Law Of Universal Gravitation Example

six.v: Newton's Universal Constabulary of Gravitation

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Learning Objectives

By the terminate of this section, you will be able to:

• Explain Earth's gravitational force.
• Describe the gravitational consequence of the Moon on Globe.
• Discuss weightlessness in space.
• Examine the Cavendish experiment

What do aching feet, a falling apple tree, and the orbit of the Moon accept in mutual? Each is caused by the gravitational force. Our feet are strained by supporting our weight—the force of Earth's gravity on usa. An apple falls from a tree because of the same force interim a few meters above Earth's surface. And the Moon orbits Earth because gravity is able to supply the necessary centripetal strength at a altitude of hundreds of millions of meters. In fact, the same forcefulness causes planets to orbit the Sun, stars to orbit the centre of the milky way, and galaxies to cluster together. Gravity is another example of underlying simplicity in nature. It is the weakest of the four basic forces found in nature, and in some ways the least understood. It is a force that acts at a distance, without concrete contact, and is expressed by a formula that is valid everywhere in the universe, for masses and distances that vary from the tiny to the immense.

Sir Isaac Newton was the first scientist to precisely define the gravitational force, and to show that it could explain both falling bodies and astronomical motions. See Figure. Simply Newton was not the start to suspect that the same force caused both our weight and the motion of planets. His forerunner Galileo Galilei had contended that falling bodies and planetary motions had the same cause. Some of Newton's contemporaries, such as Robert Hooke, Christopher Wren, and Edmund Halley, had also made some progress toward understanding gravitation. But Newton was the start to propose an exact mathematical form and to use that form to prove that the motion of heavenly bodies should be conic sections—circles, ellipses, parabolas, and hyperbolas. This theoretical prediction was a major triumph—it had been known for some time that moons, planets, and comets follow such paths, but no one had been able to propose a mechanism that caused them to follow these paths and non others.

Figure \(\PageIndex{ane}\): Co-ordinate to early accounts, Newton was inspired to make the connection between falling bodies and astronomical motions when he saw an apple fall from a tree and realized that if the gravitational force could extend above the ground to a tree, it might also achieve the Sun. The inspiration of Newton'southward apple tree is a part of worldwide folklore and may fifty-fifty be based in fact. Not bad importance is fastened to it because Newton's universal law of gravitation and his laws of movement answered very old questions well-nigh nature and gave tremendous support to the notion of underlying simplicity and unity in nature. Scientists still expect underlying simplicity to emerge from their ongoing inquiries into nature.

The gravitational force is relatively simple. Information technology is ever attractive, and information technology depends only on the masses involved and the distance between them. Stated in modernistic linguistic communication, Newton's universal law of gravitation states that every particle in the universe attracts every other particle with a forcefulness along a line joining them. The strength is directly proportional to the product of their masses and inversely proportional to the square of the altitude betwixt them.

Figure \(\PageIndex{2}\): Gravitational attraction is forth a line joining the centers of mass of these two bodies. The magnitude of the strength is the aforementioned on each, consequent with Newton's third law.

MISCONCEPT ALERT

The magnitude of the force on each object (one has larger mass than the other) is the same, consistent with Newton'due south third police.

The bodies we are dealing with tend to exist large. To simplify the situation we assume that the body acts as if its entire mass is concentrated at i specific bespeak called the center of mass (CM), which will exist further explored in Linear Momentum and Collisions. For two bodies having masses \(m\) and \(M\) with a distance \(r\) between their centers of mass, the equation for Newton's universal law of gravitation is \[ F = Grand\dfrac{mM}{r^2},\] where \(F\) is the magnitude of the gravitational force and \(G\) is a proportionality gene called the gravitational abiding. \(G\) is a universal gravitational constant—that is, it is idea to be the same everywhere in the universe. It has been measured experimentally to be

\[Chiliad = 6.673 \times x^{-11} \dfrac{N \cdot k^2}{kg^2}\]

in SI units. Note that the units of \(Yard\) are such that a force in newtons is obtained from \(F = G\frac{mM}{r^ii} \), when because masses in kilograms and distance in meters. For example, ii i.000 kg masses separated by 1.000 m will experience a gravitational attraction of \(six.673 \times ten^{-eleven} \, N\).

