Swinging+Ball+-+Peyton+Coleman+and+Nick+Gant


 * Title of Lab:** Swinging Ball Lab


 * Researchers:** Peyton Coleman and Nick Gant


 * 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 law of conservation of energy, Ei = Ef We can then state: mgh = (1/2)mv^2 Simplified: gh = (1/2)v^2

Through the diagram above we can state: L = Lcos(theta) + h Solving for h, we get the following equation: h = L - Lcos(theta) Substituting the h into the law of conservation of energy equation: g(L - Lcos(theta)) = (1/2)v^2 Simplified: v^2 = 2g(L - Lcos(theta))

Constants: Gravity, length of string, mass of sphere Variable: Angle of string (independent), Speed of sphere at base of arc (dependent)
 * Hypothesis:** Due to our research and the law of conservation of energy, we hypothesize that v^2 = -2gLcos(theta) + 2gL

Materials: wooden sphere, string, stand, photogates, angle labels
 * Procedure:**
 * 1) Adjust stand and angle labels so the string is at 0 degrees with no force other than gravity and tension acting upon it.
 * 2) Set up photogates at the base of the stand.
 * 3) Raise wooden sphere up to an angle of 5 degrees with the vertical, release sphere, and record time interval displayed on photogates.
 * 4) Repeat step 3 with increasing increments of any angle measure that can be accurately recorded with angle labels on stand.
 * 5) Calculate speed of wooden sphere by dividing the distance between photogates by the time interval.

Distance between photogates: .051m L = .945m
 * Data:**
 * Angle(Degrees) || Time (s) || Speed (m/s) || Speed ^2 (m^2/s^2) || Cos(Theta) ||
 * 0 || 0 || 0 || 0 || 1 ||
 * 5 || 0.31 || 0.16 || 0.026 || 0.996 ||
 * 10 || 0.19 || 0.28 || 0.078 || 0.985 ||
 * 15 || 0.12 || 0.43 || 0.18 || 0.966 ||
 * 20 || 0.090 || 0.57 || 0.32 || 0.940 ||
 * 25 || 0.070 || 0.70 || 0.49 || 0.906 ||
 * 30 || 0.060 || 0.85 || 0.72 || 0.866 ||
 * 35 || 0.050 || 0.96 || 0.92 || 0.819 ||
 * 40 || 0.045 || 1.0 || 1.0 || 0.766 ||
 * 45 || 0.040 || 1.2 || 1.4 || 0.707 ||
 * 50 || 0.038 || 1.3 || 1.7 || 0.643 ||
 * 55 || 0.035 || 1.5 || 2.3 || 0.574 ||
 * 60 || 0.032 || 1.6 || 2.6 || 0.500 ||


 * Data Analysis:**

The slope of the line is -5.0 which means our y-intercept is 5. Our y-intercept of 5 means that if the sphere were to be dropped at a 90 degree angle and somehow managed to follow the arc it would reach a speed of the square root of 5m/s. The theoretical slope should be equal to -2gL = -18.54 with a y-intercept of 18.54. Our experimental g becomes 2.65, giving us a percent error of 73.0%. This error could be the result of some of the following: improper setup of the photogates, inaccurate angle measurement, slight added force when releasing the sphere, or sphere collision with the photogates. Based on our percent error, we can state that our hypothesis was incorrect according to our data. Theoretically our hypothesis is justified, but experimentally it is not. A repeated experiment could be conducted with a different length string, only one photogate to minimize the chance of collision, and a mechanism to drop the ball at precise angles and with negligible added force. This experiment is an example of experimental trumping theoretical information, for another experiment would be necessary in order to attempt to justify this hypothesis once more.
 * Conclusion:**