VCE Physics Unit 1 Analysis of an Experiment An introduction to Experimental Methods in Physics
Introduction Experiments, in any field of science, are aimed at collecting results, analyzing them and finding relationships between the measured and/or collected results. The ultimate experiment is one that collects data from just a few trials which, after analysis, allows the development of a universal law (usually expressed as a mathematical equation) which is applicable anywhere, anytime. The classic example of this is Newton’s Law of Universal Gravitation which, it is said, he worked out by watching apples fall to the ground. The aim of THIS exercise is to investigate water flows from cans and develop mathematical rules or laws which can predict how water will flow out of any can, anywhere, anytime.. Before proceeding we need to take a small mathematical diversion.
A Mathematical Diversion The best way to find out whether two quantities are mathematically related is to GRAPH them. Lets call the two quantities “y” and “x”. We collected the following data. Plotting this data gives the following graph. This is a straight line graph, indicating y x. (y is directly proportional to x) The general equation for a straight line graph is y = mx + c, (where m = slope or gradient and c = y intercept). The slope = (20 – 6)/(9 – 2) = 14/7 = 2 And the y intercept = 2 So y = 2x + 2. We now have a LAW that will allow us to work out any value for y from a chosen value of x. 0 4 8 12 16 20 y x 2 4 6 8 10 y x 2 4 5 6 7 9 6 10 12 14 16 20 2 0
Cans, Water and Holes The experiment investigated the time it took for various depths of water to empty from holes (of various diameters) punched into the bottom of cans. You would expect that the time it took to empty the can would depend upon: 1. The diameter of the hole, and 2. The depth of the water To investigate the relation between time to empty and hole diameter, 4 large cylindrical cans were filled with the same volume of water and allowed to empty through holes of varying diameter . The time taken for each to empty was recorded. To investigate the relation between depth of water and time to empty, the same cans were filled to different depths. The time taken for each to empty was recorded.
Experimental Results The results of the experiments were recorded and presented in Table 1 below The times quoted were taken by a hand operated stopwatch. This timing method introduces an error of 0.1 sec for each reading. TABLE 1 Time to Empty (sec) Water Depth (cm) 30.0 10.0 4.0 1.0 Hole Diameter (cm) 1.5 73.0 43.5 26.7 13.5 2.0 41.2 23.7 15.0 7.2 3.0 18.4 10.5 6.8 3.7 5.0 6.8 3.9 2.2 1.5
A First Analysis The data collected and shown in Table 1 contains the relationships between hole diameter and time to empty and between water depth and time to empty, but they are not obvious simply by looking at the numbers. We need to ANALYSE the data. ANALYSIS No. 1 HOLE DIAMETER VERSUS TIME TO EMPTY The data (in Table No 1) has 3 variables (water depth, hole diameter and time to empty). In order to study the relation between hole diameter and time to empty we need to hold the third variable (water depth) fixed . So Table No 2 contains information on hole diameter and time to empty for a fixed depth of 30.0 cm. Water Depth 30.0 cm Hole Diameter (cm) 1.5 73.0 2.0 41.2 3.0 18.4 5.0 6.8 Time to Empty (s) Table No 2
A First Graph Just by looking at Table No 2, it seems that there is an INVERSE RELATIONSHIP between hole diameter and time. This means as hole diameter goes up, time to empty goes down. But what is the EXACT relation between diameter and time ? The only way to find out is to plot a graph. Hole diameter is the independent variable and is plotted on the horizontal axis. Time to Empty is the dependent variable and is plotted on the vertical axis. GRAPH No 1 Water Depth 30.0 cm Hole Diameter (cm) 1.5 73.0 2.0 41.2 3.0 18.4 5.0 6.8 Time to Empty (s) Table No 2 Hole Diameter (cm) Time to Empty (s) 1.0 2.0 3.0 4.0 5.0 20 40 60 80
