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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?

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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?

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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.

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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?

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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.

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(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.

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(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} } } {}

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(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.)

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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.

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

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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"} {}

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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"} {} .

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(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} {}

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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.

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Questions & Answers

How do you convert 0.0045kgcm³ to the si unit?
EDYKING Reply
how many state of matter do we really have like I mean... is there any newly discovered state of matter?
Falana Reply
I only know 5: •Solids •Liquids •Gases •Plasma •Bose-Einstein condensate
Thapelo
Alright Thank you
Falana
Which one is the Bose-Einstein
James
can you explain what plasma and the I her one you mentioned
Olatunde
u can say sun or stars are just the state of plasma
Mohit
but the are more than seven
Issa
what the meaning of continuum
Akhigbe Reply
What state of matter is fire
Thapelo Reply
fire is not in any state of matter...fire is rather a form of energy produced from an oxidising reaction.
Xenda
Isn`t fire the plasma state of matter?
Walter
How can you define time?
Thapelo Reply
Time can be defined as a continuous , dynamic , irreversible , unpredictable quantity .
Tanaya
what is the relativity of physics
Paul Reply
How do you convert 0.0045kgcm³ to the si unit?
flint
What is the formula for motion
Anthony Reply
V=u+at V²=u²-2as
flint
S=ut+½at
flint
they are eqns of linear motion
King
S=Vt
Thapelo
v=u+at s=ut+at^\2 v^=u^+2as where ^=2
King
hi
Mehadi
hello
King
Explain dopplers effect
Jennifer Reply
Not yet learnt
Bob
Explain motion with types
Bob
Acceleration is the change in velocity over time. Given this information, is acceleration a vector or a scalar quantity? Explain.
Alabi Reply
Scalar quantity Because acceleration has only magnitude
Bob
acleration is vectr quatity it is found in a spefied direction and it is product of displcemnt
bhat
its a scalar quantity
Paul
velocity is speed and direction. since velocity is a part of acceleration that makes acceleration a vector quantity. an example of this is centripetal acceleration. when you're moving in a circular patter at a constant speed, you are still accelerating because your direction is constantly changing.
Josh
acceleration is a vector quantity. As explained by Josh Thompson, even in circular motion, bodies undergoing circular motion only accelerate because on the constantly changing direction of their constant speed. also retardation and acceleration are differentiated by virtue of their direction in
fitzgerald
respect to prevailing force
fitzgerald
What is the difference between impulse and momentum?
Manyo
Momentum is the product of the mass of a body and the change in velocity of its motion. ie P=m(v-u)/t (SI unit is kgm/s). it is literally the impact of collision from a moving body. While Impulse is the product of momentum and time. I = Pt (SI unit is kgm) or it is literally the change in momentum
fitzgerald
Or I = m(v-u)
fitzgerald
Calculation of kinetic and potential energy
dion Reply
K.e=mv² P.e=mgh
Malia
K is actually 1/2 mv^2
Josh
what impulse is given to an a-particle of mass 6.7*10^-27 kg if it is ejected from a stationary nucleus at a speed of 3.2*10^-6ms²? what average force is needed if it is ejected in approximately 10^-8 s?
John
speed=velocity÷time velocity=speed×time=3.2×10^-6×10^-8=32×10^-14m/s impulse [I]=∆momentum[P]=mass×velocity=6.7×10^-27×32×10^-14=214.4×10^-41kg/ms force=impulse÷time=214.4×10^-41÷10^-8=214.4×10^-33N. dats how I solved it.if wrong pls correct me.
Melody
what is sound wave
Nworu Reply
sound wave is a mechanical longitudinal wave that transfers energy from one point to another
Ogor
its a longitudnal wave which is associted wth compresion nad rearfractions
bhat
what is power
PROMISE Reply
it's also a capability to do something or act in a particular way.
Kayode
Newton laws of motion
Mike
power also known as the rate of ability to do work
Slim
power means capabilty to do work p=w/t its unit is watt or j/s it also represents how much work is done fr evry second
bhat
what does fluorine do?
Cheyanne Reply
strengthen and whiten teeth.
Gia
a simple pendulum make 50 oscillation in 1minute, what is it period of oscillation?
Nansing Reply
length of pendulm?
bhat
what is the difference between temperature and heat transfer?
Bonga Reply
temperature is the measurement of hotness or coldness of a body... heat transfer is the movement of heat from one body to another
Doc
U get it right
Titilayo
correct
PROMISE
heat is an energy possesed by any substance due to random kinetic energy possesed by molecules while temperature is driving force which gives direction of flow heat
bhat
Practice Key Terms 4

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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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