Lab 2 - Gravitation 2 PDF

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Lab write up on Newtons Gravitational constant and how that relates mass and the constant G...

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Newton’s Universal Law of Gravitation Amber Peake Ashley Metivier 3955 March 6, 2021 Purpose: The experimental goal of Gravitation was to explore Newton’s Law of Gravitation, which says that the force between two masses is always attractive and proportional to the product of the two masses (m1 * m2) and inversely proportional to the distance, between the two centers of mass, squared (R2). In this experiment, the determination of the gravitational constant is the objective. The gravitational constant can be calculated from the exploration of the simulation using gravitation along with the collected data. The equation for Newton’s law of gravitation can be written as: 𝐹=𝐺(𝑚1𝑚2)/𝑅2 The constant G is equal to 6.67×10-11 Nm2/kg2. Procedure: Working with a partner, open up the link for the simulation, in which one person will share the computer screen with the simulation and the other person will record the data in Excel. The link was provided as : https://phet.colorado.edu/sims/html/gravity-force-lab/latest/gravity-forcelab_en.html. The simulation has two figures (people) facing one another, each holding a string attached to a round mass with a dot in the center. A ruler is underneath for measuring the distances between the two masses, according to the mass’s center. First, get the simulation set up for the data sets, the scientific notation values box needs to be checked to record the data values. Part 1 The distance (R) between the two masses is held constant as the masses are changing. The initial values are to place m1 at x = 0 and m2 at x = 5. The first data table is created with R being a constant 5 meters. The masses (m1 and m2) are changing at various amounts up to 100kg. The force (F) is recorded in Newtons and (m1*m2) is recorded in the last column. After recorded the data from the simulation plot the force on the y-axis and the product of m1*m2 on the x-axis. Part 2 Changing R while keeping the masses constant. For part two the distance between the two masses would be increasing starting at 2 m by increments of 1 m up to 10 m. The masses m1 and m2 are both constant at 10 kg. Force (F) in Newtons is recorded as well as 1/R^2 (m^-2). After

Newton’s Universal Law of Gravitation the values were collected a plot was to made of Force (F) on the y-axis and R on the x-axis. We were to find the right trend line that best fits and write about it. The next plot was F (F) on the yaxis and 1/R^2 on the x-axis. We were to find and write about the trendline that best fits the plot. Next is observing the three graphs and understanding the relationship between Force and m1m2 and R. Write statements about the observations that were made and formulate the statements into a mathematical relationship with all variables and a proportionality constant G. Using the slope of the line from the plot of F vs 1/R2 while the masses are both 10 kg, rearrange the equation and calculate for the proportionality constant G. Write in the units for G as well. For the last step, observe the arrows of force on both masses and decide if the arrows are attractive towards one another or repulsive. Tie all the information together in a nicely formatted lab report and turn in. For the experiment of Newton’s Universal Law of Gravitation there were some novel approaches that were taken. Since this lab was done with partners, on partner shared the screen of the simulation while manipulating the settings. The other person took down the data in Excel and and created data tables along with graphs. That approach seemed to work out very well; everyone was involved and everyone shared responsibilities. The only difficult moment that arose was trying to get the distance in between the two masses to be 5 meters (m). In doing this simulation there were many observations of how force (F), the masses (m1m2), and R2 depend on each other. It was also very clear that the arrows above the two masses got larger as the masses become closer which would be determined as attractive forces (F).

Analysis: Part 1 – The distance (R) between two masses (m1, m2) is held constant as the masses (m1, m2) are changing.

Table 1: Constant distance - R(m) with changing mass – m1, m2(kg), and associated force – F (N). The product of the masses(kg2) is previously calculated and recorded. R(m) m1 (kg) m2 (kg) 5 10 10 5 20 10 5 20 20 5 40 20 5 40 40 5 70 40 5 100 100

F (N) 2.67E-10 5.34E-10 1.07E-09 2.14E-09 4.27E-09 7.48E-09 2.67E-08

m1 x m2 (kg2) 100 200 400 800 1600 2800 10000

Newton’s Universal Law of Gravitation

Force vs. m1m2 3.00E-08 2.50E-08 y = 3E-12x + 1E-12

F (N)

2.00E-08 1.50E-08 1.00E-08 5.00E-09 0.00E+00 0

2000

4000

6000

m 1m 2

8000

10000

12000

(kg2)

(above) Linear plot of force (N) on y axis and the product of m1 and m2 (kg2) on x axis. The trendline that best fits the plot of Force (F) vs. m1m2 is a linear trendline. Part 2 Constant masses – m1, m2(kg) with changing distance – R(m) of increments of 1. Table 2: Masses m1 and m2 are held constant at 10kg as distance (R) moves up in increments of 1m. Associated force – F(N) is recorded, as well as, the inverse of distance squared – 1/R2 (m-2) which is previously calculated. R (m) 2 3 4 5 6 7 8 9 10

m1 (kg) 10 10 10 10 10 10 10 10 10

m2 (kg) 10 10 10 10 10 10 10 10 10

F (N) 1.67E-09 7.42E-10 4.17E-10 2.76E-10 1.85E-10 1.36E-10 1.04E-10 8.24E-11 6.81E-11

1/R2 (m-2) 0.25 0.111 0.0625 0.04 0.0277 0.0204 0.0156 0.0123 0.01

Newton’s Universal Law of Gravitation

Force vs. R 1.80E-09 1.60E-09 1.40E-09

y = 7E-09x-1.997

F (N)

