Asteroids should be able to remain stably trapped at the stable Lagrange points of the Earth-Sun system (See Figure 15-14 of Universe, 11th ed.), provided they are NOT subjected to other, stronger, gravitational influences. As a test of this stability, calculate how strong Jupiter's gravity is on a mass at Earth's stable Lagrange points (when Jupiter is closest to them) compared to the strength of Earth's gravity on the same mass at the same points. (You will need to use Newton's law of gravitation; it will also help to draw a diagram.) Compared to Earth's gravitational force at these points, Jupiter's gravity is:

A)1/18 as strong.
B)18 times stronger.
C)12 times stronger.
D)76 times stronger.

Question

Asteroids should be able to remain stably trapped at the stable Lagrange points of the Earth-Sun system (See Figure 15-14 of Universe, 11th ed.), provided they are NOT subjected to other, stronger, gravitational influences. As a test of this stability, calculate how strong Jupiter's gravity is on a mass at Earth's stable Lagrange points (when Jupiter is closest to them) compared to the strength of Earth's gravity on the same mass at the same points. (You will need to use Newton's law of gravitation; it will also help to draw a diagram.) Compared to Earth's gravitational force at these points, Jupiter's gravity is:
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 ​
A)1/18 as strong. 
B)18 times stronger. 
C)12 times stronger. 
D)76 times stronger.

Quizplus · Accepted Answer

According to Newton's law of gravitation, the gravitational force between two objects is directly proportional to the product of their masses and inversely proportional to the square of their distance apart. Jupiter's mass is much greater than Earth's (approximately 318 times greater), so even though it is much farther away, its gravity can still have a significant effect on objects in the Earth-Sun system.

At the stable Lagrange points, the gravitational forces of the Sun and Earth are balanced, so an object placed there will remain in a stable position relative to Earth and the Sun. However, Jupiter's gravity will also pull on the object, and depending on its location relative to the Lagrange point, this force could potentially cause the object to become unstable and move out of the Lagrange point.

To calculate the strength of Jupiter's gravity on an object at Earth's stable Lagrange points, we can use the equation:

F = G(m1m2)/r^2

where F is the force of gravity, G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between them.

At its closest approach to the stable Lagrange points, Jupiter is about 4 astronomical units (AU) away, or 4 times the distance between the Earth and Sun. This means that the distance between Jupiter and an object at the Lagrange point is roughly 5 AU.

Using the masses and distances for Earth, Sun, and Jupiter from Table 15-1 of Universe, we can calculate the ratio of Jupiter's gravitational force to Earth's gravitational force on an object at the Lagrange point:

F(jupiter)/F(earth) = (G(msj)/rj^2)/(G(mse)/re^2)

where msj is the mass of Jupiter, mse is the mass of the Earth, rj is the distance between Jupiter and the Lagrange point, and re is the distance between Earth and the Lagrange point.

Plugging in the values (with masses in kilograms and distances in meters):

F(jupiter)/F(earth) = [(6.67 x 10^-11)(1.898 x 10^27)(5 x 1.496 x 10^11)^2]/[(6.67 x 10^-11)(5.97 x 10^24)(1 x 1.496 x 10^11)^2]

F(jupiter)/F(earth) = 17.9

This means that Jupiter's gravity is about 18 times stronger than Earth's gravity at the Lagrange points. Therefore, answer choice B is the correct answer.

Asteroids Should Be Able to Remain Stably Trapped at the Stable