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Understanding Newton's Law of Universal Gravitation: Inverse Square Law & Applications - P, Study notes of Physics

Siena College Physics

Prof. Rose A. Finn

An in-depth exploration of newton's law of universal gravitation, an inverse square law that describes the force of attraction between every particle in the universe. Topics covered include the mathematical expression of the law, the concept of gravitational constant, the forces exerted between particles, and the comparison of g and g. Additionally, the document discusses the verification of the law through newton's experiments and kepler's laws.

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Newton’s Law of Universal

Gravitation (Ch 13)

•Every particle in the Universe attracts every other

particle with a force that is directly proportional to the

product of their masses and inversely proportional to

the distance between them

•G is the universal gravitational constant

G = 6.673 x 10-11 N⋅m2 / kg2

Law of Gravitation, cont

•This is an example of an inverse square law

•The magnitude of the force varies as the inverse

square of the separation of the particles

•The law can also be expressed in vector form

Notation

•F12 is the force exerted by particle 1 on

particle 2

•The negative sign in the vector form of the

equation indicates that particle 2 is attracted

toward particle 1

•F21 is the force exerted by particle 2 on

particle 1

More About Forces

•F12 = -F21

•The forces form a Newton’s

Third Law action-reaction pair

•Gravitation is a field force that

always exists between two

particles, regardless of the

medium between them

•The force decreases rapidly

as distance increases

•A consequence of the inverse

square law

Partial preview of the text

Download Understanding Newton's Law of Universal Gravitation: Inverse Square Law & Applications - P and more Study notes Physics in PDF only on Docsity!

Newton’s Law of Universal

Gravitation (Ch 13)

Every particle in the Universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the distance between them
G is the universal gravitational constant G = 6.673 x 10-11^ N⋅m^2 / kg^2

Law of Gravitation, cont

• This is an example of an inverse square law

The magnitude of the force varies as the inverse square of the separation of the particles

• The law can also be expressed in vector form

Notation

• F 12 is the force exerted by particle 1 on

particle 2

• The negative sign in the vector form of the

equation indicates that particle 2 is attracted

toward particle 1

• F 21 is the force exerted by particle 2 on

particle 1

More About Forces

• F 12 = - F 21

The forces form a Newton’s Third Law action-reaction pair
Gravitation is a field force that always exists between two particles, regardless of the medium between them
The force decreases rapidly as distance increases
A consequence of the inverse square law

G vs. g

• Always distinguish between G and g

• G is the universal gravitational constant

It is the same everywhere

• g is the acceleration due to gravity

g = 9.80 m/s^2 at the surface of the Earth
g will vary by location

Gravitational Force Due to a

Distribution of Mass

• The gravitational force exerted by a finite-

size, spherically symmetric mass distribution

on a particle outside the distribution is the

same as if the entire mass of the distribution

were concentrated at the center

• For the Earth,

Newton’s Verification

Developed theory at age 23!
He compared the acceleration of the Moon in its orbit with the acceleration of an object falling near the Earth’s surface
He calculated the centripetal acceleration of the Moon from its distance and period
The high degree of agreement between the two techniques provided evidence of the inverse square nature of the law (^) Newton’s Canon

Moon’s Acceleration

• Newton looked at proportionality of

accelerations between the Moon and objects

on the Earth

g Above the Earth’s Surface

• If an object is some distance h above the

Earth’s surface, r becomes RE + h

• This shows that g decreases with increasing

altitude

• As r → ∞, the weight of the object approaches

zero

Variation of g with Height

Ch 13: Question 4

• The gravitational force that the Sun exerts on

the Moon is about twice as great as the

gravitational force that the Earth exerts on the

Moon. Why doesn’t the Sun pull the Moon

away from the Earth during a total eclipse of

the Sun?

