Using the masses and mean distances found in Appendix B.6, calculate the net gravitational force

Using the masses and mean distances found in Appendix B.6,...

Key Concepts

Multi-body Gravitational Interactions

When more than two bodies interact gravitationally, the net force on a given body is determined by considering the contribution of each body separately before combining them. This approach is essential for understanding complex orbital dynamics and for accurately calculating how the gravitational forces of the Earth and the Sun combine to influence the Moon’s motion during different celestial alignments.

Celestial Alignments and Eclipses

Celestial alignments, such as those occurring during eclipses, can affect the net gravitational forces experienced by astronomical objects. Alignments like a lunar eclipse or a solar eclipse alter the directions and magnitudes of the gravitational influences from nearby massive bodies. Understanding these configurations is important when calculating the net force acting on an object like the Moon since the relative positions determine how the gravitational pulls add together or partially cancel out.

Newton's Law of Universal Gravitation

This law states that every two masses in the universe attract each other with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between their centers. This fundamental principle is used to calculate the gravitational attraction between any pair of objects, making it essential for determining the forces acting on celestial bodies like the Moon, Earth, and the Sun.

Superposition Principle

In the context of gravitational forces, the superposition principle allows us to find the net force on an object by vectorially adding the individual gravitational forces exerted by each mass independently. This concept is crucial when dealing with systems involving multiple bodies, such as calculating the net gravitational force on the Moon when it is under the influence of both the Earth and the Sun.

Transcript

00:01 In this problem, we have to find the net gravitational force on the moon during a lunar eclipse.

00:05 We know that in lunar eclipse, the earth is between moon and sun.

00:10 Suppose this is sun and this is moon and this is earth.

00:19 So we have to find the net gravitational force on the moon.

00:22 So the first force due to force on moon due to earth and the second force is force on moon due to sun.

00:33 Know that this distance is between moon and earth and this distance between earth and sun.

00:43 So we can write net force will be equal to g into mass of earth into mass of moon divided by r m square plus g mass of moon into mass of sun divided by r m e plus g mass of moon into mass of sun divided by r m e plus r distance between earth and sun square now we will put the values we know that 6 .67 into 4 into 10 to the power minus 11 into mass of moon we know that mass of moon is equal to 7 .349 into 10 to the power 2 7 .349 into 10 to the power 2 7 .349 into 10 to the power 22 into mass of earth which is 5 .974 5 .974 into 10 to the power 24 kg divided by distance between moon and earth distance between moon and earth is 3 .845 into 10 to the power 8 square plus now the mass of sun is 1 .98 into 10 to the power 30 divided by rm e plus r e s so this is equal to if we solve the distance between earth and sun is 1 .5 into 10 to the power 11 this is 1 .5 into 10 to the power 11 so we will add this is 3 .845 into 10 to the power 8 plus 1 .5 into 10 to the power 11 whole square right so we will solve this we get the value of force is equal to 6 .29 into 10 to 20 newton 6 .29 into 10 to the power 20 newton and this is the answer for part a now if we talk about the part b in this part it is given that during a solar eclipse that means moon is between earth and sun suppose this is moon and this is earth and this is sun so here one gravitational force between earth and moon which is force on moon due to earth and here force on moon due to sun so the net force will be equal to g mass of moon into mass of earth divided by r m square plus g mass of moon into mass of sun divided by distance between earth and sun minus distance between moon and earth square right so so we can write it this is equal to g mass of moon into m e divided by r moon and earth square plus mass of sun divided by r e s distance between earth and sun, minus distance between moon and earth square.

04:37 So we will put the values now.

04:39 This is equal to 6 .674 into 10 to the power minus 11 into mass of moon, which is 7 .349 into 10 to the power 22 kg into mass of earth, which is 5 .974.

04:56 Now here if we see these two forces are in the opposite direction.

05:02 So we will take negative shine here.

05:05 So we will take because this force will larger from this force...

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