An Inverse Relationship <ul><li>Graph No 1’s shape definitely indicates an inverse relationship exists between time and diameter. </li></ul><ul><li>But there are two types of inverse relations that could exist. </li></ul><ul><li>t 1/d, or </li></ul><ul><li>t 1/d 2 </li></ul><ul><li>So which one is it ? More investigation is needed. </li></ul><ul><li>This requires further manipulation of the data and further graphs. </li></ul>Table No 3. 73.0 41.2 18.4 6.8 1.5 2.0 3.0 5.0 0.67 0.50 0.33 0.20 0.44 0.25 0.11 0.04 If one of the graphs (t against 1/d or t against 1/d 2 ) produces a straight line , we will have established an exact mathematical relationship. ie. A LAW relating t and d. Time to Empty (s) Hole Diameter (d) in cm 1/d 1/d 2
A Second Graph Plotting a graph of Time to Empty against 1/d Should the point (0,0) be on the graph ? Yes, because because as d approaches infinity ( ) the value of 1/d approaches 0. An infinitely large hole will take no time to empty. This graph is NOT a straight line. Thus we must conclude that t is NOT to 1/d. GRAPH No 2. 1/d (cm) Time to Empty (s) 0.2 0.4 0.6 0.8 1.0 20 40 60 80 73.0 41.2 18.4 6.8 0.67 0.50 0.33 0.20 Time to Empty (s) 1/d
A Third Graph Plotting a graph of time to Empty against I/d 2 Within experimental limits, This graph IS a straight line. Thus we can say t 1/d 2 . We need to convert the proportionality ( ) to an equation in order to formulate the LAW which relates t and d. GRAPH No 3. 1/d 2 (cm) Time to Empty (s) 0.10 0.20 0.30 0.40 0.50 20 40 60 80 Time to Empty (s) 1/d 2 73.0 41.2 18.4 6.8 0.44 0.25 0.11 0.04
A Law Relating t and d Having determined that t 1/d 2 . We need to convert this to an equation. This is done by recognizing the graph is a straight line with general formula y = mx + c, where y = t, x = 1/d 2 , m = slope and c = y intercept. =(73.0 – 6.8)/(0.44 – 0.04). = 66.2/0.4 = 165.5 And y intercept = 0 Thus equation becomes: t = 165.5/d 2 . Thus we have developed a LAW which allows us to predict the time to empty a 30.0 cm depth of water for ANY diameter hole. Slope = Rise/Run Rise Run GRAPH No 3 . 1/d 2 (cm) Time to Empty (s) 0.10 0.20 0.30 0.40 0.50 20 40 60 80
A Second Analysis ANALYSIS No 2 WATER DEPTH VERSUS TIME TO EMPTY With a fixed hole diameter we can investigate the relationship between Water Depth (h) and Time to Empty (t): Table No 4 contains information for various depths of water and time to empty for a fixed hole diameter of 1.5 cm. Graphing this information, we get GRAPH No 4 Hole Diameter = 1.5 cm Water Depth (cm) 30.0 73.0 10.0 43.5 4.0 26.7 1.0 13.5 Time to Empty (s) Table No 4 Water Depth (cm) Time to Empty (s) 5.0 10.0 15.0 20.0 25.0 20 40 60 80 30.0
A Parabolic Relationship Graph No 5 is shaped like a parabola laid on its side. General Formula of this line is: y = x 2 . Normal Parabola “ Sideways” Parabola General Formula of this line is: y 2 = x. Or y = x So graph No 5 appears to show a relationship of the form Time to Empty (t ) square root of Depth ( h) Further investigation is needed. x y x y
A Second Graph To determine whether the relation we suspect is true, we need to plot a graph of t against h. To plot the graph we need data: The graph IS a straight line, thus our guess about the relationship is true. Hole Diameter = 1.5 cm Water Depth (cm) 5.48 73.0 3.16 43.5 2.00 26.7 1.00 13.5 Time to Empty (s) Table No 5 Depth (cm) Time to Empty (s) 1.0 2.0 3.0 4.0 5.0 20 40 60 80 6.0
A Law Relating t and d Having determined that t h, we need to convert this to an equation. The general equation for a straight line is y = mx + c, with y = t , x = h, m = slope and c = y intercept. Slope = Rise/Run =(73.0 – 13.5)/(5.48 – 1.00) = 59.5/5.48 = 10.86 And y intercept = 0 Thus the equation becomes t = 10.86 h We have now developed a LAW to predict the Time to Empty ANY depth of water from a hole 1.5 cm in diameter. Depth (cm) Time to Empty (s) 1.0 2.0 3.0 4.0 5.0 20 40 60 80 6.0 Rise Run