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Section summary

  • Newton’s universal law of gravitation: Every particle in the universe attracts every other particle with a force along a line joining them. The force is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. In equation form, this is
    F = G mM r 2 , size 12{F=G { { ital "mM"} over {r rSup { size 8{2} } } } } {}

    where F is the magnitude of the gravitational force. G size 12{G} {} is the gravitational constant, given by G = 6 . 674 × 10 –11 N m 2 /kg 2 size 12{G=6 "." "673" times "10" rSup { size 8{"-11"} } `N cdot m rSup { size 8{2} } "/kg" rSup { size 8{2} } } {} .

  • Newton’s law of gravitation applies universally.

Conceptual questions

Action at a distance, such as is the case for gravity, was once thought to be illogical and therefore untrue. What is the ultimate determinant of the truth in physics, and why was this action ultimately accepted?

Two friends are having a conversation. Anna says a satellite in orbit is in freefall because the satellite keeps falling toward Earth. Tom says a satellite in orbit is not in freefall because the acceleration due to gravity is not 9.80 m /s 2 size 12{9 "." "80"`"m/s" rSup { size 8{2} } } {} . Who do you agree with and why?

Draw a free body diagram for a satellite in an elliptical orbit showing why its speed increases as it approaches its parent body and decreases as it moves away.

Newton’s laws of motion and gravity were among the first to convincingly demonstrate the underlying simplicity and unity in nature. Many other examples have since been discovered, and we now expect to find such underlying order in complex situations. Is there proof that such order will always be found in new explorations?

Problem exercises

(a) Calculate Earth’s mass given the acceleration due to gravity at the North Pole is 9.830 m /s 2 size 12{9 "." "830"`"m/s" rSup { size 8{2} } } {} and the radius of the Earth is 6371 km from center to pole.

(b) Compare this with the accepted value of 5 . 979 × 10 24 kg size 12{5 "." "979" times "10" rSup { size 8{"24"} } `"kg"} {} .

a) 5.979 × 10 24 kg size 12{ {underline {5 cdot "979" times "10" rSup { size 8{"24"} } " kg"}} } {}

b) This is identical to the best value to three significant figures.

(a) Calculate the magnitude of the acceleration due to gravity on the surface of Earth due to the Moon.

(b) Calculate the magnitude of the acceleration due to gravity at Earth due to the Sun.

(c) Take the ratio of the Moon’s acceleration to the Sun’s and comment on why the tides are predominantly due to the Moon in spite of this number.

(a) What is the acceleration due to gravity on the surface of the Moon?

(b) On the surface of Mars? The mass of Mars is 6.418 × 10 23 kg size 12{6 "." "418" times "10" rSup { size 8{"23"} } `"kg"} {} and its radius is 3 . 38 × 10 6 m size 12{3 "." "38" times "10" rSup { size 8{6} } `m} {} .

a) 1.62 m / s 2 size 12{1 cdot "62"" m"/s rSup { size 8{2} } } {}

b) 3.75 m / s 2 size 12{1 cdot "62"" m"/s rSup { size 8{2} } } {}

(a) Calculate the acceleration due to gravity on the surface of the Sun.

(b) By what factor would your weight increase if you could stand on the Sun? (Never mind that you cannot.)

The Moon and Earth rotate about their common center of mass, which is located about 4700 km from the center of Earth. (This is 1690 km below the surface.)

(a) Calculate the magnitude of the acceleration due to the Moon’s gravity at that point.

(b) Calculate the magnitude of the centripetal acceleration of the center of Earth as it rotates about that point once each lunar month (about 27.3 d) and compare it with the acceleration found in part (a). Comment on whether or not they are equal and why they should or should not be.

a) 3.42 × 10 –5 m / s 2 size 12{3 cdot "42" times "10" rSup { size 8{"-5"} } m/s rSup { size 8{2} } } {}

b) 3.34 × 10 –5 m / s 2 size 12{3 cdot "34" times "10" rSup { size 8{"-5"} } m/s rSup { size 8{2} } } {}

The values are nearly identical. One would expect the gravitational force to be the same as the centripetal force at the core of the system.

Solve part (b) of [link] using a c = v 2 / r size 12{a rSub { size 8{c} } =v rSup { size 8{2} } /r} {} .

