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The given figure shows two circular objects, one with a larger mass M on the right side, and another with a smaller mass m on the left side. A point in the center of each object is shown, with both depicting the center of mass of the objects at these points. A line is drawn joining the center of the objects and is labeled as r. Two red arrows, one each from both the center of the objects, are drawn toward each other and are labeled as F, the magnitude of the gravitational force on both the objects.
Gravitational attraction is along a line joining the centers of mass of these two bodies. The magnitude of the force is the same on each, consistent with Newton’s third law.

Misconception alert

The magnitude of the force on each object (one has larger mass than the other) is the same, consistent with Newton’s third law.

The bodies we are dealing with tend to be large. To simplify the situation we assume that the body acts as if its entire mass is concentrated at one specific point called the center of mass    (CM), which will be further explored in Linear Momentum and Collisions . For two bodies having masses m size 12{m} {} and M size 12{M} {} with a distance r size 12{r} {} between their centers of mass, the equation for Newton’s universal law of gravitation is

F = G mM r 2 , size 12{F=G { { ital "mM"} over {r rSup { size 8{2} } } } } {}

where F size 12{F} {} is the magnitude of the gravitational force and G size 12{G} {} is a proportionality factor called the gravitational constant . G size 12{G} {} is a universal gravitational constant—that is, it is thought to be the same everywhere in the universe. It has been measured experimentally to be

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} } } over {"kg" rSup { size 8{2} } } } } {}

in SI units. Note that the units of G size 12{G} {} are such that a force in newtons is obtained from F = G mM r 2 size 12{F=G { { ital "mM"} over {r rSup { size 8{2} } } } } {} , when considering masses in kilograms and distance in meters. For example, two 1.000 kg masses separated by 1.000 m will experience a gravitational attraction of 6 . 674 × 10 11 N size 12{6 "." "673" times "10" rSup { size 8{ - "11"} } N} {} . This is an extraordinarily small force. The small magnitude of the gravitational force is consistent with everyday experience. We are unaware that even large objects like mountains exert gravitational forces on us. In fact, our body weight is the force of attraction of the entire Earth on us with a mass of 6 × 10 24 kg size 12{6 times "10" rSup { size 8{"24"} } `"kg"} {} .

Recall that the acceleration due to gravity g size 12{g} {} is about 9.80 m /s 2 size 12{9 "." 8`"m/s" rSup { size 8{2} } } {} on Earth. We can now determine why this is so. The weight of an object mg is the gravitational force between it and Earth. Substituting mg for F size 12{F} {} in Newton’s universal law of gravitation gives

mg = G mM r 2 , size 12{ ital "mg"=G { { ital "mM"} over {r rSup { size 8{2} } } } } {}

where m size 12{m} {} is the mass of the object, M size 12{M} {} is the mass of Earth, and r size 12{r} {} is the distance to the center of Earth (the distance between the centers of mass of the object and Earth). See [link] . The mass m size 12{m} {} of the object cancels, leaving an equation for g size 12{g} {} :

g = G M r 2 . size 12{g=G { {M} over {r rSup { size 8{2} } } } } {}

Substituting known values for Earth’s mass and radius (to three significant figures),

g = 6 . 67 × 10 11 N m 2 kg 2 × 5 . 98 × 10 24 kg ( 6 . 38 × 10 6 m ) 2 , size 12{g= left (6 "." "67" times "10" rSup { size 8{ - "11"} } { {N cdot m rSup { size 8{2} } } over {"kg" rSup { size 8{2} } } } right ) times { {5 "." "98" times "10" rSup { size 8{"24"} } " kg"} over { \( 6 "." "38" times "10" rSup { size 8{6} } " m" \) rSup { size 8{2} } } } } {}

and we obtain a value for the acceleration of a falling body:

g = 9 . 80 m/s 2 . size 12{g=9 "." "80"" m/s" rSup { size 8{2} } } {}
The given figure shows two circular images side by side. The bigger circular image on the left shows the Earth, with a map of Africa over it in the center, and the first quadrant in the circle being a line diagram showing the layers beneath Earth’s surface. The second circular image shows a house over the Earth’s surface and a vertical line arrow from its center to the downward point in the circle as its radius distance from the Earth’s surface. A similar line showing the Earth’s radius is also drawn in the first quadrant of the first image in a slanting way from the center point to the circle path.
The distance between the centers of mass of Earth and an object on its surface is very nearly the same 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’s law of gravitation takes Galileo’s 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 force of attraction between masses.

Take-home experiment

Take a marble, a ball, 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 behave like the other objects? Explain your observations.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
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
there is no specific books for beginners but there is book called principle of nanotechnology
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?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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
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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