1.20E-09 1.00E-09 8.00E-10 6.00E-10 4.00E-10 2.00E-10 0.00E+00 0

2

4

6

8

10

12

R (m)

(above) Non-linear plot of force (N) on the y-axis and distance – R (m) on the x-axis.

Force vs. 1/R2 1.80E-09 1.60E-09 1.40E-09 y = 7E-09x + 1E-12

F (N)

1.20E-09 1.00E-09 8.00E-10 6.00E-10 4.00E-10 2.00E-10 0.00E+00 0

0.05

0.1

0.15

1/R2

0.2

0.25

0.3

(m-2)

(above) Linear plot of force – F (N) on y-axis and inverse of R2 (m-2) on x-axis. The trendline that best fits the plot of Force (F) vs. R (m) is the power trendline. The trendline that best fits the plot for Force (F) vs. 1/R2 (m) is a linear trendline. Observing the graphs there is a linear relationship between force (F) and mass (m). As the mass increases so does the force (F). In Newton’s Law of Gravitation the relationship between force (F) and mass (m) is directly proportional to each other. For the plot of force (F) vs. R there is a

Newton’s Universal Law of Gravitation power trendline which is decreasing. As the distance between the two objects increases the force between the two objects will decrease. From Newton’s Law of Gravitation force (F) is inversely proportional to R so when on increases the other decreases. F= G(m1m2)/R2 The slope for the plot force (F) vs. 1/R2 is 7.0 x 10-11 which is also the same as F(R2). The equation for G= F (R2/m1m2). Since the slope is F(R2) then the slope can be divided by m1m2 which is 100 to calculate G. G=(7.0 x 10-9 N/m2)/(10kg x 10kg) G=7.0 x 10-11 Nm2/kg2 The gravitational force is attractive. Newton’s Law of Gravitation states that every particle attracts every other particle with a force that is directly proportional to the product of the masses and inversely proportional to the square of the distance between the centers. Conclusion: From the experiment of Newton’s Law of Gravitation it can be concluded that force (F) depends on m1m2 and R. The relationship between force (F) and m1m2 is that both are directly proportional to each other creating a linear line. Force (F) is dependent on R in the opposite way. There is an inversely proportional relationship between the two so as one increases the other decreases. This relates to Newton’s Law of Gravitation from the equation: F=G(m1m2)/R2 with G as 6.67 x 10-11 Nm2/kg2. From this equation it can be predicted that force (F) is directly proportional to the product of the two masses of m1*m2. In this equation m1m2 is in the numerator which corresponds the direct relationship, as force increases so would the masses. From the equation the relationship between force (F) and the distance between the two masses, R, can be determined to be inversely proportional since the R is in the denominator which suggests that as one increases the other decreases. For the experiment of Newton’s Law of Gravitation data was collected from a simulation. The quantities measured were force (F) and the product of the two masses, m1m2. In Part 1, the distance between the two masses (R) stayed the same while the masses changed. In Part 2, the distance between the two masses (R) changed while the masses stayed the same. The measurements of the quantities taken were from the data that was collected while performing the relationships as it relates to Newton’s Law of Gravitation. Below is a comparison table for the Gravitational constant with an expected value of 6.67 x 10-11 Nm2/kg2 and a calculated value from the simulation and data that as collected as 7.0 x 10-11 Nm2/kg2.

Newton’s Universal Law of Gravitation Gravitational Constant Expected (Nkg-2m2) Calculated (Nkg-2m2) 6.67E-11 7.00E-11

The calculated gravitational constant was determined experimentally from the data that was collected from the simulation and was calculated to be 7.0 x 10-11 Nm2/kg2. This is similar and close the expected gravitational constant which is 6.67 x 10-11 Nm2/kg2. Although the constants are similar, there is some error to be accounted for in the calculated value. In using simulations, it is possible that there are built in errors that is uncontrollable. It is not certain, but likely for the simulation itself to have a built-in error. Another error that could be accounted for is the alignment on the ruler with the masses. It is possible that the distance between the two masses was not exactly 5 meters. It could have been off as it was difficult to align it perfectly with the 5 meter distance. For the experiment of Newton’s Law of Gravitation, there could be some improvements moving forward to account for less error. For this experiment, the simulation could provide unrounded numbers for force (F) which could help to account for less errors in the calculation of the gravitational constant. Overall, the experiment didn’t have many sources of error and was fairly close to the expected gravitational constant....