Newtons Second Law - The TA\'s name was Michael Schott. This is a full lab report. PDF

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The TA's name was Michael Schott. This is a full lab report....

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Woods

Newton’s Second Law Lab Report

Dana Woods

Lab Partner: Katherine Andersh Course: PHYS181-015 TA: Michael Schott Due Date: 5:00 PM on 9/29/16

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Abstract For this experiment, we measured the acceleration of the cart as the hanging mass (the force) changed in the first part and then as the cart mass changed in part two. In part one, it was found that as the mass of the hanger increased, the acceleration increased. This was seen by observing the changes in time and velocity; as hanging mass increased, time decreased while velocity increased. For the hanging mass versus acceleration slope derivation, we got a slope of 9.0805 m/s2 kg-1 for time and a slope of 0.0316 m/s2 kg-1 for velocity. The trend for part two was as the total mass increased, the acceleration decreased. Again, the two variables of time and velocity could be observed to observe this. As the mass of the cart increased, the time increased and the velocity decreased. The derivations for the slopes for the inverse of total mass versus acceleration were 0.3524 kg m/s2 for time and 1.8392 kg m/s2 for velocity. It was observed that as the force increased, acceleration increased, but if the mass increased, acceleration decreased.

Introduction Newton’s second law describes the relationship between the mass of a system, the acceleration of the system, and the net force being applied to the system. According to the law, the net force is equal to the product of the mass and acceleration. If the equation is rearranged, acceleration is seen to equal the force divided by the mass. Stated in words, the acceleration is directly proportional to force (as force increases, acceleration increases and vice versa) and inversely proportional to mass (as mass increases, acceleration decreases and vice versa). The goal of this lab was to verify this law by changing the force of the system while keeping mass constant, then changing the mass while keeping force constant. By observing the changes and quantitatively determining accelerations for each condition, we were able to verify that Newton’s second law could be applied to our system, and that the relationship between the variables is consistent with what the law says.

Theory Newton’s second law was used in this lab in order to determine how acceleration changed as both force and mass changed. The relationship between force, mass, and acceleration can be seen in the equation (1) where F is the magnitude of the net forces, m is the mass of the system, and a is the magnitude of the acceleration. Because we are seeing how acceleration is affected, we can rearrange the equation like so (2) in order to put acceleration on one side of the equation and force and mass on the other. According to this equation, acceleration is directly proportional to force, but inversely proportional to mass.

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Woods Next, for our specific system, we had to determine where the mass and force came from. First, we had to find out the total mass consisted of. There were two components: the hanger and the cart. To figure out the total mass of the system the formula (3) was used, where Mcart is the mass of the cart, mhanging is the mass of the hanger, and Mtotal is the total mass of the system. The force was then determined. Because of the way the apparatus was set up, the weight of the hanging mass acted as the force acting on the system. The force is shown by (4) where g is the acceleration due to gravity. Equations 3 and 4 can be substituted in for F and m Equation 2 to determine the force and mass, respectively, for our specific system, which in turn will determine the theoretical acceleration. (5) For our lab, acceleration was measured indirectly through the time and velocity measurements we got using the Xplorer GLX software connected to the photogates on the track. The first method was using the time the cart was between the photogates. Because the cart is only moving in the horizontal direction, and because it is subject to a constant acceleration, the position equation (6) can be used, where xf is the final position, x0 is the initial position, v0 is the initial velocity, and t is the time. This can be simplified to (7) because the initial position (the first photogate location) was considered zero and the cart started from rest, so the initial velocity is zero. Because we need to find the time between the photogates, we can solve for time to get (8) so that we are able to plot a t2 versus xf graph and determine the acceleration from the slope. The second method utilized the final velocity reading from the GLX software. Again, our cart moved in one dimension with a constant acceleration, so the velocity kinematics equation (9) can be used, where vf is the final velocity. As previously stated, the initial velocity is zero, so

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(10) becomes the equation. Because we want to relate it back to the time method, we can square the entire equation (11) and then substituting the t2 value from Equation 8. This gives us the final equation of (12) and now we can make a vf2 versus xf to determine the acceleration. By evaluating the data from these equations and the graphs we make with them, we can determine the relationship between force, mass, and acceleration for our specific system.