Astrology, that unlikely and vague pseudoscience, makes much of the position of the planets at the moment of one’s birth. The only known force a planet exerts on Earth is gravitational.

(a) Calculate the magnitude of the gravitational force exerted on a 4.20 kg baby by a 100 kg father 0.200 m away at birth (he is assisting, so he is close to the child).

(b) Calculate the magnitude of the force on the baby due to Jupiter if it is at its closest distance to Earth, some 6 . 29 × 10 11 m size 12{6 "." "29" times "10" rSup { size 8{"11"} } `m} {} away. How does the force of Jupiter on the baby compare to the force of the father on the baby? Other objects in the room and the hospital building also exert similar gravitational forces. (Of course, there could be an unknown force acting, but scientists first need to be convinced that there is even an effect, much less that an unknown force causes it.)

a) 7.01 × 10 –7 N size 12{7 cdot "01" times "10" rSup { size 8{"-7"} } N} {}

b) 1.35 × 10 –6 N size 12{1 cdot "35" times "10" rSup { size 8{"-6"} } N} {} , 0.521 size 12{0 cdot "521"} {}

The existence of the dwarf planet Pluto was proposed based on irregularities in Neptune’s orbit. Pluto was subsequently discovered near its predicted position. But it now appears that the discovery was fortuitous, because Pluto is small and the irregularities in Neptune’s orbit were not well known. To illustrate that Pluto has a minor effect on the orbit of Neptune compared with the closest planet to Neptune:

(a) Calculate the acceleration due to gravity at Neptune due to Pluto when they are 4 . 50 × 10 12 m size 12{4 "." "50" times "10" rSup { size 8{"12"} } `m} {} apart, as they are at present. The mass of Pluto is 1 . 4 × 10 22 kg size 12{1 "." 4 times "10" rSup { size 8{"22"} } `"kg"} {} .

(b) Calculate the acceleration due to gravity at Neptune due to Uranus, presently about 2 . 50 × 10 12 m size 12{2 "." "50" times "10" rSup { size 8{"12"} } `m} {} apart, and compare it with that due to Pluto. The mass of Uranus is 8 . 62 × 10 25 kg size 12{8 "." "62" times " 10" rSup { size 8{"25"} } " kg"} {} .

(a) The Sun orbits the Milky Way galaxy once each 2 . 60 x 10 8 y size 12{2 "." "60 x 10" rSup { size 8{8} } " y"} {} , with a roughly circular orbit averaging 3 . 00 x 10 4 size 12{3 "." "00 x 10" rSup { size 8{4} } } {} light years in radius. (A light year is the distance traveled by light in 1 y.) Calculate the centripetal acceleration of the Sun in its galactic orbit. Does your result support the contention that a nearly inertial frame of reference can be located at the Sun?

(b) Calculate the average speed of the Sun in its galactic orbit. Does the answer surprise you?

a) 1.66 × 10 –10 m / s 2 size 12{1 cdot "66" times "10" rSup { size 8{"-10"} } m/s rSup { size 8{2} } } {}

b) 2.17 × 10 5 m/s size 12{2 cdot "17" times "10" rSup { size 8{"5"} } m/s} {}

Unreasonable Result

A mountain 10.0 km from a person exerts a gravitational force on him equal to 2.00% of his weight.

(a) Calculate the mass of the mountain.

(b) Compare the mountain’s mass with that of Earth.

(c) What is unreasonable about these results?

(d) Which premises are unreasonable or inconsistent? (Note that accurate gravitational measurements can easily detect the effect of nearby mountains and variations in local geology.)

a) 2.937 × 10 17 kg size 12{2 cdot "94" times "10" rSup { size 8{"17"} } kg} {}

b) 4.91 × 10 –8 size 12{4 cdot "92" times "10" rSup { size 8{"-8"} } } {}

of the Earth’s mass.

c) The mass of the mountain and its fraction of the Earth’s mass are too great.

d) The gravitational force assumed to be exerted by the mountain is too great.

Questions & Answers

what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
how to synthesize TiO2 nanoparticles by chemical methods
what's the program
what chemical
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Physics 110 at une. OpenStax CNX. Aug 29, 2013 Download for free at http://legacy.cnx.org/content/col11566/1.1
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