This is an extraordinarily small force. The small magnitude of the gravitational force is consistent with everyday experience. We are unaware that fifty-fifty large objects like mountains exert gravitational forces on u.s.. In fact, our trunk weight is the force of attraction of the entire Earth on usa with a mass of \(half-dozen \times 10^{24} \, kg\).

Recall that the acceleration due to gravity \(g\) is nigh \(9.80 \, 1000/s^ii\) on Earth. We can at present make up one's mind why this is so. The weight of an object mg is the gravitational force betwixt it and World. Substituting mg for \(F\) in Newton's universal law of gravitation gives

\[mg = One thousand\dfrac{mM}{r^2}, \] where \(m\) is the mass of the object, \(M\) is the mass of Earth, and \(r\) is the distance to the center of Globe (the distance betwixt the centers of mass of the object and Earth). See Figure. The mass \(m\) of the object cancels, leaving an equation for \(g\):

\[g = Chiliad\dfrac{M}{r^2}. \]

Substituting known values for Globe's mass and radius (to three significant figures),

\[k = \left(six.673 \times 10^{-11} \, \dfrac{Northward \cdot m^2}{kg^2} \right) \times \dfrac{v.98 \times 10^{24} \, kg}{(6.38 \times ten^six \, m)^2},\]

and nosotros obtain a value for the acceleration of a falling body: \[g = 9.80 \, m/s^ii.\]

Figure \(\PageIndex{3}\): The distance between the centers of mass of Globe and an object on its surface is very about the aforementioned as the radius of Earth, because Earth is so much larger than the object.

This is the expected value and is independent of the body's mass. Newton'south constabulary of gravitation takes Galileo'due south observation that all masses fall with the same acceleration a step further, explaining the observation in terms of a force that causes objects to fall—in fact, in terms of a universally existing forcefulness of attraction between masses.

TAKE HOME EXPERIMENT

Have a marble, a brawl, and a spoon and drop them from the same height. Do they hit the floor at the same time? If you drop a piece of paper as well, does it acquit like the other objects? Explicate your observations.

MAKING CONNECTIONS

Attempts are however being made to sympathize the gravitational forcefulness. Equally we shall run across in Particle Physics, modern physics is exploring the connections of gravity to other forces, infinite, and fourth dimension. General relativity alters our view of gravitation, leading united states to think of gravitation as bending space and time.

In the following case, we make a comparison similar to one made by Newton himself. He noted that if the gravitational force caused the Moon to orbit Earth, then the dispatch due to gravity should equal the centripetal acceleration of the Moon in its orbit. Newton plant that the 2 accelerations agreed "pretty about."

Example \(\PageIndex{1}\): World'due south Gravitational Force Is the Centripetal Force Making the Moon Move in a Curved Path

1. Discover the dispatch due to Earth's gravity at the distance of the Moon.
2. Calculate the centripetal dispatch needed to keep the Moon in its orbit (bold a circular orbit almost a fixed Earth), and compare information technology with the value of the acceleration due to Globe's gravity that you accept simply found.

Strategy for (a)

This calculation is the same as the one finding the acceleration due to gravity at Globe's surface, except that \(r\) is the distance from the center of Earth to the center of the Moon. The radius of the Moon's nearly circular orbit is \(3.84 \times x^8 \, m\).