Procedure

Figure 1. Apparatus for experiment. Showing cart attached to hanger by a string. Cart is on a track with two photogates connected to a computer. Weights used to add mass to cart and/or hanger.

The experiment was done using the apparatus shown in Figure 1. The cart was placed on a frictionless track and was attached to a hanger by a string. The hanger hung off the side of the table. The weights were used to add mass to the cart and/or the hanger. There were two photogates on the track connected to a computer that gave us values for time and velocity when the index card sticking up from the cart triggered the sensors on the photogates. The first photogate remained stationary, while the second photogate moved depending on which distance we were testing. The cart was released as close to the first photogate sensor as possible, and was pulled down the track due to the hanging mass. For the first part of the experiment, the hanging mass varied, while the total mass of the system remained the same. The total mass was fixed at 0.6 kg. The distance between the photogates started out at 0.15 m and the hanging mass was 0.02 kg. This meant that the mass of

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Woods the cart needed to be 0.58 kg. The index card attached to the cart was put as close to the sensor on the first photogate as possible (this was done to get our initial velocity as close to zero as possible). Once the “run” was started on the computer, the cart was released and was stopped after it passed through both photogates. To calculate the time between photogates, the time it passed through the first photogate was subtracted from the time it passed through the second photogate (times displayed on the computer). The final velocity was displayed on the computer as well. This was done three times for each hanging mass at each distance. The average time and average velocity were determined for these three trials. Time squared and velocity squared were determined by squaring the value for average time and the value for average velocity, respectively. This was repeated for hanging masses of 0.04 kg, 0.06 kg, and 0.08 kg at distances of 0.3 m, 0.45 m, and 0.6 m. For the second part of the lab, the hanging mass now stayed the same, but the total mass of the system changed, meaning that the mass of the cart was changing. The distance between the photogates started out at 0.15 m again, and the total mass was 0.6 kg. Because the hanging mass remained at 0.02 kg throughout this part of the lab, this meant the cart mass for this part was 0.58 kg. The same method for bringing the cart down the track was executed during this part of the experiment. The values were all calculated in the same way as well. Three trials were done for each total mass at each distance. This procedure was repeated for total masses of 0.85 kg, 1.1 kg, and 1.35 kg at distances of 0.3 m, 0.45 m, and 0.6 m. After the data was collected, it was compiled into tables according to the different masses. These tables were then made into graphs to determine how force and mass affected the acceleration of our system.

Sample Calculation and Results The data for the time between the photogates and the final velocities for part one can be seen in Tables 1-4. The same data for the second part of the experiment can be seen in Tables 58. All of these calculations were done using the Xplorer GLX software. All of the described calculations in this section were done the same for each data set. It explain how this was done, I will use the data set of a 0.02 kg hanging mass, with a constant mass of 0.6 kg and a distance of 0.15 m. First, the average time and velocity needed to be calculated from the three trials. The equation 0.31 m/s + 0.31 m/s + 0.31 m/s 3

0.31 m/s

(13)

can be used, where Rave is the average, for either velocity or time (velocity for the sample calculation), Rsum is the sum of the velocities or times, and n is the total number of velocities or times. According to the equations derived the theory section, the average velocities and times needed to be squared in order to correctly determine the acceleration. (0.87 s)

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0.7569 s

2

(14) 5

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(0.31)

2

2

0.0961 m /s

2

(15)

These calculations were done for all 32 runs of the experiment. The acceleration could be calculated using Equation 5 from the theory section. This would give us a theoretical acceleration according to the masses in our system. However, we measured the acceleration indirectly using the squared times and velocities found above. This process can be seen in Graphs 1-4 for the first part and Graphs 5-8 for the second part. As a sample, we can look at the first part of the experiment. In Graph 1, the hanging mass was graphed with each one’s corresponding squared time. According to the Equation 8, the acceleration is inversely proportional to time squared. So, the inverse slope of each of the hanging mass lines was determined. 2 -1 2 (6.4255 s /m) = 0.1556 m/s

