Swinging+Ball-+Holly+Therell+and+Swathi+Ganesh

= Title of Lab: Conservation of Energy Inquiry Lab- Swinging Ball =

Researchers:
Swathi Ganesh and Holly Therrell

Research Question:
How does the initial angle with the vertical relate to the speed of a swinging ball at its lowest point?

Research:
Initial Energy = Final Enerrgy mgh = .5mv 2 Since the masses divide out, we are left with gh = .5v 2 If the length of the string is L, using a trigonometric ratio we can say the initial height is equal to L-Lcos(theta). Therefore, g(L-Lcos(theta)) = .5mv 2 gL-gLcos(theta) = .5v 2 v 2 = -2gLcos(theta) + 2gL

Hypothesis:
According to our research, we predict that the velocity squared will have a negative linear relationship with the cosine of our measured angle according to the equation: v 2 = -2gLcos(theta) + 2gL in which v is the final velocity, g is acceleration due to gravity, L is the length of the string, and (theta) is the angle with respect to the vertical. Given this equation we would expect the slope and y-intercept to be equal but opposite of one another.

Materials:
meter stick, ball with a diameter of 0.035 m, string to which ball is attached, pendulum stand, protractor attached to top of pendulum stand, photogate

1) Measure the length of the string of the pendulum. 2) Place one photogate directly in front of the pendulum stand where the lowest point of the pendulum would swing, and where the "laser: of the photogate will hit the ball at its center.  3) Drop the ball, keeping the string straight, from various angles and catch the ball after it passes the photogate and swings to the other side. Repeat angles if possible.  4) Record the time measured by the photogate.  5) Divide the diameter of the ball (in meters) by the measurement given by the photogate (in seconds).
 * Procedure: **

**Data:** Ball diameter: 0.035 m Length of string: 0.915 m
 * Angle (⁰) || Time (s) || Velocity (m/s) ||
 * 0⁰ || 0.0000 || 0.0000 ||
 * 5⁰ || 0.0560 || 0.6250 ||
 * 5⁰ || 0.0640 || 0.5469 ||
 * 10⁰ || 0.0520 || 0.6731 ||
 * 10⁰ || 0.0464 || 0.7543 ||
 * 15⁰ || 0.0308 || 1.136 ||
 * 15⁰ || 0.0315 || 1.111 ||
 * 20⁰ || 0.0255 || 1.373 ||
 * 20⁰ || 0.0261 || 1.341 ||
 * 25⁰ || 0.0215 || 1.628 ||
 * 25⁰ || 0.0201 || 1.741 ||
 * 30⁰ || 0.0177 || 1.977 ||
 * 30⁰ || 0.0181 || 1.934 ||
 * 35⁰ || 0.0162 || 2.160 ||
 * 35⁰ || 0.0152 || 2.303 ||
 * 40⁰ || 0.0143 || 2.448 ||
 * 40⁰ || 0.0143 || 2.448 ||
 * 45⁰ || 0.0124 || 2.823 ||
 * 45⁰ || 0.0124 || 2.823 ||
 * 50⁰ || 0.0119 || 2.941 ||
 * 50⁰ || 0.0118 || 2.966 ||
 * 55⁰ || 0.0109 || 3.211 ||
 * 55⁰ || 0.0110 || 3.182 ||
 * 60⁰ || 0.0102 || 3.431 ||
 * 60⁰ || 0.0101 || 3.465 ||