Solution for (a)

Substituting known values into the expression for \(thousand\) found to a higher place, remembering that \(M\) is the mass of World not the Moon, yields

\[1000 = G\dfrac{Thousand}{r^two} = \left(6.67 \times 10^{-xi} \dfrac{North \cdot one thousand^2}{kg^ii} \right) \times \dfrac{5.98 \times ten^{24} \, kg}{iii.84 \times x^8 \, m)^2} \]

\[= ii.lxx \times 10^{-3} \, thousand/s^2.\]

Strategy for (b)

Centripetal dispatch tin be calculated using either form of

\[a_c = \dfrac{five^2}{r}\]

\[a_c = r\omega^2\]

We choose to use the second class:

\[a_c = r\omega^2,\]

where \(\omega\) is the angular velocity of the Moon virtually Earth.

Solution for (b)

Given that the flow (the time it takes to make one complete rotation) of the Moon's orbit is 27.iii days, (d) and using

\[1 \, d \times 24 \dfrac{hr}{d} \times 60 \dfrac{min}{60 minutes} \times 60 \dfrac{s}{min} = 86,400 \, southward\]

nosotros run across that

\[a_c = r\omega^2 = (iii.84 \times 10^8 \, k)(2.66 \times ten^{-6} \, rad/s^2) \]

\[ = 2.72 \times 10^{-three} \, m/s^2. \]

The direction of the dispatch is toward the center of the World.

Word

The centripetal dispatch of the Moon found in (b) differs by less than 1% from the acceleration due to Earth's gravity plant in (a). This agreement is approximate because the Moon'due south orbit is slightly elliptical, and Earth is non stationary (rather the Earth-Moon system rotates about its middle of mass, which is located some 1700 km beneath Earth'southward surface). The clear implication is that Globe'south gravitational strength causes the Moon to orbit Globe.

Why does Earth non remain stationary as the Moon orbits it? This is because, as expected from Newton's third police force, if Earth exerts a force on the Moon, so the Moon should exert an equal and opposite force on Earth (meet Figure). We do non sense the Moon'due south effect on Globe's motion, because the Moon's gravity moves our bodies correct along with Globe but there are other signs on Globe that clearly show the effect of the Moon's gravitational force as discussed in Satellites and Kepler'southward Laws: An Statement for Simplicity.

Figure \(\PageIndex{four}\): (a) Earth and the Moon rotate approximately once a month around their mutual center of mass. (b) Their center of mass orbits the Sunday in an elliptical orbit, simply Earth'south path around the Sun has "wiggles" in it. Similar wiggles in the paths of stars have been observed and are considered direct evidence of planets orbiting those stars. This is of import because the planets' reflected lite is ofttimes as well dim to exist observed.

Tides

Ocean tides are one very observable result of the Moon'southward gravity acting on Earth. Figure is a simplified drawing of the Moon'due south position relative to the tides. Because water easily flows on Earth'due south surface, a high tide is created on the side of Earth nearest to the Moon, where the Moon'due south gravitational pull is strongest. Why is there too a high tide on the opposite side of Globe? The answer is that Earth is pulled toward the Moon more than the water on the far side, because World is closer to the Moon. And then the water on the side of Earth closest to the Moon is pulled away from Earth, and Earth is pulled away from water on the far side. As Earth rotates, the tidal bulge (an effect of the tidal forces between an orbiting natural satellite and the primary planet that it orbits) keeps its orientation with the Moon. Thus there are two tides per twenty-four hour period (the actual tidal period is nearly 12 hours and 25.2 minutes), because the Moon moves in its orbit each day too).

Figure \(\PageIndex{v}\): The Moon causes ocean tides past alluring the water on the near side more than than Earth, and by attracting World more than the h2o on the far side. The distances and sizes are not to calibration. For this simplified representation of the Earth-Moon organization, at that place are two loftier and two depression tides per mean solar day at whatsoever location, because Earth rotates under the tidal bulge.

The Lord's day also affects tides, although it has about one-half the upshot of the Moon. All the same, the largest tides, chosen bound tides, occur when Globe, the Moon, and the Sun are aligned. The smallest tides, called neap tides, occur when the Dominicus is at a \(90^o\) bending to the Earth-Moon alignment.

