Assuming that toward the top of the page is the positive y axis, write the forces acting on the ball in terms of their components in the form of 10 pts

Other by Geoffrey
Figure 2: A displaced ball hanging from a string.
  1. Activity Based Questions

 

a)  On the diagram on the right to draw a free body (or vector) diagram with arrows showing the direction of each of the forces on the mass, m, including the gravitational force, , the force the string exerts, , and a horizontal Coulomb force due to the other ball, .   Use “Insert>Shapes” to draw the arrows.

Note: The diagram is not to scale. Actually the string length L>>R/2 and  is small enough so that we can make the approximation that

where R/2 is half the distance between the centers of the two   balls.

 

  1. b) Assuming that toward the top of the page is the positive y axis, write the forces acting on the ball in terms of their components in the form of 10 pts

 

 

 

  1. c) Show that when q is small and the balls are in equilibrium, the net force on each of them is zero and the magnitude of the Coulomb force on each ball is given by.  Show your derivation here.   10 pts

 

 

 

  1. d) Write the equation needed to calculate the distance between the centers of the balls, R, as a function of the locations of the center of each ball where X is the left ball’s location and X2 is the right ball’s location.    10 pts

 

 

 

  1. e) Open the Logger Pro experiment file <cmbl>, which has the movie inserted. Click on the “R” column header and add the appropriate equation for R using the Calculated Column option. You should find that R seems to decrease linearly in time between about 100 s and 520 s. Does that seem to be the case? If the answer is no, check your work. 5 pts

 

 

 

  1. f) If the balls are far enough apart to behave more or less like equal point charges, then Fcoul should be given by Fcoul = kq2/R2. Write an equation that can be used to determine the amount of charge, q, on each ball as a function of m, R, and L as well and the gravitational and Coulomb constants, g and k. Show your derivation here.   10 pts

 

 

 

  1. g) Your next task is to use the same Logger Pro file and the new column feature once again to determine the magnitude of the charge on each ball for times between 100 s and 520 s. Pay attention to the notes in the box in the <cmbl> file, and note that L = 1.68 m, m1 ≈ m2 ≈ 0.0029 kg and k = 8.99 x 109 N•m2/C2. If you enter the equation needed to calculate q properly, you should find that q also seems to decrease linearly in time between about 100 s and 520 s. Use the Examine tool to select the times of interest. Then perform a linear fit to determine rate of charge decrease and write it in the space below with appropriate units. Warning: The charges in Coulombs are quite small. If your linear fit shows 0.000 for slope, you should double click on the box that displays the fit parameters and choose four significant figures rather than four decimal places for the Display Precision. Also explain which fit parameter gives you the value of . 10 pts

 

= ______________ C/s

 h) Copy and paste your graph, including the fit, here. 20 pts

 

i) Recall that your final challenge is to estimate how many minutes it should take for the balls’ centers to be one diameter apart (i.e., when R = 0.0377 m) so the balls touch ‑ ending the discharge process. Your supervisor has suggested that you find the slope and intercept for the linear portion of the R vs. Time graph (from t = 100 s to t = 520 s) using the Examine tool and then the Linear Fit Then you can use the linear discharge equation to find the time in seconds at which R = 0.0377 m. You also realize that you need to calculate that time in minutes instead of seconds. Show your work in the space that follows and report your estimated number of minutes to two significant figures. 10 pts

 3Reflections on Your Findings

4a) Can you think of reasons why the charge on each ball decreases over time and where the charges might go? 10 pts

 

b) The linear model you used for times between 100 s and 520 s matched your data very well. During the first 100 s the rate of loss of charge differs from your linear model. This suggest that some other path for charge loss may be important when each ball carries even more excess charge. Can you think of a physical reason why this might happen? 10 pts