Swinging+Ball+-+Amir+and+Ellen

Title of Lab: Swinging Ball

Researchers: Amir Raheem and Ellen Apple

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

Research: Due to the conservation of energy, the initial energy will equal the final energy. So, mgh = (1/2)mv 2, where h is the initial height and v is the final velocity. Since net work = the change in kinetic energy, mg(L - Lcos(theta)) = (1/2)mv 2. The masses divide out. Multiply by 2 to get rid of the fraction. So, 2g(L - Lcos(theta)) = v 2 also written v 2 = -2gLcos(theta) + 2gL Then to get cos(theta) by itself, 2gL(1 - cos(theta)) = v 2



Hypothesis If the angle of the ball's height is larger, then the ball will have higher maximum velocity with a V 2 to cos(theta) relationship.

Materials: 1 Physics stand, 1 pendulum board, 1 string (with ball attached), 1 ruler, and 2 photogates (with a time measurement device)

Constants: distance between the photogates' sensors (about 2.6 cm), length of the string with the ball (which we didn't find)

Variables: Independent variable -- angle(theta) (particularly cos(theta) Dependent variable -- velocity (m/s)

Procedure: 1. Set up the Physics stand, and attach the pendulum board (the board should be placed to the side so that one of the sides with 30 degrees is parallel to the stand so that you can record data at higher angle measures) to it. Then, attach the ball and string to the pendulum board. 2. Set up the photogates so that the sensors are about 2.6 cm apart (the photogates will be right next to each other). You should also check to make sure that the ball and string activate the sensors. 3. Move the ball on the string to a 5 degree angle, and start the interval setting for the photogates. (Make sure the ball doesn't hit the sides of the photogate or anything else during the trial.) 4. Record the time given, and reset it. Repeat these steps for every angle (increase in 5 degree increments) until you get to 60 degrees. 5. Once you've recorded the times, find the velocity in cm/s for each angle using the distance between the sensors (2.6 cm) and the time gathered for that angle. (To get the velocity in m/s, divide the resulting velocities above by 100.)

Data Here is the original data collected Here is the new data which forms a negative sloping line From this graph we take the velocity and square it, and take the cos of the angle.
 * Theta || V ||
 * 0 || 0 ||
 * 5 || 0.257 ||
 * 10 || 0.5 ||
 * 15 || 0.7 ||
 * 20 || 0.974 ||
 * 25 || 1.22 ||
 * 30 || 1.59 ||
 * 35 || 1.75 ||
 * 40 || 1.912 ||
 * 45 || 2.222 ||
 * 50 || 2.407 ||
 * 55 || 2.766 ||
 * 60 || 2.889 ||
 * Cos(Theta) || V 2 ||
 * 0.5 || 8.346321 ||
 * 0.573576 || 7.650756 ||
 * 0.642787 || 5.793649 ||
 * 0.707106 || 4.937284 ||
 * 0.766044 || 3.655744 ||
 * 0.819152 || 3.0625 ||
 * 0.866025 || 2.5281 ||
 * 0.906307 || 1.4884 ||
 * 0.939692 || 0.948676 ||
 * 0.965925 || 0.49 ||
 * 0.984807 || 0.25 ||
 * 0.996194 || 0.066049 ||
 * 1 || 0 ||

Data Analysis The point (0,1) states that with a 0 degree angle, the velocity at the bottom of the curve is also 0. Using the line of best fit, the Y-intercept appears to be 14. This states that the average rate of change of cos(theta) vs. Velocity 2 is -14 m 2 /s 2. Using the equation v 2 = -2gLcos(theta) + 2gL we found that the expected slope is -13 m 2 /s 2. giving us an error 92% accuracy. Another way of finding the slope is -2gl.

Conclusion The data supports our hypothesis that the cosine of the angle is linearly related to the velocity squared with only 8% inaccuracy. The causes of error could be associated with technology inefficiency, incorrect readings of the angle.