Lab Report 4

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Northeastern University *

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1141

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Mechanical Engineering

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Apr 3, 2024

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Report for Experiment #7 Work and Energy On An Air Track Abstract In this experiment, we used an air track with a glider on top of this track. The purpose of allowing the glider to simulate flight was achieved by opening the air vents. The change of glider position with time was tested and recorded in two separate experiments. Then the speed and the square of the speed are calculated from the time and the position. The acceleration of gravity is then calculated from the work energy theorem. Introduction In this experiment, we studied the movement of an object on an inclined horizontal plane. Ultrasonic pulses were sent through a motion sensor to track the position of the track at any time and record it. Using the data obtained in the experiment, we can then investigate the principles of work and energy. We mainly tested the motion on horizontal plane, inclined plane, and of a Glider Connected to a Hanging Weight. According to the work-energy theorem, the total work done by the force in a certain distance equals to the change in kinetic energy. The equation of kinetic energy is :

K = 1 2 mv 2 The next equation that we can find from the work-energy theorem is the work done on an object: W = F x ∆ X Due to the work-energy theorem, we can combine these two equation to get: F x ∆ X = 1 2 mf 2 + 1 2 m i 2 By using these equations, we can gain the equation of the final velocity of the motion along the inclined plane which is v 2 = 2 gsinθ ( x− x 0 ) . The velocity of the a Glider connected to a Hanging weight could be determined by using this function v 2 = 2 m' g m + m ' ( x− x 0 ) In investigation 1, the glider was putted on an inclined plane. We used the motion sensor to track the position due to the time of the glider and used these data to calculated g. In investigation 2, the glider was connected to a hanging weight and made it move through the horizontal plane. The same method to determine g. Both of the investigations were used to test the theorem of work- energy.. Investigation 1 The setup of investigation 1 were linear air track with glider and air pulley, PASCO PASPort USB Link, motion sensor, and a small block. At the beginning of the experiment, we first opened the air source of the air track in order to let the glider not be affected by the friction generated by the contact with the air track. Then we placed a small block of wood on the bottom of the track to raise it up. Once the track was set up properly, the glider was placed 20 cm away from the motion sensor and released as soon as the program was recorded. The sensor would record the position of the glider every 0.05 seconds. The time and distance travelled was recorded on the computer. After I gained the data on the computer, I putted them into the excel and drew a graph below.

Since the data which was recorded are too large, I tried to determined the first two maximum points and than only used the data from start to the second maximum points to minimum the size of the data. The maximum points means the time that the glider hit the base of the air track. Then, I used these data to create a second table. sin θ g(away) g(towards) g(avg) percentage difference 0.036058 8.300587 11.0045867 9.652587 0.016046211 Positio n (m) Run #3 error of positio n Time (s) Auto Velocity(m/ s) error of velocit y V^2 Error of V^2 x(avg) error of x(avg) 0.169 0.002 0 0.14 0.0566 0.019 6 0.0158 0.172 5 0.0014 0.176 0.002 0.05 0.16 0.0566 0.025 6 0.0181 0.18 0.0014 0.184 0.002 0.1 0.18 0.0566 0.032 4 0.0204 0.188 5 0.0014 0.193 0.002 0.15 0.22 0.0566 0.048 4 0.0249 0.198 5 0.0014 0.204 0.002 0.2 0.22 0.0566 0.048 0.0249 0.209 0.0014

