Method Part A
An old 35mm film container was used to hold Canadian $1 coins (loonies); a paper clip was bent into a “V” shape and attached to the top of the container by electrical tape. A rubber band was place inside the paperclip to enable the container to be hung from a nail to measure the elasticity of the rubber band. The rubber band measured 56mm with no loonies in the container. The ruler was attached to the wall to reduce the amount of random error. One by one the loonies were added to the container and the results were recorded. Table 1 has the results with the number of loonies being substituted by their individual masses.
Table 1. Extension of a rubber band with loonies
In Table 1, the mass is the mass added not including the container and rubber band. The total mass is the total mass of the loonies in the container. The results regarding length was the raw reading from the ruler and the extension is just the subtraction of the ruler reading from the starting length. The total weight was the total mass multiplied by 9.81 m/s2. The results were graphed using Graphical Analysis and a linear regression was performed to see the relationship between the extension and the force applied. Figure 1 is the graph that was produced with the linear regression line. The line was the best fit for this particular model.
The total weight is the applied force so the graph is a representation of Hooke’s Law. F=-kx. With this being the case then k (proportionality constant) is 31.95 N/m, which is the slope of the regression line.
Analysis Part A
The Graphical Analysis fit could have been better. Since Hooke’s law is a linear law a linear regression was performed. Not all of the dots aligned as good as was hoped. The errors were probably due to random errors in measurement.
According to the results the rubber band should be able to stretch to 1 meter when 32.01 N of force is applied (32.01 N = 31.948 N/m (1m) + 0.05871 N). To be able to achieve this then 32.01 N / 9.81 m/s2 = 3.263 kg of mass needs to be added. This translates into 3.263 kg X (1000 g / 1 kg) = 3263 g, 3263 g / 6.97 g = 468 loonies needed. This wouldn’t work because the rubber band would break before it reached a meter. To test this the rubber band was stretched until it broke and it was nowhere near 1 meter.
When holding the container with the loonies it was quite evident that there was a downward force being exerted. When the same container was attached to the rubber band and pulled downward the rubber band presented an upward force against the container. The force of the rubber band can be expressed by a force constant, k. The formula to find it out is F = -kx, with x the extension in meters. This would make the units of k, N/m, with N referring to the Newton kg.m/s2. The force constant that the rubber band exerted against the container with loonies was 0.4786 = -k . 0.012
Method Part B
A pendulum was made from the same container used in part A by tying thread to it and hanging it over an open door to allow the necessary swing. The length of the string was changed between tests. Each test consisted of ten full oscillations, which were timed and recorded. Figure 2 is the graphed results from the test comparing the time for ten oscillations and the length of the string. The graph was created using Graphical Analysis software.
Table 2. Raw data from the experiment
Analysis Part B
This part of the experiment looked at a pendulum to see the relationship between the length of string and the time it took to do ten full oscillations. Unlike the rubber band experiment the results produced a curve. The best-fit curve was produced from a power regression using the Graphical Analysis software and the Ti-83 plus calculator to cut down on systematic error. This fit was much better than the linear regression because more points lie on the curve and it also supports what the textbook says.
Part C
In both experiments the possibility of errors are a concern; a concern that must not be forgotten. Systematic error is caused by the mis-collection of data or an improper model. One type of error that is always found is random error. It is the combination of errors that are important and calculated differently depending on the circumstances. Relative error gives more meaning to the importance of a random error. It is much easier to see the influence of a particular error when it is compared to the whole to make a percentage of error. The general function to determine error is a derivative of the function multiplied by the error. For example the error formula of y = x5 then the error formula is 5x4 Δx. The error formula for y = √X is (1 / (2√X)) ΔX. Shortcuts can be used when determining the relative error of z when z = xy by adding the relative errors of x and y (Δz = Δx/x + Δy/y). This can be done because of the following proof: Δz = xΔy + yΔx and z = xy then Δz/z = (xΔy + yΔx) / xy. The shortcut Δz/z = Δx/x + Δy/y is the same equation when a common denominator is calculated as xy. The same holds true for division z = x/y with the same result being Δz/z = Δx/x + Δy/y. Traditionally the equation would initially look like Δz/z = ((xΔy + yΔx)/y2) / (x/y), which is also the same as saying ((xΔy + yΔx) / y2) X (y/x). After the y/x is multiplied through then the same equation as the former is produced which is Δz/z = (xΔy + yΔx) / xy therefore proving the results.
Method Part D
Part D of the experiment looked at the quadratic function and its unique properties. The quadratic function of y = 10 + 30t – 4.9t2 was graphed using Graphical Analysis. This function represents a ball thrown upwards at 30 m/s with gravity working against it causing a downward motion. Figure 3 shows the resulting graph. This is just a theoretical situation and does not represent data collected.
Analysis Part D
The graphed quadratic function helps to understand the quadratic relationship more closely. It is a very useful function in physics because it is often seen when using motion. The roots are a very important part of the function. These are the points at which the curve crosses through the horizontal axis when y = 0. To figure this out the quadratic function can be rewritten as: x = -b/2a ± √b2 – 4ac / 2a. The graph is helpful to quickly see where the roots are. The Graphical Analysis software allowed me to zoom into the roots closer than what figure 3 allows to be seen. Visually the roots looked to be –0.32 and 6.44 but with a calculator the equation was a little more accurate for the first root at –0.317. Using the proper number of significant figures then the answers would be the same whether calculated or visually enhanced.
Looking at the equation more closely shows an interesting equation within the bigger one. Before the “±” symbol is –b / 2a, which is the same for both roots. This is important because the quadratic equation is symmetrical and the –b / 2a equation points to the apex of the curve, like a mid-point. Another note on this point is that the point is also where the slope = 0, which is when the ball would begin falling back to the earth.
Conclusion
This lab looked at some of the different physical relationships that are current theories or laws. The experiments verified these relationships.