(16)

These acceleration values were graphed with the corresponding hanging masses in Graph 2. The velocity method was also used to determine the acceleration of the cart. Graph 3 shows the average velocity squared over the various distances for each hanging mass. Velocity squared is seen to be directly proportional to the acceleration, so the slopes for each hanging mass directly translated to the accelerations, as seen in Graph 4. The second part of the lab was done in this same manner (Graphs 5-8). The only difference was that Graphs 6 and 8 graphed the inverse of the total mass with the accelerations for each. -1 -1 (0.6 kg) = 1.6667 kg

(17)

As stated earlier, the acceleration could be calculated using Equation 5. I will be using the same data set that was used for Equations 13-15. 2 0.02 kg (9.8 m/s ) 0.58 kg + 0.02 kg

0.3267 m/s

2

(18)

Because this acceleration is different than the one seen in Equation 16, there is some error present. This error is explained in the discussion.

Discussion and Conclusions

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Woods Our biggest source of error in this lab was the fact that the string was not placed in the proper hole at the end of the track, so it was at an angle as the cart was going down the track. The more accurate value for the force would be the xcomponent of the force we found. Therefore, by using the Pythagorean theorem, our actual force should be the force we found multiplied by cosine of the angle (Fcosθ). It is a bit more complicated than this, considering that as the cart moved down the track, the angle changed. However, this is just a simplified way that would make our data more accurate. Another source of error is from releasing the cart on the track. The index card needed to be as close to the sensor as possible to the first photogate sensor so the initial velocity can be as close to zero as possible. Our goal was just to get the initial velocity under 0.5 m/s and just consider it “zero.” However, this was harder to do, especially when the cart’s acceleration was high. We ended up having measurements that had an initial velocity of around 0.8 m/s. The closer we got it to zero, the more accurate our measurements were. Because the initial velocity reading was never zero, this created error in our times as well as velocities, and therefore is reflected in the accelerations. Each time we put mass on the carts, we tried to make sure the exact correct mass was present. Most of the time we were able to do this with the weights we were given. However, there were two or three times where the measurement was 0.05 g less than what it was supposed to be. While this is a rather small number and most likely did not cause a large amount of error, it still could have contributed to the overall sum of the errors of the experiment.

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Distanc e (m)

0.15 0.15 0.15 0.3 0.3 0.3 0.45 0.45 0.45 0.6 0.6 0.6

Constant Total Mass – 0.02 kg Hanging Mass Total Mass: 0.6 kg Hanging Mass: 0.02 kg Cart Mass: 0.58 kg Time Final Average Average of Average of Average of Between Velocity of Time Time Velocity Velocity Squared Photogates (m/s) (s) Squared (m/s) (m^2/s^2) (s) (s^2) 0.87 0.31 0.87 0.7569 0.31 0.0961 0.87 0.31 0.87 0.31 1.32 0.42 1.32 1.7424 0.42 0.1764 1.31 0.42 1.33 0.42 1.61 0.53 1.61 2.5921 0.53 0.2809 1.62 0.53 1.61 0.53 1.91 0.61 1.92 3.6864 0.6 0.36 1.92 0.6 1.92 0.59

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Woods Table 1. Time and velocity data for hanging mass of 0.02 kg. Three trials taken for each distance of 0.15 m, 0.3 m, 0.45 m, and 0.6 m.