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4 5 0.215 0.002 0.25 0.24 0.0566 0.057 6 0.0272 0.221 0.0014 0.227 0.002 0.3 0.26 0.0566 0.067 6 0.0294 0.233 5 0.0014 0.24 0.002 0.35 0.26 0.0566 0.067 6 0.0294 0.246 5 0.0014 0.253 0.002 0.4 0.3 0.0566 0.09 0.0339 0.260 5 0.0014 0.268 0.002 0.45 0.32 0.0566 0.102 4 0.0362 0.276 0.0014 0.284 0.002 0.5 0.34 0.0566 0.115 6 0.0385 0.292 5 0.0014 0.301 0.002 0.55 0.34 0.0566 0.115 6 0.0385 0.309 5 0.0014 0.318 0.002 0.6 0.36 0.0566 0.129 6 0.0407 0.327 0.0014 0.336 0.002 0.65 0.38 0.0566 0.144 4 0.0430 0.345 5 0.0014 0.355 0.002 0.7 0.4 0.0566 0.16 0.0453 0.365 0.0014 0.375 0.002 0.75 0.42 0.0566 0.176 4 0.0475 0.385 5 0.0014 0.396 0.002 0.8 0.44 0.0566 0.193 6 0.0498 0.407 0.0014 0.418 0.002 0.85 0.44 0.0566 0.193 6 0.0498 0.429 0.0014 0.44 0.002 0.9 0.48 0.0566 0.230 4 0.0543 0.452 0.0014 0.464 0.002 0.95 0.48 0.0566 0.230 4 0.0543 0.476 0.0014 0.488 0.002 1 0.5 0.0566 0.25 0.0566 0.500 0.0014

5 Since the table was too large, I only put the data that I collected for the first second of recording. The velocity was calculated by the equation that : V = x n + 1 −x n ∆t In which x is the position and t is the time.Then, we could simply determine the v square and the error of it by the following equation: δ v 2 = √ 8 v 2 ( δ x ∆t ) The error of x was 0.002 which I gained from the lab manual. After that, I can calculated the average position by using: X avg = x n + 1 + x n 2 The error of the average position was: δ X avg = √ 2 2 δ X Based on what I got after the calculation, I made a graph of v^2 and X avg . There were two lines in this graph. One of it represented the motion towards the sensor and the other one represented the motion away from the sensor. These two motion did not have the some slope due to the change of direction of the friction.

In this slope, I could get an equation of v^2 which is v 2 = Bx + C Based on what the equation that I wrote before v 2 = 2 gsinθ ( x− x 0 ) . The slope B is equal to 2gsin θ . The value of sin θ can be determined by: sin θ = h d In which h equaled to the height of the wooden block which is 3.75cm, and d is the length of the track which is 104cm. The value of sin θ is 0.036. The uncertainty of the sin θ could be determine by the following equation: δ sin θ sin θ = √ ( δh h ) 2 +( δd d ) 2 Which is 0.15. Due to the relationship between B,g, and si nθ . The g(away) that I calculated was 8.3 m/s^2. The g(towards) was 11 m/s^2. Then, I gained a average value of two g which is 9.65 m/s^2. The average value of g is pretty closed to the really g. The error may caused by the error of the air track which was not perfectly frictionless. I then determined the percentage difference of the g that I tested compared to the really g by using: %Dif = ¿ The percentage difference is 1.6%. Investigation 2 The setup in survey 2 is a little different from the setup in survey 1. In survey 2, I will keep the air track horizontal while fixing a weight to the end of a thin rope that will also be fixed to the glider. The same steps as in Survey 1 for recording data are repeated, but this time the glider will be released at a distance of 40 cm from the sensor. The mass of the weight was 0.028kg. Based on the data, I found the plot of position vs. Time.

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Then, same to investigation 1, I chose the first two maximum points to minimize the data. g(away) g(towards) g(avg) percentage difference m'(kg) m(kg) 5.5947 14.6775 10.1361 -0.03324159 0.028 0.14 Position (m) Run #7 Time (s) Auto error of x Velocity(m/ s) error of velocity V^2(m/ s)^2 X(avg) 1.26 1.1 0.002 -1 0.0025 1.000 1.235 1.21 1.15 0.002 -0.86 0.0024 0.740 1.189 1.167 1.2 0.002 -0.74 0.0023 0.548 1.149 1.13 1.25 0.002 -0.6 0.0022 0.360 1.115 1.1 1.3 0.002 -0.5 0.0021 0.250 1.088 1.075 1.35 0.002 -0.42 0.0020 0.176 1.065 1.054 1.4 0.002 -0.24 0.0020 0.058 1.048 1.042 1.45 0.002 -0.14 0.0019 0.020 1.039 1.035 1.5 0.002 -0.02 0.0018 0.000 1.035 1.036 1.6 0.002 0.12 0.0017 0.014 1.039 1.042 1.65 0.002 0.14 0.0017 0.020 1.046 1.049 1.7 0.002 0.2 0.0016 0.040 1.054 1.059 1.75 0.002 0.26 0.0016 0.068 1.066 1.072 1.8 0.002 0.3 0.0015 0.090 1.080 1.087 1.85 0.002 0.34 0.0015 0.116 1.096 1.104 1.9 0.002 0.42 0.0015 0.176 1.115 1.125 1.95 0.002 0.42 0.0014 0.176 1.136 1.146 2 0.002 0.5 0.0014 0.250 1.159