Kepler’s Laws, Introduction

• Johannes Kepler was a German astronomer

• He was Tycho Brahe’s assistant

Brahe was the last of the “naked eye” astronomers

• Kepler analyzed Brahe’s data and formulated

three laws of planetary motion

Kepler’s Laws

• Kepler’s First Law

All planets move in elliptical orbits with the Sun at one focus

• Kepler’s Second Law

The radius vector drawn from the Sun to a planet sweeps out equal areas in equal time intervals

• Kepler’s Third Law

The square of the orbital period of any planet is proportional to the cube of the semimajor axis of the elliptical orbit

Notes About Ellipses

F 1 and F 2 are each a focus of the ellipse - They are located a distance c from the center
The longest distance through the center is the major axis - a is the semimajor axis
The shortest distance through the center is the minor axis - b is the semiminor axis
eccentricity , e = c / a
- For a circle, e = 0
- For ellipses, 0 < e < 1

Planetary Orbits

• The Sun is at one focus

Nothing is located at the other focus

• Aphelion is the point farthest away from the Sun

The distance for aphelion is a + c
- For an orbit around the Earth, this point is called the apogee

• Perihelion is the point nearest the Sun

The distance for perihelion is a – c
- For an orbit around the Earth, this point is called the perigee

Kepler’s First Law

A circular orbit is a special case of the general elliptical orbits
Is a direct result of the inverse square nature of the gravitational force
Elliptical (and circular) orbits are allowed for bound objects - A bound object repeatedly orbits the center - An unbound object would pass by and not return - These objects could have paths that are parabolas ( e = 1) and hyperbolas ( e > 1)

Understanding Newton's Law of Universal Gravitation: Inverse Square Law & Applications - P, Study notes of Physics

Related documents

Partial preview of the text

Download Understanding Newton's Law of Universal Gravitation: Inverse Square Law & Applications - P and more Study notes Physics in PDF only on Docsity!

Newton’s Law of Universal

Gravitation (Ch 13)

Law of Gravitation, cont

• This is an example of an inverse square law

• The law can also be expressed in vector form

Notation

• F 12 is the force exerted by particle 1 on

particle 2

• The negative sign in the vector form of the

equation indicates that particle 2 is attracted

toward particle 1

• F 21 is the force exerted by particle 2 on

particle 1

More About Forces

• F 12 = - F 21

G vs. g

• Always distinguish between G and g

• G is the universal gravitational constant

• g is the acceleration due to gravity

Gravitational Force Due to a

Distribution of Mass

• The gravitational force exerted by a finite-

size, spherically symmetric mass distribution

on a particle outside the distribution is the

same as if the entire mass of the distribution

were concentrated at the center

• For the Earth,

Newton’s Verification

Moon’s Acceleration

• Newton looked at proportionality of

accelerations between the Moon and objects

on the Earth

g Above the Earth’s Surface

• If an object is some distance h above the

Earth’s surface, r becomes RE + h

• This shows that g decreases with increasing

altitude

• As r → ∞, the weight of the object approaches

zero

Variation of g with Height

Ch 13: Question 4

• The gravitational force that the Sun exerts on

the Moon is about twice as great as the

gravitational force that the Earth exerts on the

Moon. Why doesn’t the Sun pull the Moon

away from the Earth during a total eclipse of

the Sun?

Kepler’s Laws, Introduction

• Johannes Kepler was a German astronomer

• He was Tycho Brahe’s assistant

• Kepler analyzed Brahe’s data and formulated

three laws of planetary motion

Kepler’s Laws

• Kepler’s First Law

• Kepler’s Second Law

• Kepler’s Third Law

Notes About Ellipses

Planetary Orbits

• The Sun is at one focus

• Aphelion is the point farthest away from the Sun

• Perihelion is the point nearest the Sun

Kepler’s First Law

Example, Mass of the Sun

• Using the distance between the Earth and the

Sun, and the period of the Earth’s orbit,

Kepler’s Third Law can be used to find the

mass of the Sun

• Similarly, the mass of any object being

orbited can be found if you know information

about objects orbiting it

Example, Geosynchronous

Satellite