6.5 Newton's Law of Universal Gravity - School of Physics 2e

learning target

At the end of this unit you will be able to:

Explain the gravity of the earth.
Describe the gravitational influence of the Moon on the Earth.
Discuss space weightlessness.
Consider the Cavendish experiment

What do sore feet, falling apples, and lunar orbits have in common? Each of them is caused by gravity. Our feet are strained from supporting our weight - the gravity of the Earth is on us. An apple falls from a tree because it exerts the same force a few meters above the Earth's surface. The Moon orbits the Earth because gravity is able to provide the necessary centripetal force over a distance of hundreds of millions of meters. In fact, the same force keeps planets orbiting the sun, stars orbiting the center of the Milky Way, and galaxies clump together. Gravity is another example of nature's underlying simplicity. It is the weakest of the four fundamental forces found in nature, and in some ways the least understood. It is a force acting at a distance, without physical contact, expressed in a formula that applies everywhere in the universe, for masses and distances from small to infinite.

Sir Isaac Newton was the first scientist to precisely define gravity and demonstrate that it could explain falling bodies and astronomical motion. I understandFigure 6.17. But Newton wasn't the first to suspect that the same force is responsible for our weight and planetary motion. Galileo's predecessor, Galilei, believed that objects fall and planets move for the same reason. Some of Newton's contemporaries, such as Robert Hooke, Christopher Wren, and Edmund Halley, also made some progress in understanding gravity. But Newton was the first to come up with the precise mathematical form by which he showed that the motion of celestial bodies must have conics—circles, ellipses, parabolas, and hyperbolas. This theoretical prediction turned out to be a huge victory—moons, planets, and comets have been known to follow such paths for some time, but no one had been able to come up with a mechanism that would make them follow these paths rather than others. Other prominent scientists and mathematicians at the time, especially those outside of Britain, were reluctant to accept Newton's principles. It wasn't until the work of another eminent philosopher, author, and scientist, Émilie du Châtelet, that Newtonian gravity became a precise and universal law. Du Châtelet, who laid the foundations for understanding the principles of the conservation of energy and the masslessness of light, translated and extended Newton's seminal work. He also used calculus to explain gravity, which helped gain recognition.

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picture6.17 According to early records, when Newton saw an apple fall from a tree and realized that if gravity could reach the earth and enter a tree, it was inspired to connect the falling body to astronomical motion, and it could also reach the sun. Newton's apple inspiration is part of world folklore and may even be based in reality. Newton's law of gravitation and his laws of motion answered age-old questions about nature and greatly supported the concepts behind nature's simplicity and unity. Like many revolutionary discoveries, it was not immediately accepted. Renowned French scientist and philosopher Émilie du Châtelet helped establish Newton's theories in France and continental Europe.

Gravity is relatively simple. It's always attractive, and it just depends on the masses involved and the distance between them. Serving in modern languages,Newton's law of universal gravitationPoint out that every particle in the universe attracts all other particles by force along the lines connecting them. The force is proportional to the product of their masses and inversely proportional to the square of the distance between them.

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picture6.18 The gravitational force lies on the line connecting the centers of mass of the two objects. According to Newton's third law, every force has the same magnitude.

Misunderstanding notice

According to Newton's third law, the magnitude of the force acting on each object (one object has more mass than the other) is the same.

The cadavers we handle are usually large. For simplicity, let's assume that an object behaves as if its entire mass is concentrated in an object calledcenter of gravity(CM), which will be inLinear Momentum and Collisions.For two objects with mass $rice$ I $rice$ and distance $R$ Between their centers of mass, the equation for Newton's law of universal gravitation is as follows

$eat = G \frac{mm}{R^{2}},$

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Where $eat$ is the magnitude of gravity i $G$ is the scaling factorgravitational constant. $G$ It's a universal constant for gravity -- that is, it's thought to be the same throughout the universe. It has been experimentally measured that it does

$G = 6 . 674 \times 10^{- 11} \frac{no \cdot {rice}^{2}}{{Kilogram}^{2}}$

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in SI units. note its unit $G$ is the resultant force in Newtons $eat = G \frac{mm}{R^{2}}$ , taking into account the mass in kilograms and the distance in meters. For example, two objects with a mass of 1000 kg and a distance of 1000 m are attracted by gravity $6 . 674 \times 10^{- 11} no$ . This is an extremely low power. The magnitude of gravity is consistent with everyday experience. We ignore the fact that even large objects like mountains exert a gravitational pull on us. In fact, the weight of our body is its pulling forcethe whole earthweight to us $6 \times 10^{24} Kilogram$ .