1.171 2.05 0.002 0.52 0.0013 0.270 1.184 Since the data is still to large, I only put the part that I used to make the graph below. Based on the information in this graph, we can discover that the slope of two series is different. This was because of the change in direction of the friction. To solve g, the velocity square is still equal to Bx+C based on the graph. The velocity square is also equal to v 2 = 2 m' g m ' + m ( x− x 0 ) we can get the equation of B B = 2 m' g m' + m Then, it is easy to find g. For the motion towards the sensor, g equals to 14.68 m/s^2. For the motion away from the sensor, g equals to 5.59 m/s^2. The average g then could be calculated which is 10.14 m/s^2. It is still pretty closed to the real value of g even each of the g were really far away from the true value. The percentage difference of g in investigation 2 is 3.3%. I think the work energy was been verified based on the two closed value of g which were tested. The small difference of the two tested g may based on the friction. The value of C could be calculated by the equation: C = 2 m' g m' + m ∗ x 0 Since the value of velocity at x0 is equal to 0. Conclusion

In Investigation 1, I let the gilder slide down an inclined air track. Also use motion sensor to record position and time. thus get the velocity we need. then use the kinetic energy formula to find the experimental value of gravity and compare it with the actual value. Through a series of calculations, I came up with an average value of gravitational acceleration of 9.65 m/s^2, which is 1.6% different from the actual value of gravity of 9.81 m/s^2. Potential uncalculated errors include the motion sensor not being properly calibrated resulting in too much friction. An improvement to this experiment would be to conduct multiple trials allowing for a wider range of data and more accurate graphs, resulting in more accurate gravity values in the experiment. In Investigation 2, I adjusted the inclined plane to be horizontal while hanging a weight on the tail of the glider. I still used the motion sensor to record the time and position. From the data obtained, I calculated an average gravitational acceleration of 10.14 m/s^2 as an average, which is 3.3% different from the true value of gravity of 9.81 m/s^2. Potential unaccounted errors again include excessive friction due to motion sensors not being properly calibrated. Improvements to this investigation could be made by selecting more parts from the data, as the two parts I have chosen are both very different from the actual values if viewed separately, and may therefore have led to some errors. Questions 1. As the speed of the glider increases, the accuracy of the motion sensor will decrease. Motion sensors detect speed by reflecting sound waves. At high speeds, these reflected sound waves are too fast causing errors in the motion sensor and therefore reducing the accuracy of the motion sensor. 2. The meaning of friction is a force that impedes the motion of an object. The effect of friction on energy in both directions is to reduce the total energy because some of the energy is consumed by friction. 3. 4.Since we know that v 2 = 2 m' g m ' + m ( x− x 0 ) . g = g avg ( m + m' 2 m' ) . If m’ ->infinity on the top and bottom of the equation would cancel out, the acceleration will be 0. 5.The change in potential energy for investigation 2 is negative because during the descent of the glider, the potential energy is converted into kinetic energy. The kinetic energy before the collision with the bumper is very high, which means that the kinetic energy of the system increases. This is because the bumper collision dissipates some energy and each time it bounces, the energy decreases. Fn Fn Downward Friction Fg Upward Friction Fg

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