Data Analysis:
The slope of the best fit line is approximately -23.47 m^2/s^2/cos θ, while the theoretical value, found by plugging in the corresponding values to 2gL (L being the length of the string), is approximately -17.95 m^2/s^2/cos θ. This gives us a percent error of 30.75%, which could have been due to imprecise angle measurements or the imprecise measurement of the diameter of the ball.
 * Cos θ || Velocity^2 (m^2/s^2) ||
 * 1 || 0.0000 ||
 * 0.9962 || 0.3906 ||
 * 0.9962 || 0.2991 ||
 * 0.9848 || 0.4531 ||
 * 0.9848 || 0.5690 ||
 * 0.9659 || 1.291 ||
 * <span style="font-family: 'Times New Roman',serif; font-size: 12pt;">0.9659 || <span style="font-family: 'Times New Roman',serif; font-size: 12pt;">1.234 ||
 * <span style="font-family: 'Times New Roman',serif; font-size: 12pt;">0.9397 || <span style="font-family: 'Times New Roman',serif; font-size: 12pt;">1.885 ||
 * <span style="font-family: 'Times New Roman',serif; font-size: 12pt;">0.9397 || <span style="font-family: 'Times New Roman',serif; font-size: 12pt;">1.798 ||
 * <span style="font-family: 'Times New Roman',serif; font-size: 12pt;">0.9063 || <span style="font-family: 'Times New Roman',serif; font-size: 12pt;">2.650 ||
 * <span style="font-family: 'Times New Roman',serif; font-size: 12pt;">0.9063 || <span style="font-family: 'Times New Roman',serif; font-size: 12pt;">3.031 ||
 * <span style="font-family: 'Times New Roman',serif; font-size: 12pt;">0.8660 || <span style="font-family: 'Times New Roman',serif; font-size: 12pt;">3.909 ||
 * <span style="font-family: 'Times New Roman',serif; font-size: 12pt;">0.8660 || <span style="font-family: 'Times New Roman',serif; font-size: 12pt;">3.740 ||
 * <span style="font-family: 'Times New Roman',serif; font-size: 12pt;">0.8192 || <span style="font-family: 'Times New Roman',serif; font-size: 12pt;">4.666 ||
 * <span style="font-family: 'Times New Roman',serif; font-size: 12pt;">0.8192 || <span style="font-family: 'Times New Roman',serif; font-size: 12pt;">5.304 ||
 * <span style="font-family: 'Times New Roman',serif; font-size: 12pt;">0.7660 || <span style="font-family: 'Times New Roman',serif; font-size: 12pt;">5.993 ||
 * <span style="font-family: 'Times New Roman',serif; font-size: 12pt;">0.7660 || <span style="font-family: 'Times New Roman',serif; font-size: 12pt;">5.993 ||
 * <span style="font-family: 'Times New Roman',serif; font-size: 12pt;">0.7071 || <span style="font-family: 'Times New Roman',serif; font-size: 12pt;">7.969 ||
 * <span style="font-family: 'Times New Roman',serif; font-size: 12pt;">0.7071 || <span style="font-family: 'Times New Roman',serif; font-size: 12pt;">7.969 ||
 * <span style="font-family: 'Times New Roman',serif; font-size: 12pt;">0.6428 || <span style="font-family: 'Times New Roman',serif; font-size: 12pt;">8.649 ||
 * <span style="font-family: 'Times New Roman',serif; font-size: 12pt;">0.6428 || <span style="font-family: 'Times New Roman',serif; font-size: 12pt;">8.797 ||
 * <span style="font-family: 'Times New Roman',serif; font-size: 12pt;">0.5736 || <span style="font-family: 'Times New Roman',serif; font-size: 12pt;">10.31 ||
 * <span style="font-family: 'Times New Roman',serif; font-size: 12pt;">0.5736 || <span style="font-family: 'Times New Roman',serif; font-size: 12pt;">10.13 ||
 * <span style="font-family: 'Times New Roman',serif; font-size: 12pt;">0.5 || <span style="font-family: 'Times New Roman',serif; font-size: 12pt;">11.77 ||
 * <span style="font-family: 'Times New Roman',serif; font-size: 12pt;">0.5 || <span style="font-family: 'Times New Roman',serif; font-size: 12pt;">12.01 ||

The significance of the y-intercept, in this case about 23 m^2/s^2, is that this is approximately equal to the additive inverse of the slope of the line. This is as a result of the equation v 2 = -2gLcos(theta) + 2gL, which shows that both the slope and the y-intercept have the same numerical value. This is also as a result of the x-intercept being at 1, and the x-intercept is the point that we know for sure since this the velocity when the ball is dropped from 0 degrees (whose cosine is 1) has to be 0.

**Conclusion:** When we compare our experimental slope to our theoretical slope, we have an error of 30.75%. Though this error is higher than desired, the fact that the slope of the line of best fit is nearly equal to the additive inverse of it's y intercept (there is a 2% error) indicates that our data does support the hypothesis because we expected our slope and y intercept to be equal but opposite since they were -2gL and 2gL respectively. A significant source of error in our experiment that resulted in such low accuracy compared to our expectations was probably the imprecise nature of our angle measures and the diameter of our ball. Could we repeat the experiment, we would like to have more time to carefully measure each angle and do many repetitions at each angle value. However, at this point we have significant support for the hypothesis that v^2 and the cosine of the angle share a negative linear relationship using the equation previously discussed.