# 8.4 Newton’s universal law of gravitation  (Page 7/11)

 Page 7 / 11

## 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
$F=G\frac{\text{mM}}{{r}^{2}}\text{,}$

where F is the magnitude of the gravitational force. $G$ is the gravitational constant, given by $G=6\text{.}\text{674}×{\text{10}}^{\text{–11}}\phantom{\rule{0.25em}{0ex}}\text{N}\cdot {\text{m}}^{2}{\text{/kg}}^{2}$ .

• 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 $9.80 m{\text{/s}}^{2}$ . 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 $9.830 m{\text{/s}}^{2}$ and the radius of the Earth is 6371 km from center to pole.

(b) Compare this with the accepted value of $5\text{.}\text{979}×{\text{10}}^{\text{24}}\phantom{\rule{0.25em}{0ex}}\text{kg}$ .

a) $5.979×{\text{10}}^{\text{24}}\phantom{\rule{0.25em}{0ex}}\text{kg}$

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 $6.418×{\text{10}}^{\text{23}}\phantom{\rule{0.25em}{0ex}}\text{kg}$ and its radius is $3\text{.}\text{38}×{\text{10}}^{6}\phantom{\rule{0.25em}{0ex}}\text{m}$ .

a) $1.62\phantom{\rule{0.5em}{0ex}}\text{m}/{\text{s}}^{2}$

b) $3.75\phantom{\rule{0.5em}{0ex}}\text{m}/{\text{s}}^{2}$

(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) $3.42×{\text{10}}^{\text{–5}}\phantom{\rule{0.25em}{0ex}}\text{m}/{\text{s}}^{2}$

b) $3.34×{\text{10}}^{\text{–5}}\phantom{\rule{0.25em}{0ex}}\text{m}/{\text{s}}^{2}$

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 ${a}_{c}={v}^{2}/r$ .

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 $6\text{.}\text{29}×{\text{10}}^{\text{11}}\phantom{\rule{0.25em}{0ex}}\text{m}$ 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) $7.01×{\text{10}}^{\text{–7}}\phantom{\rule{0.25em}{0ex}}\text{N}$

b) $1.35×{\text{10}}^{\text{–6}}\phantom{\rule{0.25em}{0ex}}\text{N}$ , $0.521$

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 $4\text{.}\text{50}×{\text{10}}^{\text{12}}\phantom{\rule{0.25em}{0ex}}\text{m}$ apart, as they are at present. The mass of Pluto is $1\text{.}4×{\text{10}}^{\text{22}}\phantom{\rule{0.25em}{0ex}}\text{kg}$ .

(b) Calculate the acceleration due to gravity at Neptune due to Uranus, presently about $2\text{.}\text{50}×{\text{10}}^{\text{12}}\phantom{\rule{0.25em}{0ex}}\text{m}$ apart, and compare it with that due to Pluto. The mass of Uranus is $8\text{.}\text{62}×{\text{10}}^{\text{25}}\phantom{\rule{0.25em}{0ex}}\text{kg}$ .

(a) The Sun orbits the Milky Way galaxy once each $2\text{.}{\text{60 x 10}}^{8}\phantom{\rule{0.25em}{0ex}}\text{y}$ , with a roughly circular orbit averaging $3\text{.}{\text{00 x 10}}^{4}$ 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) $1.66×{\text{10}}^{\text{–10}}\phantom{\rule{0.25em}{0ex}}\text{m}/{\text{s}}^{2}$

b) $2.17×{\text{10}}^{\text{5}}\phantom{\rule{0.25em}{0ex}}\text{m/s}$

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) $2.937×{\text{10}}^{\text{17}}\phantom{\rule{0.25em}{0ex}}\text{kg}$

b) $4.91×{\text{10}}^{\text{–8}}$

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.

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
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?
what is a peer
What is meant by 'nano scale'?
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
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
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?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
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