Lab 1 Graphical Analysis PDF

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Lab 1 - Graphical Analysis...

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Phys1113 – Lab #1 Graphical Analysis September 21, 2020

Objective The objective of this experiment is to explore how graphical data analysis techniques can be used to determine functional relationships between experimentally measured quantities. Specifically, we are to find the mathematical relationship between the length extension of a wooden ruler (independent variable) and the amount of displacement. Apparatus Materials used: A 600 gram mass, a C-clamp, a safety block, and 2 1-metre sticks One of the stick (stick A) is used as a cantilever beam and will deflect under the action of the 600-g mass. The other stick (stick B) is used as a ruler to measure amount of displacement of stick A. Stick A will be clamped onto the bench at one end and extends out from the bench. Please see image below for apparatus set up:

To conduct this experiment, the mass will be kept constant, while stick A will be extended from 40cm to 90cm inclusively in increments of 5cm. At each extension, the stick’s initial vertical position without mass and its displacement will be recorded.

Theory In this experiment, since we are suspending a mass on the end of a wooden stick that is clamped onto a bench, hence the gravitational force will be the only force that is acting on the stick, resulting in a displacement. In class, we learnt that gravity is at 9.80m/s 2 [downward]. With that knowledge, I would assume the displacement vs extension distance would have a power law relationship. Results

Uncertainty in the position measurement is ~0.5cm. Analysis y vs L graph

y vs L2 graph

y vs L3 graph

y vs L4 graph

My initial guess for the functional relationship between y vs L was a non-linear, exponential relationship, meaning that as the distance for the extension increases, the value of deflection would increase exponentially. The cartesian plot graph for y vs L showed my initial guess was correct, as the graph appears similar to a positive exponential growth. As I plotted the y vs L2 and y vs L3 graphs, the graph became more linear. As I plotted y vs L4, the curve seemingly became a negative exponential decay graph. log(y) vs log(L) graph

In the log(y) vs log(L) graph, the following information was given:

The slope of the log-log graph is A, which is 2.955±0.04415 (±0.04415 would be the margin of error for this slope). The functional relationship between log (y) and log (L) appeared to have a linear relationship, as the line of best-fit matched most of the dots on the graph, and there does not appear to have any outliers in this graph. Discussion In the log-log graph, the slope is 2.955±0.04415, this is related to the exponent n in the power relationship between length extension and amount displacement – as mentioned before, the graph y vs L had a non-linear relationship, but as we plotted y vs L 2 and y vs L3 graphs, we see the curve becoming more linear, but never a straight line. The y vs L3 graph had a curve most resembles a straight line, and that correlates to the slop of the log-log graph of 2.955 where it is very close to 3. As mentioned in the results section, the uncertainty in the position measurement is ~0.5cm. This is reflected from the slope’s error of margin (0.4415). In the initial y vs L graph, the graph appeared to show an exponential relationship. As we us bigger values for L, the graph appeared more linear because there is deflection value remained the same (same amount of deflection over greater amount of extension). The information from the log plot (having a relatively good linear fit) showed that when we use the log of extension and log of deflection, the relationship become more linear. This would help support that when we place a weight on the end of the ruler and extend it away from the table ledge, the amount of deflection would grow exponentially until

the ruler cannot support the weight anymore and snap or if the weight reaches the floor and prevents further displacement of the stick. Conclusion The objective of this experiment was to determine the functional relationship between the length extension of a wooden ruler and the amount of deflection when there is a mass hung on the end of the ruler. The graphs from the results proved there is a non-linear functional relationship between these two variables, and a linear relationship for the log of length extension and log of amount of deflection....