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Determine the mass of the earth knowing that the mean radius of the moon's orbit about the earth is $238,910 \mathrm{mi}$ and that the moon requires 27.32 days to complete one full revolution about the earth.
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Physics 101 Mechanics
Chapter 12
Kinetics of Particles: Newton’s Second Law
Newton's Laws of Motion
Cornell University
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University of Sheffield
University of Winnipeg
Lectures
03:28
Newton's Laws of Motion are three physical laws that, laid the foundation for classical mechanics. They describe the relationship between a body and the forces acting upon it, and its motion in response to those forces. These three laws have been expressed in several ways, over nearly three centuries, and can be summarised as follows: In his 1687 "Philosophiæ Naturalis Principia Mathematica" ("Mathematical Principles of Natural Philosophy"), Isaac Newton set out three laws of motion. The first law defines the force F, the second law defines the mass m, and the third law defines the acceleration a. The first law states that if the net force acting upon a body is zero, its velocity will not change; the second law states that the acceleration of a body is proportional to the net force acting upon it, and the third law states that for every action there is an equal and opposite reaction.
09:37
Isaac Newton (4 January 1643 – 31 March 1727) was an English mathematician, physicist, astronomer, theologian, and author (described in his own day as a "natural philosopher") who is widely recognised as one of the most influential scientists of all time and a key figure in the scientific revolution. His book Philosophiæ Naturalis Principia Mathematica ("Mathematical Principles of Natural Philosophy"), first published in 1687, laid the foundations of classical mechanics. Newton also made seminal contributions to optics, and he shares credit with Gottfried Wilhelm Leibniz for developing the infinitesimal calculus.
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from the gravitational force Law. We know that the force between two massive objects is equal to the universal gravitational constant G. And in this case, the masses off the two objects are capital in the mass of the earth hands little m the mass of the moon over the separation between them squared r squared. And we also know that these letters force a normal force That x between the object results in central promotion. So from Newton's second law, F N is equal to M A N T normal component of acceleration, which is, um, the squid off our so V squared is the tangential acceleration and artists separation between the two planets between the Earth and the moon. So we can now equate these two forces. So what this means is that G capital M little M over R squared is equal to little m The squared off the art, remember, little M is the mass of the moon. So in short, we have the earth with mess Mm. And it has radius big art. And the separation between the art and the moon is little are so if you look at this equation, the small aims cancel on either side and we get that The mass, the capital M physical toe are over g times v squared. And we if we assume that this that this orbit off the moon is circular about the earth and the velocity at which the Earth the moon orbits Earth is equal to its the conference two pi r over its period talk. So what? This means that we can rewrite the mess off the earth. The creation we had above to our over G and will replace V by two pi r off tour and that's squared. So this becomes one over the gravitational constant G multiplied by two pi over tour. All squared and we have are times are square becomes our cube. Now we note that were given values for the period off the moon's orbit about the earth. And so that's talk. Tor is 27 point 32 days and so we convert this two seconds and we get 2.3 six 04 times 10 to the power six seconds. We also have the separation between the Earth and the moon. Little are and are we had given as 200 38,000 900 and 10 miles. So we convert this 2 ft and we get this to be one point 26144 times, 10 to the nine feet. So now we can substitute our values into the equation above. And hence we confined the mass off the earth with this data. So the mass of the Earth M is one of the G, which is 34.4 times 10 to the minus nine, and that's feet to the powerful over pound second to the power for multiplied bye. Two pi over tall, which is two point 3604 times 10 to the six seconds in that squared multiplied by our cube in feet, is 1.26144 times 10 to the nine. And that's feet. It's a cute, and we get the mass off the earth in these units to be approximately 413 times 10 to the power 21 and that's in pound second squid, a foot
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