Distanc e (m)

0.15 0.15 0.15 0.3 0.3 0.3 0.45 0.45 0.45 0.6 0.6 0.6

Constant Total Mass – 0.04 kg Hanging Mass Total Mass: 0.6 kg Hanging Mass: 0.04 kg Cart Mass: 0.56 kg Time Final Average Average of Average of Average of Between Velocity of Time Time Velocity Velocity Squared Photogates (m/s) (s) Squared (m/s) (m^2/s^2) (s) (s^2) 0.58 0.47 0.58 0.3364 0.47 0.2209 0.58 0.47 0.58 0.47 0.98 0.57 0.98 0.9604 0.57 0.3249 0.98 0.57 0.98 0.58 1.08 0.81 1.08 1.1664 0.81 0.6561 1.08 0.81 1.08 0.81 1.27 0.93 1.28 1.6384 0.93 0.8649 1.28 0.92 1.28 0.93

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Woods Table 2. Time and velocity data for hanging mass of 0.04 kg. Three trials taken for each distance of 0.15 m, 0.3 m, 0.45 m, and 0.6 m.

Distance (m)

0.15 0.15 0.15 0.3 0.3 0.3 0.45 0.45 0.45 0.6 0.6 0.6

Constant Total Mass – 0.06 kg Hanging Mass Total Mass: 0.6 kg Hanging Mass: 0.06 kg Cart Mass: 0.54 kg Time Final Average Average of Average of Average of Between Velocity of Time Time Velocity Velocity Squared Photogates (m/s) (s) Squared (m/s) (m^2/s^2) (s) (s^2) 0.46 0.59 0.46 0.2116 0.59 0.3481 0.46 0.59 0.46 0.59 0.68 0.83 0.68 0.4624 0.83 0.6889 0.68 0.84 0.68 0.83 0.87 1.01 0.87 0.7569 1.01 1.0201 0.87 1.01 0.86 1.01 1.02 1.16 1.02 1.0404 1.16 1.3456 1.02 1.16 1.01 1.17

Table 3. Time and velocity data for hanging mass of 0.06 kg. Three trials taken for each distance of 0.15 m, 0.3 m, 0.45 m, and 0.6 m.

Distance (m)

0.15 0.15 0.15 0.3 0.3 0.3 0.45 0.45 0.45 0.6 0.6 0.6

Constant Total Mass – 0.08 kg Hanging Mass Total Mass: 0.6 kg Hanging Mass: 0.08 kg Cart Mass: 0.52 kg Time Final Average Average of Average of Average of Between Velocity of Time Time Velocity Velocity Squared Photogates (m/s) (s) Squared (m/s) (m^2/s^2) (s) (s^2) 0.39 0.68 0.39 0.1521 0.67 0.4489 0.39 0.67 0.39 0.67 0.59 0.98 0.59 0.3481 0.98 0.9604 0.59 0.97 0.59 0.98 0.74 1.18 0.75 0.5625 1.18 1.3924 0.75 1.18 0.75 1.19 0.88 1.35 0.89 0.7921 1.24 1.5376 0.9 1.04 0.89 1.34

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Woods Table 4. Time and velocity data for hanging mass of 0.08 kg. Three trials taken for each distance of 0.15 m, 0.3 m, 0.45 m, and 0.6 m.

Distance (m)

0.15 0.15 0.15 0.3 0.3 0.3 0.45 0.45 0.45 0.6 0.6 0.6

Constant Hanging Mass – 0.6 kg Total Mass Total Mass: 0.6 kg Hanging Mass: 0.1 kg Cart Mass: 0.5 kg Time Final Average Average of Average of Average of Between Velocity of Time Time Velocity Velocity Squared Photogates (m/s) (s) Squared (m/s) (m^2/s^2) (s) (s^2) 0.37 0.73 0.37 0.1369 0.73 0.5329 0.37 0.72 0.37 0.73 0.56 1.03 0.56 0.3136 1.03 1.0609 0.57 1.03 0.56 1.04 0.71 1.26 0.71 0.5041 1.26 1.5876 0.71 1.26 0.71 1.25 0.85 1.43 0.88 0.7744 1.35 1.8225 0.84 1.4 0.95 1.23

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Woods Table 5. Time and velocity data for total mass of 0.6 kg. Three trials taken for each distance of 0.15 m, 0.3 m, 0.45 m, and 0.6 m.

