# Newton's Gravitational Law

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Core 1 Quantitative Assignment 1
Background: When wanting to find the mass of a massive object, the following is needed, Newton’s gravitational law (F = ma = mv2/r). Newton’s gravitational law can be used to calculate the mass which is represented by (M). This can be calculated if the velocity represented by (v), at orbital radius represented by (r), is known for the equation. When these two laws are put together they create the formula

M = v2 r / G

what G represents is the gravitational constant (6.673 X 10 – 11 m3 Kg – 1 s – 2). For a circular orbit, the velocity (v) is determined by its circumference of (2r) this is divided by the time it takes to complete an entire orbit also referred to as a period (P), this is entirely
Calculations: In order to calculate the mass of “Object X” from the orbit of its moon the first step needed to be taken is to have specified units for distance (m), time (s), and mass (kg). This needs to be done because they are the units Kepler’s Third Law require in order to be able to obtain the mass of “Object X” properly. The units given in the introduction are in kilometers and days. Therefor the radius has to be converted to Kepler’s Third Law values, so from kilometers to meter and days get converted to seconds.
1. r = 57, 783 kilometers

r = 57, 783kilometers x 1,000 m/ 1 km = 57,783,000
Firs is to multiply the number of days, which in this case is 32.17 days by 24 hours over 1 day. Then the days cancel out because they are the same unit measure and hours stay. Then again this process is repeated for minutes and hours until eventually what stays remaining is seconds. Finally, what stays remaining is multiplied across which should be 32.17 times 24 times 60 times 60 and then it is divided by 1 giving the final answer.
After these values are obtained and changed into the proper unit of distance and time, meters and seconds, they can be plugged into Kepler’s Third Law:
The first step to calculating the equation is to take the first half and multiple across. Meaning obtain pi and then square its value and multiply it by four. Since pi is a constant that is used that shows a circle’s circumference to it’s diameter ration, it is represented by they symbol , its numerical value is 3.141592654, but in most cases its shortened down to 3.142 M = (42r3) / (G P2) M = (4)(3.142)2 (5.778 x 107 m)

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