Question

Tidal forces are gravitational forces exerted on different parts of a body by a second body. Their effects are particularly visible on the earth's surface in the form of tides. To understand the origin of tidal forces, consider the earth-moon system to consist of two spherical bodies, each with a spherical mass distribution. Let re be the radius of the earth, m be the mass of the moon, and G be the gravitational constant.

1) Earth is subject not only to the gravitational force of the Moon but also to the gravitational pull of the Sun. However, Earth is much farther away from the Sun than it is from the Moon. In fact, the center of Earth is at an average distance of 1.5×1011m from the center of the Sun. Given that the mass of the Sun is 1.99×1030kg, which of the following statements is correct?

A) The force exerted on Earth by the Sun is weaker than the corresponding force exerted by the Moon.
B) The force exerted on Earth by the Sun is stronger than the corresponding force exerted by the Moon.
C) The force exerted on Earth by the Sun is of the same order of magnitude of the corresponding force exerted by the Moon.

Answer 1

B. The force exerted on Earth by the Sun is stronger than the corresponding force exerted by the Moon.

Step-by-step explanation:

The gravitational force between two planets is given by the following formula:

`F = G \cdot (m\cdot M)/(r^(2))`

Where:

`G` - Gravitational constant.

`m`,
`M` - Masses of planets.

`r` - Distance between planets.

The following is the description of the forces exerted on Earth as both as by the Sun and by the Moon:

Earth - Moon

`F = \left(6.674* 10^(-11)\,(m^(3))/(kg\cdot s^(2)) \right)\cdot \left[((5.972* 10^(24)\,kg)\cdot (7.348* 10^(22)\,kg))/((384000000\,m)^(2)) \right]`

`F = 1.986* 10^(20)\,N`

Earth - Sun

`F = \left(6.674* 10^(-11)\,(m^(3))/(kg\cdot s^(2)) \right)\cdot \left[((5.972* 10^(24)\,kg)\cdot (1.99* 10^(30)\,kg))/((1.5* 10^(11)\,m)^(2)) \right]`

`F = 3.525* 10^(22)\,N`

The force exerted on Earth by the Sun is greater than the force by the Moon. Hence, the answer is B.

Answer 2

B) True.

Step-by-step explanation:

In this exercise to answer the statements we must calculate the force on the earth by the two bodies.

In both cases, the law of universal gravitation describes the process.

F = G m₁ m₂ / r²

Let's calculate each force

- bodies Moon - Earth

Let's call m₁ the mass of the earth (m₁ = me), m₂ the mass of the moon (m₂ = m), the distance from the earth to the moon is r₁ = 3.84 10⁸m and the radius of the earth is re. The force on the tide that is a body on the surface of the Earth have a distance

R₁ = r₁ -re

R₁ = 3.84 10⁸ - 6.37 10⁶

R₁ = 3.77 10⁸ m

Let's calculate

F₁ = G me m / R₁²

F₁ = (G me) 7.36 10²² / (3.77 10⁸)²

F₁ = (G me) 5.2 10⁵

- bodies Earth -Sun

Let's call the mass of the sun M (m2 = M) the distance from the sun to the earth is 1.5 10¹¹ m, so the distance to the surface of the earth

R₂ = r₂ - re

R₂ = 1.5 10¹¹ - 6.37 10⁶

R₂ = 1.5 10¹¹ m

The radius of the earth is too small compared to the earth-sun distance

Let's calculate

F₂ = G me M / R₂²

F₂ = (G me) 1.99 10³⁰ / (1.5 10¹¹)²

F₂ = (G me) 8.8 10⁷

Let's see the statements:

A) False. It´s oppsote

B) True. In the previous part it has a differentiated 10² orders of magnitude

C) False. We saw that they are very different