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Chapter 2: Problem 30

The Sun's mass is very much larger than the Moon's, yet the tides on Earth produced by the Sun are much lower than those caused by the Moon. Why?

Short Answer

Expert verified
Answer: Although the Sun has a much larger mass than the Moon, the greater distance between the Earth and the Sun diminishes its gravitational effect on the Earth, resulting in lower tides caused by the Sun. The tidal force formula shows a cubic relationship with distance, which means the effect of the Sun's gravity reduces significantly due to its greater distance from Earth compared to the Moon.

Step by step solution

01

Understand the concept of tides

Tides occur due to the gravitational forces exerted by the Moon and the Sun on Earth's surface. When the gravitational pull from these celestial bodies is strong enough, it causes the water in Earth's oceans to rise and fall, creating high and low tides.
02

Apply the formula for gravitational force

We can use the formula to calculate the gravitational force between two masses, which is given by: F = G * (m1 * m2) / r^2 Where: - F is the gravitational force - G is the gravitational constant, approximately 6.674 * 10^-11 N(m/kg)^2 - m1 and m2 are the masses of the two objects (in this case, Earth and the Moon or the Sun) - r is the distance between the centers of the two objects' masses
03

Consider the mass and distance of the Moon and the Sun

The mass of the Sun is approximately 1.989 * 10^30 kg, while the mass of the Moon is about 7.342 * 10^22 kg. The average distance from the Earth to the Sun is about 149.6 million kilometers (1.496 * 10^11 m), while the average distance from the Earth to the Moon is about 384,400 kilometers (3.844 * 10^8 m).
04

Calculate the tidal force

Tidal force is the difference in gravitational force acting on different points of an object. For Earth, this is typically focused on the difference between the force acting on the point closest to the celestial body (Sun or Moon) and the point furthest away. The tidal force formula is given by: T = (2 * G * M * m * R) / r^3 Where: - T is the tidal force - G is the gravitational constant - M is the mass of the celestial body (Sun or Moon) - m is the mass of the Earth - R is the radius of the Earth (approximately 6.371 * 10^6 m) - r is the distance from Earth's center to the celestial body's center
05

Compare the tidal forces exerted by the Sun and the Moon

We can now plug in the values for the Sun and the Moon into the tidal force formula: Tidal force of the Sun on Earth: Tsun = (2 * G * Msun * mearth * R) / rsun^3 Tidal force of the Moon on Earth: Tmoon = (2 * G * Mmoon * mearth * R) / rmoon^3 Although the Sun has a much larger mass than the Moon, the distance between the Earth and the Sun is much greater than the distance between the Earth and the Moon. Due to the cubic relationship with distance in the tidal force formula, the greater distance between the Earth and the Sun diminishes the Sun's gravitational effect on the Earth, resulting in lower tides caused by the Sun despite its larger mass.

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Most popular questions from this chapter

The first "Lunar Olympics" is to be held on the Moon inside a huge dome. Of the usual Olympic events-track and field, swimming, gymnastics, and so on -which would be drastically affected by the Moon's lower gravity? In which events do you think Earth-based records would be broken? In which events would the performances be no better-or perhaps worse-than on Earth?

A person places a hand on a closed book resting on a table and then presses downward while pushing outward. Either the book slides across the table or the hand slides across the book. What determines which event happens? Which type(s) of friction is (are) involved?

Imagine that two identical containers, one filled with marbles and the other filled with Styrofoam beads, are released simultaneously from rest from the same height - say, \(3 \mathrm{~m}\) -above Earth's surface. (a) Which container, if either, experiences the greater gravitational force? Justify your answer. (b) Which container, if either, experiences the greater acceleration from gravity? Again, justify your answer.

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