Distanc e (m)

0.15 0.15 0.15 0.3 0.3 0.3 0.45 0.45 0.45 0.6 0.6 0.6

Constant Hanging Mass – 0.85 kg Total Mass Total Mass: 0.85 kg Hanging Mass: 0.1 kg Cart Mass: 0.75 kg Time Final Average Average of Average of Average of Between Velocity of Time Time Velocity Velocity Squared Photogates (m/s) (s) Squared (m/s) (m^2/s^2) (s) (s^2) 0.43 0.62 0.43 0.1849 0.62 0.3844 0.43 0.63 0.43 0.62 0.66 0.88 0.66 0.4356 0.88 0.7744 0.66 0.88 0.66 0.88 0.83 1.07 0.82 0.6724 1.08 1.1664 0.84 1.08 0.8 1.08 1.13 1.06 1.08 1.1664 1.03 1.0609 1.13 1.07 0.98 0.95

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Woods Table 5. Time and velocity data for total mass of 0.85 kg. Three trials taken for each distance of 0.15 m, 0.3 m, 0.45 m, and 0.6 m.

Distanc e (m)

0.15 0.15 0.15 0.3 0.3 0.3 0.45 0.45 0.45 0.6 0.6 0.6

Constant Hanging Mass – 1.1 kg Total Mass Total Mass: 1.1 kg Hanging Mass: 0.1 kg Cart Mass: 1 kg Time Final Average Average of Average of Average of Between Velocity of Time Time Velocity Velocity Squared Photogates (m/s) (s) Squared (m/s) (m^2/s^2) (s) (s^2) 0.49 0.55 0.49 0.2401 0.55 0.3025 0.49 0.55 0.49 0.55 0.75 0.78 0.75 0.5625 0.78 0.6084 0.75 0.78 0.75 0.78 0.95 0.95 0.95 0.9025 0.94 0.8836 0.95 0.94 0.95 0.94 1.11 1.05 1.11 1.2321 1 1 1.11 0.98 1.1 0.97

Table 7. Time and velocity data for total mass of 1.1 kg. Three trials taken for each distance of 0.15 m, 0.3 m, 0.45 m, and 0.6 m.

Distanc e (m)

0.15 0.15 0.15 0.3 0.3 0.3 0.45 0.45 0.45 0.6 0.6 0.6

Constant Hanging Mass – 1.35 kg Total Mass Total Mass: 1.35 kg Hanging Mass: 0.1 kg Cart Mass: 1.25 kg Time Average Average of Final Average of Average of Between Time Velocity of Time Velocity Velocity Squared Photogates (s) Squared (m/s) (m/s) (m^2/s^2) (s) (s^2) 0.54 0.5 0.54 0.2916 0.5 0.25 0.54 0.5 0.54 0.49 0.82 0.7 0.82 0.6724 0.7 0.49 0.82 0.7 0.82 0.7 1.03 0.87 1.01 1.0201 0.83 0.6889 0.98 0.76 1.03 0.87 1.23 0.87 1.23 1.5129 0.87 0.7569 1.23 0.84 1.23 0.9

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Woods Table 8. Time and velocity data for total mass of 1.35 kg. Three trials taken for each distance of 0.15 m, 0.3 m, 0.45 m, and 0.6 m.

Distance vs. Time Squared - Varying Hanging Mass 4

Average Time Squared (s^2)

3.5

f(x) = 6.43 x − 0.22 R² = 1

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0.02 kg Hanger Linear (0.02 kg Hanger) 0.04 kg Hanger Linear (0.04 kg Hanger) 0.06 kg Hanger Linear (0.06 kg Hanger) 0.08 kg Hanger Linear (0.08 kg Hanger)

2.5 2 1.5 1 0.5

f(x) = 2.74 x − 0 R² = 0.97 f(x) = 1.85 x − 0.08 R² = 1 f(x) = 1.42 x − 0.07 R² = 1

0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

Distance (m) Graph 1. Photogate distance versus average squared time for various hanging masses. The colored lines indicate the best fit line for each hanging mass data set. The greater the hanging mass, the shorter the time the cart was between the photogates.<...