Universe: Stars And Galaxies
Universe: Stars And Galaxies
6th Edition
ISBN: 9781319115098
Author: Roger Freedman, Robert Geller, William J. Kaufmann
Publisher: W. H. Freeman
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Chapter 4, Problem 58Q
To determine

(a)

The gravitational force exerted by the moon on a 1 kg rock placed at the point on Earth’s surface that is closest to the moon.

Expert Solution
Check Mark

Answer to Problem 58Q

The gravitational force exerted by the moon on a 1 kg rock at the closest point on the Earth’s surface is 3.43×105N.

Explanation of Solution

Given:

Distance from the moon to the center of the Earth, d1=384400km.

The diameter of Earth, D=12756km

The mass of rock, m=1 kg

Formula used:

The gravitational force between two objects is given by,

F=Gm1m2r2

Where, G is the universal gravitation constant, m1 and m2 are masses of the two objects, and r is the distance between them.

Calculation:

We take the formula for gravitational force, F=Gm1m2r2

Here, m1 is the mass of the moon, which we know as m1 = 7.349×1022 kg, and, m2 is the mass of the rock so m2 = 1 kg. Value of G is also known to be 6.67×1011Nm2/kg2.

First, we calculate the value of r, which is the distance between the moon and the closest point on Earth’s surface.

This is given by,

r=d1D2

r=384400127562=378022 Km = 378022×103 m.

Putting in all the values in the formula for gravitational force, we get,

F=(6.67×1011)(7.349× 10 22)(1) ( 378022× 10 3 )2F=3.43×105N

Conclusion:

Thus, the gravitational force exerted by the moon on a 1 kg rock at the closest point on the Earth’s surface is 3.43×105N.

To determine

(b)

The gravitational force exerted by the moon on a 1 kg rock placed at the point on Earth’s surface that is farthest to the moon.

Expert Solution
Check Mark

Answer to Problem 58Q

The gravitational force exerted by the moon on a 1 kg rock at the farthest point on the Earth’s surface is 3.20×105N

Explanation of Solution

Given:

Distance from the moon to the center of the Earth, d1=384400km.

The diameter of Earth, D=12756km and the mass of rock = 1 kg.

Formula used:

F=Gm1m2r2, which has been explained in the above section.

Calculation:

Again, we have m1 = 7.349×1022 kg, m2 = 1 kg and G = 6.67×1011Nm2/kg2

Here, r is the distance between the moon and the farthest point on the Earth’s surface. This is given by,

r=d1+D2

r=384400+127562=390778 km = 390778×103 m.

Substituting all values in F=Gm1m2r2, we get,

F'=(6.67×1011)(7.349× 10 22)(1)390778× 103F'=3.20×105N

Conclusion:

Thus, the gravitational force exerted by the moon on a 1 kg rock at the farthest point on the Earth’s surface is

3.20×105N.

To determine

(c)

The difference between the two forces F and F', calculated in the sections (a) and (b) respectively.

Expert Solution
Check Mark

Answer to Problem 58Q

Difference between the two forces F and F' is 0.23×105N.

Explanation of Solution

Given:

F=3.43×105N and F'=3.20×105N

Formula used:

The difference between the two forces is calculated by

ΔF=FF'

Calculation:

The difference between the two forces is,

ΔF=FF'ΔF=3.43×1053.20×105ΔF=0.23×105N

The tidal force, i.e., the difference between the two forces F and F' is 0.23×105N. As we can see, the magnitude of this force is very small. Since the tidal force is of such a small magnitude, so it causes only a small deformation on Earth.

Conclusion:

Thus, the difference between the two forces F and F' is 0.23×105N.

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