Remember the acceleration of gravity $G$ it's about $9,80 meters {/small}^{2}$ on the ground. Now we can determine why this is the case. product weightmgis the gravitational force between it and the earth. exchangemgfor $eat$ Give Newton's law of universal gravitation

$mg = G \frac{mm}{R^{2}},$

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Where $rice$ is the mass of the object, $rice$ is the mass of the earth i $R$ is the distance from the center of the Earth (the distance between the object's center of mass and the Earth). I understandFigure 6.19.Lots of $rice$ The object cancels, leaving the equation $G$ :

$G = G \frac{rice}{R^{2}} .$

experiment at home

Take a ball, a ball and a spoon and drop them from the same height. Will they land at the same time? If a piece of paper is dropped on the ground, does it behave like any other object? Explain your observations.

build connection

People are still trying to understand gravity. as we will inparticle physicsModern physics studies gravity in relation to other forces, space and time. General relativity changed the way we think about gravity, causing us to think of it as the curvature of space and time.

In the example below, we do a comparison similar to what Newton himself did. He pointed out that if gravity is causing the moon to orbit the Earth, then the acceleration due to gravity should be equal to the centripetal acceleration of the moon in its orbit. Newton found that the two accelerations "almost" matched.

example 6.6

Earth's gravity is the centripetal force that moves the moon along a curved path

(a) Find the acceleration due to gravity at a certain distance from the moon.

(b) Calculate the centripetal acceleration required to keep the Moon in orbit (assuming it is in a circular orbit around the stationary Earth) and compare it to the gravitational acceleration just calculated.

(a) Strategy

This calculation is the same as for the acceleration due to gravity at the Earth's surface, except that $R$ is the distance from the center of the Earth to the center of the Moon. The radius of the moon's near-circular orbit is $3 . 84 \times 10^{8} rice$ .

Solution for (a)

Substitute known values into for expressions $G$ Found above, let's remember this $rice$ It's the Earth's mass that's fading, not the Moon's

$\begin{array}{l} G & = & G \frac{rice}{R^{2}} = (6 . 67 \times 10^{- 11} \frac{no \cdot {rice}^{2}}{{Kilogram}^{2}}) \times \frac{5 . 98 \times 10^{24} Kilogram}{(3 . 84 \times 10^{8} rice)^{2}} \\ = & 2 . 70 \times 10^{- 3} {Agitation.}^{2} \end{array}$

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(b) Strategy

Centripetal acceleration can be calculated using any form

$\begin{matrix} {one}_{Do} = \frac{w^{2}}{R} \\ {one}_{Do} = {oh}^{2} \end{matrix}} .$

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Solution for (b)

Since the period of the lunar orbit (the time it takes to complete one revolution) is 27.3 days, (d) and using

$1 day \times 24 \frac{time}{Hello} \times 60 \frac{minute}{time} \times 60 \frac{small}{minute} = 86 400 SEK$

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we saw

$oh = \frac{Man I}{Man Ton} = \frac{2 pieces radian}{(27 . three days) (86.400 seconds/day)} = 2 . 66 \times 10^{- 6} \frac{radian}{small} .$

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The centripetal acceleration is

$\begin{array}{l} {one}_{Do} & = & {oh}^{2} = (3 . 84 \times 10^{8} rice) (2 . 66 \times 10^{- 6} rad/s)^{2} \\ = & 2.72 \times 10^{- 3} {Agitation.}^{2} \end{array}$

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The direction of acceleration is towards the center of the earth.

discuss

The lunar centripetal acceleration found in (b) differs by less than 1% from the Earth's gravitational acceleration found in (a). This correspondence is approximate because the Moon's orbit is slightly elliptical and the Earth is not stationary (instead, the Earth-Moon system rotates around its center of mass, which is located about 1,700 km below the Earth's surface). The clear implication is that the Earth's gravitational pull causes the Moon to orbit the Earth.

