Swinging+Ball-+Nathan+Devan,+Jonathan+Hannings,+Nathan+Campbell

Swinging Ball Lab
 * Title of Lab**:

**Researchers**: (The Nathan Team) Nathan Devan, (Jo)Nathan Hannings, Nathan Campbell

**Research Question**: How does the initial angle with the vertical relate to the speed of a swinging ball at its lowest point? **Research**: Gravity, a conservative force, is the primary force acting upon the ball as it accelerates along its path of motion. Because of this fact, the law of conservation of mechanical energy applies to this situation.

__Law of Conservation of Mechanical Energy __: //If only conservative forces are acting, the total mechanical energy neither increases nor decreases in any process, it remains constant.// //Equation//: mechanical energy (ME) = potential energy (PE) + kinetic energy (KE)

Based on the law of conservation of mechanical energy, we can deduce the mechanical energy at the moment of release (its highest point) is equal to the energy at the ball’s lowest point.

Therefore, the initial potential energy= the final kinetic energy

PE(initial) = KE(final) [at its lowest point]

since PE = mgh and KE= 1/2mv^2, we can deduce that: mgh = 1/2mv^2 By eliminating the masses, our equation: gh = 1/2v^2 Solving for v^2, our equation: v^2 = 2gh


 * ( http://en.allexperts.com/q/Physics-1358/2011/4/Physics-pendulum-1.htm) **

Based on the above image, we can determine that cos( **θ**)= (L - h)/L (L is the length of the string and the radius of the ball's circular path of motion) Solving for h, we get the equation: h = L - L(cos **θ**)

Since L remains constant during the experiment, we can substitute it into the previous equation: v^2= 2g(L - L(cos **θ**)) Distributing 2g gives us the equation, __ v^2= -2gL(cos **θ)** + 2gL __

The cosine of the angle will be inversely proportional to the speed of the ball, based on the equation v^2= -2gL(cos **θ)** + 2gL
 * Hypothesis ** :

// Materials: // 1. A stand with a horizontal peg on top. 2. string 3. ball 4. photogate equipment 5. ruler 6. protractor
 * Procedure ** :

// Constants: // Length of the string Diameter of the ball Gravity

//Variables:// Speed (our dependent variable) Angle (our independent variable)

// Procedure: // 1. Set up the vertical stand 2. Attach the ball to the end of the string, then tie the other end of the string to the vertical stand 3. Measure the length of the string 4. Position the photogate sensor so that the ball passes through at its lowest point of its arc 5. Measure the diameter of the ball 6. Let the ball swing from a 10 degree angle through the photogate and record the time it takes 7. Repeat step 6 at 10 degree intervals (20,30, 40, etc.) 8. To find the speed, divide the diameter of the ball by the time at each angle.

Length of the string: 84 cm (0.84m) Diameter of the ball: 3.25 cm (0.0325 m)
 * Data ** :
 * Angle (degrees) || Time #1 (seconds) || Time #2 (seconds) || Speed #1 (m/s) || Speed #2 (m/s) || Average Speed ||
 * 10 || 0.0671 || 0.0675 || 0.48 || 0.48 || 0.48 ||
 * 20 || 0.0289 || 0.0331 || 1.12 || 0.98 || 1.05 ||
 * 30 || 0.0209 || 0.0218 || 1.56 || 1.49 || 1.5 ||
 * 40 || 0.0162 || 0.0162 || 2.01 || 2.01 || 2.01 ||
 * 50 || 0.0128 || 0.0131 || 2.54 || 2.48 || 2.5 ||
 * 60 || 0.0110 || 0.0113 || 2.95 || 2.88 || 2.9 ||
 * We derived the velocity by dividing the diameter (0.0325 m) of the ball by the time it took for the ball to pass through the photogate.

Next, we determine the cos**θ** values as well as the speed^2 values
 * Cosine(**θ**) || Average Speed^2 ||
 * 0.985 || 0.23 ||
 * 0.940 || 1.10 ||
 * 0.866 || 2.25 ||
 * 0.766 || 4.04 ||
 * 0.643 || 6.25 ||
 * 0.500 || 8.41 ||


 * Data Analysis:**

Our experimental slope was found to be -16.98
The line of best fit crosses the x-axis at a point close to (0.98, 0 m^2/s^2). This intercept makes sense as when cosine(theta) reaches 1 (which is the cosine of 0 degrees), it is at rest. If error were eliminated, the x-intercept would be much closer to (1, 0 m^2/s^2). Our y-intercept is found to be approximately 17.01. This number is close to opposite the slope number, signifying that it is the rate of change between the x and y-intercepts. The y-intercept signifies the greatest velocity that the ball can attain, when the angle is 90 degrees (its highest point).

Based on our research, our theoretical slope would equal 2gL (with L being the length of the string) Theoretical Slope=2gL=2(9.81m/s^2)(0.84m)= -16.4 Percent Error=3.5%

Our data supported our hypothesis, as a negative linear relationship can be seen, and the slope follows the 2gL model. As the angle increases, the cosine of that angle will decrease, and the velocity^2 will increase. Our error may be a little high, but we believe it is a fair representation of our hypothesis. Error in our experiment could have been improved through several methods: making sure no outside force affects the ball when it is released, taking measurements at smaller intervals, performing multiple trials at each angle, making sure the angles are more exact, measuring the diameter of the ball more accurately, making sure the string had no slack before being released. The experiment could have been improved by using different string lengths or different objects (other than a ball) to confirm our hypothesis more strongly.
 * Conclusion ** :