Figure \(\PageIndex{half dozen}\): (a, b) Spring tides: The highest tides occur when World, the Moon, and the Sun are aligned. (c) Neap tide: The everyman tides occur when the Lord's day lies at \(xc^o\) to the World-Moon alignment. Note that this figure is not fatigued to calibration.

Tides are not unique to Globe but occur in many astronomical systems. The most extreme tides occur where the gravitational strength is the strongest and varies well-nigh rapidly, such as nearly black holes (run across Figure). A few likely candidates for black holes have been observed in our galaxy. These have masses greater than the Sun but accept diameters only a few kilometers across. The tidal forces well-nigh them are so corking that they tin can actually tear thing from a companion star.

Figure \(\PageIndex{seven}\): A black hole is an object with such potent gravity that not even light can escape information technology. This black hole was created past the supernova of i star in a two-star system. The tidal forces created by the blackness pigsty are so great that it tears affair from the companion star. This thing is compressed and heated as information technology is sucked into the black hole, creating light and X-rays observable from Earth.

"Weightlessness" and Microgravity

In contrast to the tremendous gravitational force near black holes is the apparent gravitational field experienced by astronauts orbiting Earth. What is the effect of "weightlessness" upon an astronaut who is in orbit for months? Or what about the event of weightlessness upon establish growth? Weightlessness doesn't hateful that an astronaut is non beingness acted upon past the gravitational forcefulness. There is no "zero gravity" in an astronaut's orbit. The term just means that the astronaut is in free-fall, accelerating with the acceleration due to gravity. If an elevator cablevision breaks, the passengers inside volition be in free fall and will feel weightlessness. Yous can feel short periods of weightlessness in some rides in amusement parks.

Figure \(\PageIndex{eight}\): Astronauts experiencing weightlessness on board the International Space Station. (credit: NASA).

Microgravity refers to an environs in which the apparent internet acceleration of a trunk is small compared with that produced by Earth at its surface. Many interesting biology and physics topics take been studied over the past three decades in the presence of microgravity. Of immediate concern is the event on astronauts of extended times in outer infinite, such as at the International Space Station. Researchers have observed that muscles will atrophy (waste material away) in this environment. There is also a corresponding loss of bone mass. Study continues on cardiovascular accommodation to space flying. On Earth, claret force per unit area is usually higher in the feet than in the caput, because the college column of blood exerts a downward forcefulness on it, due to gravity. When standing, lxx% of your blood is below the level of the center, while in a horizontal position, just the opposite occurs. What difference does the absence of this pressure differential take upon the center?

Some findings in human physiology in infinite tin be clinically important to the management of diseases dorsum on World. On a somewhat negative note, spaceflight is known to affect the human immune system, possibly making the crew members more than vulnerable to infectious diseases. Experiments flown in space likewise have shown that some leaner grow faster in microgravity than they practice on World. All the same, on a positive notation, studies point that microbial antibody production can increase past a factor of two in space-grown cultures. One hopes to be able to understand these mechanisms so that like successes tin can exist achieved on the ground. In another area of physics space research, inorganic crystals and protein crystals have been grown in outer infinite that have much higher quality than any grown on Earth, so crystallography studies on their construction can yield much amend results.

Plants accept evolved with the stimulus of gravity and with gravity sensors. Roots grow downwards and shoots grow upwards. Plants might exist able to provide a life support system for long duration space missions by regenerating the atmosphere, purifying water, and producing food. Some studies have indicated that plant growth and development are non afflicted by gravity, merely at that place is withal doubt nigh structural changes in plants grown in a microgravity surround.