Why doesn't the earth stand still when the moon revolves around the earth? This is because, according to Newton's third law, if the earth exerts a force on the moon, the moon must exert an equal and opposite force on the earth.Figure 6.20). We cannot see the Moon's influence on Earth's motion because the Moon's gravity keeps our bodies in perfect alignment with Earth's motion, but there are other signs on Earth that clearly show the Moon's gravity at work, as inSatellites and Kepler's Laws: An Argument for Simplicity.

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picture6.20 (a) The Earth and Moon rotate around their common center of mass about once a month. (b) Their centers of mass revolve around the Sun in elliptical orbits, but there is a "wobble" in the Earth's path around the Sun. Similar motions in the paths of stars have been observed and considered to be direct evidence of planets orbiting these stars. This is important because the light reflected from planets is usually too faint to be seen.

tidal

Ocean tides are a very visible result of the Moon's gravitational pull on Earth.Figure 6.21is a simplified diagram of the Moon's position relative to the tides. Because water flows easily across the Earth's surface, tides form on the side of Earth closest to the Moon, where the Moon's gravitational pull is strongest. Why are the tides also on the other side of the earth? The answer is that the Earth is more attracted to the Moon than the water on the other side because the Earth is closer to the Moon. Therefore, the water on the side of the Earth near the Moon is sucked away from the Earth, and the Earth is sucked away from the water on the other side. As the Earth rotates, the tidal bulge (the effect of tidal forces between an orbiting natural satellite and the parent planet it orbits) maintains its orientation relative to the Moon. So there are two tides per day (the actual tidal period is about 12 hours 25.2 minutes, since the moon also orbits it every day).

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picture6.21 The Moon pulls more on the water on the near side than on the Earth, which pulls on the Earth more than on the far side, causing ocean tides. Distances and sizes are not to scale. For this simplified representation of the Earth-Moon system, there are two high tides and two low tides at any point each day because the Earth rotates under the tidal bulge.

The sun also affects the tides, although its influence is only about half that of the moon's. However, the largest tides, called spring tides, occur when the Earth, Moon, and Sun align. The smallest tide, called low tide, occurs when the sun is at point a $90º$ Align the Earth-Moon angle.

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picture6.22 (a, b) High tides: Highest tides occur when the Earth, Moon, and Sun are aligned. (c) Tides: The lowest tides occur when the sun is at $90º$ In the Earth-Moon arrangement. Note that this figure is not drawn to scale.

Tides are not unique to Earth, but occur in many astronomical systems. The most extreme tides occur where gravity is strongest and where changes are fastest, such as near black holes (see Figure 1).Figure 6.23). Several potential black hole candidates have been observed in our galaxy. They are more massive than the sun, but only a few kilometers across. The tidal forces near them are so strong that they can tear apart material from the companion star.

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picture6.23 A black hole is a celestial body with such a strong gravitational force that not even light can escape. This black hole was created by a stellar supernova in a binary star system. The tidal forces generated by black holes are so strong that they can drag material away from companion stars. This material is compressed and heated as it is drawn into the black hole, producing the light and X-rays that can be observed from Earth.

"Weightlessness" and microgravity

The apparent gravitational field experienced by astronauts orbiting Earth, as opposed to the enormous gravitational pull near a black hole. How did the "disaster" affect astronauts who spent months in orbit? What effect does zero gravity have on plant growth? Weightlessness doesn't mean astronauts are immune to gravity. There is no "zero gravity" for astronauts in orbit. The term simply means that the astronaut is in free fall, accelerated by the acceleration of gravity. If the elevator cable breaks, the passengers inside will free fall and experience zero gravity. A brief period of weightlessness may occur in some amusement park rides.

