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Making connections

Attempts are still being made to understand the gravitational force. As we shall see in Particle Physics , modern physics is exploring the connections of gravity to other forces, space, and time. General relativity alters our view of gravitation, leading us to think of gravitation as bending space and time.

In the following example, 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 acceleration due to gravity should equal the centripetal acceleration of the Moon in its orbit. Newton found that the two accelerations agreed “pretty nearly.”

Earth’s gravitational force is the centripetal force making the moon move in a curved path

(a) Find the acceleration due to Earth’s gravity at the distance of the Moon.

(b) Calculate the centripetal acceleration needed to keep the Moon in its orbit (assuming a circular orbit about a fixed Earth), and compare it with the value of the acceleration due to Earth’s gravity that you have just found.

Strategy for (a)

This calculation is the same as the one finding the acceleration due to gravity at Earth’s surface, except that r size 12{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 × 10 8 m size 12{3 "." "84" times "10" rSup { size 8{8} } `m} {} .

Solution for (a)

Substituting known values into the expression for g size 12{M} {} found above, remembering that M size 12{M} {} is the mass of Earth not the Moon, yields

g = G M r 2 = 6 . 67 × 10 11 N m 2 kg 2 × 5 . 98 × 10 24 kg ( 3 . 84 × 10 8 m ) 2 = 2 . 70 × 10 3 m/s. 2

Strategy for (b)

Centripetal acceleration can be calculated using either form of

a c = v 2 r a c = 2 } . size 12{ left none matrix { a rSub { size 8{c} } = { {v rSup { size 8{2} } } over {r} } {} ##a rSub { size 8{c} } =rω rSup { size 8{2} } } right rbrace "." } {}

We choose to use the second form:

a c = 2 , size 12{a rSub { size 8{c} } =rω rSup { size 8{2} } } {}

where ω size 12{ω} {} is the angular velocity of the Moon about Earth.

Solution for (b)

Given that the period (the time it takes to make one complete rotation) of the Moon’s orbit is 27.3 days, (d) and using

1 d × 24 hr d × 60 min hr × 60 s min = 86,400 s size 12{ω= { {Δθ} over {Δt} } = { {2π" rad"} over { \( "27" "." "3 d" \) \( "86,400 s/d" \) } } =2 "." "66" times "10" rSup { size 8{ - 6} } { {"rad"} over {s} } } {}

we see that

ω = Δ θ Δ t = rad ( 27 . 3 d ) ( 86,400 s/d ) = 2 . 66 × 10 6 rad s . size 12{ω= { {Δθ} over {Δt} } = { {2π" rad"} over { \( "27" "." "3 d" \) \( "86,400 s/d" \) } } =2 "." "66" times "10" rSup { size 8{ - 6} } { {"rad"} over {s} } } {}

The centripetal acceleration is

a c = 2 = ( 3 . 84 × 10 8 m ) ( 2 . 66 × 10 6 rad/s ) 2 = 2.72 × 10 3 m/s. 2 alignl { stack { size 12{a rSub { size 8{c} } =rω rSup { size 8{2} } = \( 3 "." "84" times "10" rSup { size 8{8} } " m" \) \( 2 "." "66" times "10" rSup { size 8{ - 6} } " rad/s" \) rSup { size 8{2} } } {} #" "=2 "." "72" times "10" rSup { size 8{ - 3} } " m/s" rSup { size 8{2} } {} } } {}

The direction of the acceleration is toward the center of the Earth.

Discussion

The centripetal acceleration of the Moon found in (b) differs by less than 1% from the acceleration due to Earth’s gravity found in (a). This agreement is approximate because the Moon’s orbit is slightly elliptical, and Earth is not stationary (rather the Earth-Moon system rotates about its center of mass, which is located some 1700 km below Earth’s surface). The clear implication is that Earth’s gravitational force causes the Moon to orbit Earth.

Why does Earth not remain stationary as the Moon orbits it? This is because, as expected from Newton’s third law, if Earth exerts a force on the Moon, then the Moon should exert an equal and opposite force on Earth (see [link] ). We do not sense the Moon’s effect on Earth’s motion, because the Moon’s gravity moves our bodies right along with Earth but there are other signs on Earth that clearly show the effect of the Moon’s gravitational force as discussed in Satellites and Kepler's Laws: An Argument for Simplicity .

Questions & Answers

what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
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.
Bharti
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
Daniel
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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