Lab 8 Report - Grade: A PDF

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Report for Lab 8 PHY241 Max Brown...

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Lab #8: Ballistic Pendulum Introduction: The purpose of this experiment was to determine the initial velocity of a steel ball fired from a launcher using two different methods. A few of the technical terms included in this report are defined here. Velocity is the time rate of change of an object’s position, with an attached vector indicating the velocity’s direction. The center of mass is the point within the bounds of an object equivalent to the average of the object’s mass. The problem we evaluated in our experiment was to determine the initial velocity of the ball based on the maximum angle the pendulum reached and the change in height of the center of mass of the pendulum when the ball was fired into it. We found that the average initial velocity of our steel ball launched with the “long range” setting was 4.90 ± 0.01 m/s. This was calculated using the maximum angle of the pendulum’s swing, the radius of the pendulum (pivot to center of mass), and the maximum height difference of the center of mass of the pendulum during the swing. For comparison, the initial velocity of the steel ball was calculated using an entirely different method involving shooting the steel ball at the same “long range” setting into a cart. A CBR would record the cart’s velocity automatically. In this second method, the average initial velocity was found to be 4.46 ± 0.01 m/s. This was calculated using the velocity of the ball and the card after the collision, the mass of the ball, and the mass of the ball combined with the cart.

Procedure: In this lab we realized that we had some systematic uncertainty. This was due to the fact that we did not measure the center of mass correctly. This through the rest of our calculations off a little which is one of the reasons the velocity calculated in method 1 is not identical to the velocity calculated in method 2. Another possible source for our velocities not being identical was due to random uncertainty. In this particular lab there are many random uncertainties. See following list 1. Friction between ball and ballistic pendulum barrel. a. Every time the ball is shot out of the barrel of the ballistic pendulum there is friction. This is not always the same for every test shot. b. There is nothing you can do to minimize this uncertainty. 2. In method 2 a. Ballistic pendulum shooter was not lined up with cart exactly which could cause a component of the velocity to be in the direction perpendicular to the track rather than all of it being in the parallel direction like it should be.

b. To minimize this we tried to measure as accurately as possible the center and line it up. Method 1 procedure In method 1 the initial velocity of the ball was calculated by using conservation of energy. The initial energy being kinetic energy after the ballistic pendulum is fired, and the final energy being stored in gravitational potential energy. 1. Equation a. KEi = GPEf i. 1/2mv^2=mgh b. Using this formula solve for v because that is the initial speed at which the ball was fired. 2. The change in height (h) is the y-component of the pendulum's displacement. It starts at an initial position, and the displacement is the vector leading from the initial position to the final position. Then you just take the sin of the displacement to get the change in height (h).

Method 2 procedure Conservation of Momentum was used for our method 2 using the CBR, Cart with foam catcher, and the ballistic pendulum. 1. 2. 3. 4. 5. 6. 7.

Measure mass of cart and mass of ball Level track Place ballistic shooter at one end of track Place CBR on track about 1 meter away from the ballistic shooter. Shoot ball into cart with the foam catcher on top. Measure velocity of the cart using CBR Calculations a. Use conservation of momentum of a perfectly elastic collision b. M1Vi=(M1+M2)Vf c. Solve for Vi d. Vi is the initial speed of the ball

Analysis: A summary of our results is presented in Graph 1. The initial and final velocities in both methods are compared to show trends and uncertainties. Graph 2 shows the change in the

pendulum’s height with respect to the maximum angle achieved. This graph is included in order to show the uncertainty in the change in height due to the inaccuracy in measuring the pendulum’s center of mass.

Velocity of Ball Before and After Collision 6.00

Velocity (m/s)

5.00 4.00 3.00 2.00 1.00 0.00

1

2

3

4

5

Trial V before collision (m/s) [Method 1] V before collision (m/s) [Method 2]

V after collision (m/s) [Method 1] V after collision (m/s) [Method 2]

Graph 1. This graph shows the velocity before and after the collisions calculated using both methods. All data points for method two are below the respective points for method one, which indicates some form of systematic error during the experiment. This was likely due to the center of mass being calculated inaccurately in method one, shifting the data upward. It could also be systematic error in the CBR or manual timed actions in method two.

