In the Earth-Moon system, there is a point where the...

In the Earth-Moon system, there is a point where the gravitational forces balance. Assume that the mass of the Moon is $\frac{1}{81}$ that of the Earth. a) At what point, on a line between the Earth and the Moon, is the gravitational force exerted on an object by the Earth exactly balanced by the gravitational force exerted on the object by the Moon? b) Is this point one of stable or unstable equilibrium? c) Calculate the ratio of the force of gravity due to the Sun acting on an object at this point to the force of gravity due to Earth and, separately, to the force of gravity due to the Moon.

In the Earth-Moon system, there is a point where the gravitational forces balance. Assume that the mass of the Moon is $\frac{1}{81}$ that of the Earth.
a) At what point, on a line between the Earth and the Moon, is the gravitational force exerted on an object by the Earth exactly balanced by the gravitational force exerted on the object by the Moon?
b) Is this point one of stable or unstable equilibrium?
c) Calculate the ratio of the force of gravity due to the Sun acting on an object at this point to the force of gravity due to Earth and, separately, to the force of gravity due to the Moon.

Key Concepts

Newton's Law of Universal Gravitation

This concept states that every two masses attract each other with a force proportional to the product of their masses and inversely proportional to the square of the distance between them. It forms the basis for calculating gravitational forces in any system, allowing us to set up equations that determine where forces balance.

Gravitational Equilibrium

Gravitational equilibrium occurs at a point where the net gravitational forces acting on an object are zero. In a two-body system, this is the location where the gravitational pull from one mass exactly cancels the pull from the other, leading to a condition of force balance.

Stability of Equilibrium

This concept distinguishes between stable and unstable points of equilibrium. A stable equilibrium means that small displacements result in restoring forces that return the object to equilibrium, whereas an unstable equilibrium implies that any small displacement causes forces that further displace the object from its balanced position.

Lagrange Points

Lagrange points are positions in a two-body system where the gravitational forces and the orbital motion combine to create equilibrium points. These special positions, such as the L1 point, are critical in understanding the dynamics of objects influenced by two major celestial bodies.

Comparative Gravitational Forces

This concept involves analyzing and comparing the magnitudes of gravitational forces exerted by different celestial bodies on an object. It is essential for understanding how secondary influences, such as a third body's gravitational pull, affect the net force in a multi-body gravitational system.

Transcript

00:01 This question belongs to the gravitational chapter and it says that in earth moon system there is a point where gravitational forces balance.

00:07 So if gravitational forces are balanced, assuming that mass of the moon m m that is equals to 1 by 81 to the mass of the earth.

00:21 So this is given and for the part a we have to calculate the point on the line between earth and moon where gravitational force are balanced.

00:29 So suppose this is the diagram.

00:31 For this question in which this is earth and this is moon and this point is l1 where gravitational forces are equal.

00:38 So we have to calculate this value x and this is the capital r distance between earth and moon which is equals to which is equals to 3 .8 double 4 multiplied by 10 to the power 5 kilometer.

00:52 So if gravitational forces are balanced then earth force due to earth will be equals to the force due to moon on this object present at the l1.

01:01 So we can write that we can write that f e equals to f m so substituting values so g m multiplied by small m mass of the mass of the particle or object divided by x square this will be equals to g multiplied by mass of the moon multiplied by small m divided by r minus x whole square so substituting values so we will get that g m m multiplied by small m divided by by x square that is equal to g made by mass of the moon which is given as 1 by 81 times of the earth multiplied by small m divided by r minus x whole square okay so from here after solving we will get that x that is equal to 9 by 10 of capital r so substituting value of capital r so 9 by 10 into capital r which is equals to 3 .8 double 4 multiplied by 10 to the power 5 so, from here after solving x, this will be equals to x, this will be equals to 3 .46 multiplied by 10 to the power 5 kilometer.

02:10 So this becomes the answer for the part a of the question at this distance from the earth, the net gravitational forces are balanced.

02:18 So for the part b is this point one of stable or unstable equilibrium.

02:23 So we can say that this point is of unstable equilibrium, unstable equilibrium, unstable equilibrium equilibrium because if an object at the neutral point is pushed towards the earth or moon the object will have a net gravitational force in the direction of force the net force will be in the direction of push so that's why the object will start will keep moving toward the push direction so hence it will not return back so it means the equilibrium condition is the unstable equilibrium condition now, moving to the next part c, we have to calculate the ratio of the force of gravity due to sun acting on the object on this point to the force of gravity to the earth...

Please give Ace some feedback