Swinging+Ball-+Sneha+Mittal,+Yuki+Kurosu+&+Daniel+Hayes

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

Researchers: Sneha Mittal, Yuki Kurosu, and Daniel Hayes

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

Research: The principle of the conservation of mechanical energy states that if only conservative forces are acting on a system, __E__ neither increases nor decreases in any process. Mechanical energy is conserved, **E** i **= E** f.

Total mechanical energy is the sum of kinetic and potential energies at any moment. Since vi = 0 and hf = 0, can be rewritten as and further simplified to
 * (mv** i **^2)/2 + mgh** i **= (mv** f **^2)/2 + mgh** f
 * mgh** i **= (mv** f **^2)/2**
 * gh** i **= (v** f **^2)/2.**

Considering the graphics depicted, hi can be solved for using trigonometry. Using substitution,
 * L = L* cos(**θ** ) + h i **
 * h i = L- L* cos(**θ** ) **
 * g (L- L* cos(**θ** )) = (v f ^2)/2 **
 * 2g (L- L* cos(**θ** )) = (v f ^2) **
 * v f ^2 = -2gLcos(**θ** ) + 2gL **

Hypothesis: Theta and v f are directly proportional according to the following equation:
 * v f ^2 = -2gLcos(**θ** ) + 2gL **.

Materials: - Swinging ball stand with angle measurements on top - String attached to ball - Photogates

Procedure: 1. Set the two photogates 5.5 cm from each other centered around the ball's lowest point. 2. Measure the string to be 69.8 cm in length. 3. Lift the ball to an angle of 10 degrees with the vertical. 4. Record the time interval between the two photogates. 5. Repeat Steps 3-4 two more times to ensure precision. 6. Repeat Steps 3-5 in angle increments of 10 degrees up to 60 degrees. 7. Calculate the velocity using the data.

Data:

The three trials produced the following results: The velocity was calculated to two significant figures using the average values for time and the equation **d= rt**:
 * Angle (degrees) || Time 1(sec) || Time 2 (sec) || Time 3 (sec) || Average Time (sec) ||
 * 0 || 0 || 0 || 0 || 0 ||
 * 10 || .1133 || .1022 || .1027 || .1061 ||
 * 20 || .0531 || .0578 || .0595 || .0568 ||
 * 30 || .0389 || .0364 || .0378 || .0377 ||
 * 40 || .0300 || .0300 || .0302 || .0301 ||
 * 50 || .0247 || .0286 || .0286 || .0273 ||
 * 60 || .0232 || .0235 || .0238 || .0235 ||
 * Angle (degrees) || Velocity (m/s) ||
 * 0 || 0 ||
 * 10 || 0.52 ||
 * 20 || 0.97 ||
 * 30 || 1.5 ||
 * 40 || 1.8 ||
 * 50 || 2.0 ||
 * 60 || 2.3 ||

Data Analysis:

The graph of angle vs. velocity takes the following form:  When velocity squared (m/s) 2 is plotted against cos(theta), the best-fit line is linear.



Using the hypothesized equation **v f ^2 = -2gLcos(**θ** ) + 2gL, ** the slope (**- 2gL** ) was anticipated to be -13 m^2/s^2 and the y-intercept (**2gL**) was to be 13 m^2/s^2 (additive inverses of each other). T he linear regression equation given by Microsoft Excel states the slope to be -11 m^2/s^2 and the y-intercept of the graph to be 11 m^2/s^2. Since the slope and y-intercept are, in fact, additive inverses of each other, the data can be concluded precise. When compared to the hypothesized slope and y-intercept, the percent accuracy is determined to be 85%.

Conclusion: Based on the precise data and percent accuracy, one can conclude that the experimental results supported the proposed hypothesis. The 15% error could be a factor of experimental error; in future experiments, the procedure may be revised to include better technology which enables proper alignment of the ball through the photo gate.

Experimentation and data analysis lead to a fuller understanding of basic physical forces that enable technological innovation; engineers, architects, and many others apply these findings in every day life. Hence, physics is essential, needed for the betterment of society and life as whole.