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picture6.24 Astronauts experience weightlessness aboard the International Space Station. (Source: NASA)

microgravityRefers to an environment in which the apparent net acceleration of the body is small compared to the acceleration produced by the Earth at its surface. Over the past three decades, many interesting topics in biology and physics have been studied in the presence of microgravity. A looming issue is the effect on astronauts of prolonged periods in space, such as on the International Space Station. The scientists observed that in this environment, the muscles atrophy (atrophy). There is also corresponding bone loss. Research on the adaptation of the cardiovascular system to spaceflight continues. On Earth, blood pressure is generally higher in the legs than in the head because the higher blood column exerts a downward gravitational force on it. When you're standing, 70% of your blood is below the level of your heart, and when you're lying down, the opposite is true. What does not having this pressure differential mean for the heart?

Some discoveries in human physiology in space may have clinical implications for treating diseases on Earth. On the downside, spaceflight is known to affect the body's immune system, potentially making crew members more susceptible to infectious diseases. Experiments in space have also shown that some bacteria grow faster in microgravity than on Earth. On the bright side, however, studies have shown that the production of microbial antibiotics can be doubled in cultures grown in space. It is hoped that these mechanisms will be understood in order to achieve similar success in this field. In another area of space physics research, inorganic and protein crystals have grown in space to a much higher mass than anything grown on Earth, so crystallographic studies of their structures can yield better results.

Plants evolve under the influence of gravity stimuli and gravity sensors. Roots grow downward and grow upward. Plants can provide life support systems for long-duration space missions by regenerating the atmosphere, purifying water and producing food. Some studies have shown that gravity does not affect plant growth and development, but there is still uncertainty about the structural changes in plants grown in microgravity environments.

The Cavendish Experiment: Then and Now

As mentioned earlier, the gravitational constant $G$ Experimentally established. The definition was first accurately proposed by the British scientist Henry Cavendish (1731-1810) in 1798, more than 100 years after Newton's law of gravitation was published. its measurement $G$ is very basic and important as it determines the strength of one of the four natural forces. Cavendish's experiment was very difficult because he used theFigure 6.25.It is worth noting that the value $G$ It is within 1% of the best modern value.

important outcome of knowledge $G$ is the exact value for which the Earth's mass can be finally obtained. This is done by measuring the acceleration of gravity as accurately as possible and then calculating the Earth's mass $rice$ The relationship given from Newton's law of gravitation

$mg = G \frac{mm}{R^{2}},$

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Where $rice$ is the mass of the object, $rice$ is the mass of the earth i $R$ is the distance from the center of the Earth (the distance between the object's center of mass and the Earth). I understandFigure 6.18.Lots of $rice$ The object cancels, leaving the equation $G$ :

$G = G \frac{rice}{R^{2}} .$

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Reorder to solve $rice$ phrase

$rice = \frac{{gram}^{2}}{G} .$

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so $rice$ can be calculated because all quantities on the right, including the radius of the Earth $R$ Known from direct measurement. we'll see insideSatellites and Kepler's Laws: An Argument for Simplicitythat know $G$ It also allows determination of astronomical masses. Interestingly, of all the fundamental constants of physics, $G$ is by far the least well-defined.

The Cavendish experiment has also been used to study other aspects of gravity. A more interesting question is whether gravity depends on matter and mass—for example, does a kilogram of lead exert the same gravitational force as a kilogram of water. Hungarian scientist Roland von Eötvös initiated this research in the early 20th century. He found that, on the scale of five parts per billion, gravity does not depend on matter. Such experiments continue today and have improved upon Eötvös' measurements. Cavendish-style experiments, such as those performed by Eric Adelberg of the University of Washington and others, also place strict constraints on the possibility of a fifth force and verify an important prediction of general relativity - Gravitational energy contributes to rest mass. The measurements being made there use torsion balances and parallel plates (not the spheres used by Cavendish) to study how Newton's law of gravitation works at submillimeter distances. Do gravitational effects deviate from the inverse square law on such a small scale? So far, no deviations have been observed.

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picture6.25 Cavendish used such a device to measure the gravitational force between two suspended spheres ( $rice$ ) are on the stand ( $rice$ ) by observing the amount of twist (twist) created in the fiber. The distance between the masses can be varied to control the force-distance relationship. Modern experiments of this type continue to study gravity.

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