Change in Pendulum Height vs. Angle 0.062

Change in Height (m)

0.060

f(x) = 0 x − 0.06

0.058 0.056 0.054 0.052 0.050 37.90

38.00

38.10

38.20

38.30

38.40

Angle (degrees) Change in height (m)

Linear (Change in height (m))

38.50

38.60

Graph 2. This graph shows the change in height of the pendulum’s center of mass with respect to the maximum angle of the pendulum’s swing. Two of the trials had duplicate points, thus, only three data points are shown. The error bars represent the uncertainty due to the way the pendulum’s center of mass was calculated.

Figure 1 and Figure 2 show the data values calculated and used in methods one and two, respectively.

Method 1 Trial 1 2 3 4 5

Angle (degrees) 38.00 38.25 38.50 38.00 38.25

Δh (m) 0.060 0.061 0.061 0.060 0.061

Velocity after collision (m/s) 1.08 1.09 1.10 1.08 1.09

Velocity before collision (m/s) 4.87 4.90 4.93 4.87 4.90

Initial Angle (degrees) 1 1 1 1 1

Radius of Pendulum (m) 0.282 0.282 0.282 0.282 0.282

Method 2 Trial 1 2 3 4 5

Velocity after collision (m/s) 0.427 0.397 0.352 0.427 0.422

Mass Ball & Cart (kg) 0.727 0.727 0.727 0.727 0.727

Mass ball (kg) 0.066 0.066 0.066 0.066 0.066

Velocity before collision (m/s) 4.70 4.37 3.88 4.70 4.65

Figure 1. This table shows the initial angle of the pendulum (at rest), the maximum angle the pendulum reached, the change in height of the pendulum, the radius of the pendulum (i.e. the distance from the pivot point to the center of mass), the velocity of the ball and pendulum after the inelastic collision, and the velocity of the ball before the collision. The radius of the pendulum was determined by finding its center of mass and measuring the distance from the pivot point to the center of mass. The radius was used, along with the maximum angle, to calculate the change in height of the pendulum (r – r*cos[(θ*π)/180], where θ is the maximum angle). This change in height was then used, along with the acceleration due to gravity, to find the velocity of the ball after the collision (vafter = sqrt[2*9.8*0.282 kg]). This velocity, along with the mass of the ball and pendulum, was used to calculate the velocity of the ball before the collision (vbefore = [(mb + mp)*vafter]/mball). The mass of the ball is 0.066 ± 0.001 kg and the mass of the pendulum is 0.231 ± 0.001 kg, and were both measured using a triple-beam balance.

Figure 2. This table shows the mass of the ball, the combined mass of the ball and cart, the velocity of the ball after the collision with the cart, and the velocity of the ball before the collision. The masses were measured using a triple-beam balance. The velocity of the cart and ball after the inelastic collision was measured using a CBR. The velocity of the ball before the

collision was calculated using the conservation of momentum equation (vbefore = [0.727 kg * vafter] / 0.066 kg).

Conclusion: In graph 1, we noticed that nothing agreed with the tread lines or errors bars because we did not have any in our first graph. Not only that but we did not notice some systematic error in our data that we had. Everything seemed to be pretty constant and non-changing throughout our experiment that we did. In graph 2, we did notice that the tread lines did agree with the data by passing through error bars. The errors bars in graph two all lined up with our data points on the graph and complete the graph. With that, we did not notice any systematic errors when completing this lab and experiment. All of our data was good, the graph was completed and done well, and all of our data was reasonable and had no issues with any equipment or tools throughout the lab. The graphs that we provided from this lab are very relevant when playing a role into answering the research question. The graph plays the role of having the proof to back up our answer, having the data that we can look back on for reference and for being a reliable source of information....