Graphing. C= d (1) Under constant acceleration, the relationship between the distance s an object moves and the time t it takes is given by
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1 Graphing Name Section Physics itself is all about mathematical relationships between variables. In class, you will study some of the more important relationships that have been found to exist. In the lab, you will perform experiments in order to confirm these relationships, and graphing is the tool that you will use. In this experiment, you will confirm two different relationships; that between circumference and diameter in a circle, and that between distance and time in an object undergoing a constant acceleration. Theory In the Lab Reference Manual (web site) you will find a section on Confirming Functional Relationships; you should read through this before doing the pre-lab. You will also want to look at Proper Form before plotting any graphs. The relationship between the circumference C and diameter d of a circle is given by where π is a dimensionless constant. C= d (1) Under constant acceleration, the relationship between the distance s an object moves and the time t it takes is given by s= 1 2 at2 (2) where a is the rate of acceleration. Here, we assume that the object starts at s = 0 with no initial velocity. Apparatus Disks, Vernier caliper, Meterstick, String, Scissors, Tape, Metal track, Support, Steel sphere, Stopwatch. Procedure Circumference and Diameter 1. Take a disk and measure its diameter with the vernier caliper. Record the data in Table Carefully measure the circumference of the disk using the materials provided. Do this 3 times and determine the average circumference. Record all data. 3. Repeat Steps 1 2 for each disk. 4. Graph the Average Circumference vs. Diameter (Graph 1). Plot the straight line of best fit over your data. Determine the slope of the line plotted. Fa07 Page 1 of 6
2 Measured Diameter Table 1 Disk Data Measured Circumference Average Circumference Distance and Time 1. Place the support under one end of the track and adjust its height so that you will be able to comfortably time the sphere as it travels to the bottom of the track when released at a distance of 20.0cm up from the bottom of the track. 2. Release the sphere 3 separate times from this location; each time, measure the amount of travel time for the sphere. Record these times in Table 2, and determine the average. 3. Repeat Step 2 at different distances from the bottom of the track. Spread these distances out relatively evenly over the total ramp distance that you have available. 4. Determine the square of each of the average times in Table Graph the Average Time Squared vs. Distance (Graph 2). Plot the straight line of best fit over your data. Determine the slope of the line plotted. Distance Traveled Table 2 Distances and Times Measured Times (s) Average Time (s) Average Time Squared (s 2 ) 20.0 Fa07 Page 2 of 6
3 Questions 1. How did you measure the circumference of the disks? What made this method the best, in your opinion? 2. According to Equation 1, what is the relationship between circumference and diameter for the circular disks? In other words, what what? 3. Is Equation 1 in the form of a straight line? If so, write it below; if not, rewrite it below in straight-line form. In your equation, identify the dependent variable, the independent variable, the slope, and the y-intercept (if applicable). 4. Does Graph 1 confirm the relationship between circumference and diameter you stated in Question 2? How? Fa07 Page 3 of 6
4 5. You determined the slope of the line in Graph 1. What is the percent error between this value and what you identified as the theoretical slope of the line in Question 3? 6. According to Equation 2, what is the relationship between distance and time for a uniformly accelerated object? In other words, what what? 7. In Equation 2, the distance traveled s is the dependent variable. Why then, in Graph 2, is it the independent variable? 8. Why did you plot the square of the time (and not just time) in Graph 2? Fa07 Page 4 of 6
5 9. What is the theoretical slope of the line you plotted in Graph 2 according to Equation 2? 10. Does Graph 2 confirm the relationship between distance and time you stated in Question 6? How? 11. What is the slope of the line in Graph 2 (including units)? How could you use this value to determine the acceleration of the sphere? Calculate the acceleration below. Fa07 Page 5 of 6
6 Pre-Lab: Graphing Name Section 1. For each of the power functions below: a) write in the form of a straight line b) state what proportionality exists between the variables c) state what you would need to plot (what vs. what) in order to get a straight line graphically d) state the theoretical slope of the line (from the equation). You may assume that y is the dependent variable, x is the independent variable, and everything else is a constant. i. y=cx ii. y= c x2 d iii. y= 3 x 2. How do you calculate the slope of a straight line? Fa07 Page 6 of 6
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