensor to measure distance would increase the precision for small displacements.

Conclusion

Ultimately Hooke’s law was verified as per the initial objective. The linear relationship between incremental loads against the extension of the spring is observed in figure 2. Read More