The Cavendish Experiment: Then and Now

Every bit previously noted, the universal gravitational constant \(G\) is determined experimentally. This definition was first done accurately by Henry Cavendish (1731–1810), an English language scientist, in 1798, more 100 years later on Newton published his universal constabulary of gravitation. The measurement of \(1000\) is very basic and important because information technology determines the force of one of the 4 forces in nature. Cavendish's experiment was very hard because he measured the tiny gravitational attraction between two ordinary-sized masses (tens of kilograms at nearly), using apparatus similar that in Effigy. Remarkably, his value for \(K\) differs past less than 1% from the best mod value.One important event of knowing \(G\) was that an authentic value for Earth's mass could finally be obtained. This was done by measuring the dispatch due to gravity as accurately every bit possible so calculating the mass of Earth \(K\) from the relationship Newton's universal law of gravitation gives

\[mg = Thousand\dfrac{mM}{r^2},\]

where \(m\) is the mass of the object, \(M\) is the mass of World, and \(r\) is the distance to the center of Earth (the altitude between the centers of mass of the object and Globe). See Figure. The mass \(g\) of the object cancels, leaving an equation for \(thousand\):

\[g = M\dfrac{Thousand}{r^2}. \]

Rearranging to solve for \(M\) yields

\[Yard = \dfrac{gr^2}{G}.\]

and then \(G\) tin can be calculated considering all quantities on the right, including the radius of Earth \(r\), are known from direct measurements. Nosotros shall encounter in Satellites and Kepler'due south Laws: An Statement for Simplicity that knowing \(G\) too allows for the conclusion of astronomical masses. Interestingly, of all the key constants in physics, \(1000\) is past far the least well determined.

The Cavendish experiment is as well used to explore other aspects of gravity. 1 of the nigh interesting questions is whether the gravitational forcefulness depends on substance too as mass—for example, whether one kilogram of lead exerts the same gravitational pull as 1 kilogram of water. A Hungarian scientist named Roland von Eötvös pioneered this inquiry early on in the 20th century. He found, with an accuracy of five parts per billion, that the gravitational force does not depend on the substance. Such experiments keep today, and accept improved upon Eötvös' measurements. Cavendish-type experiments such equally those of Eric Adelberger and others at the University of Washington, accept also put astringent limits on the possibility of a 5th force and have verified a major prediction of general relativity—that gravitational energy contributes to residuum mass. Ongoing measurements there utilize a torsion balance and a parallel plate (not spheres, as Cavendish used) to examine how Newton's police of gravitation works over sub-millimeter distances. On this small-scale, practise gravitational furnishings depart from the inverse foursquare law? And then far, no difference has been observed.

Figure \(\PageIndex{9}\): Cavendish used an apparatus like this to mensurate the gravitational attraction between the two suspended spheres \((k)\) and the two on the stand up \((M)\) by observing the amount of torsion (twisting) created in the cobweb. Altitude between the masses can be varied to bank check the dependence of the force on distance. Modern experiments of this type go on to explore gravity.

Summary

• Newton's universal police force of gravitation: Every particle in the universe attracts every other particle with a strength along a line joining them. The force is direct proportional to the product of their masses and inversely proportional to the square of the distance between them. In equation course, this is

\[F = 1000\dfrac{mM}{r^two} \]

where F is the magnitude of the gravitational forcefulness. \(G\) is the gravitational constant, given by \(G = 6.63 \times ten^{-11} \, N \cdot m^ii/kg^2\).

• Newton'south law of gravitation applies universally.

Glossary

gravitational constant, Grand

a proportionality factor used in the equation for Newton's universal law of gravitation; it is a universal constant—that is, it is thought to be the same everywhere in the universe

heart of mass

the betoken where the unabridged mass of an object tin can exist thought to be concentrated

microgravity

an environment in which the credible net acceleration of a body is modest compared with that produced by Earth at its surface

Newton's universal police of gravitation

every particle in the universe attracts every other particle with a force along a line joining them; the forcefulness is directly proportional to the product of their masses and inversely proportional to the square of the distance betwixt them

Law Of Universal Gravitation Example,

Source: https://phys.libretexts.org/Bookshelves/College_Physics/Book%3A_College_Physics_(OpenStax)/06%3A_Uniform_Circular_Motion_and_Gravitation/6.05%3A_Newtons_Universal_Law_of_Gravitation

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