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The Law of Universal Gravitation

- Physics
- Montwood High School
- R. Casao

Newtons Universal Law of Gravity

- Legend has it that Newton was struck on the head

by a falling apple while napping under a tree.

This prompted Newton to imagine that all bodies

in the universe are attracted to each other in

the same way that the apple was attracted to the

Earth. - Newton analyzed astronomical data on the motion

of the Moon around the Earth and stated that the

law of force governing the motion of the planets

has the same mathematical form as the force law

that attracts the falling apple to the Earth.

Newtons Universal Law of Gravity

- Newtons law of gravitation 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 square of the distance

between them. - If the particles have masses m1 and m2 and are

separated by a distance r, the magnitude of the

gravitational force is

Newtons Universal Law of Gravity

- G is the universal gravitational constant, which

has been measured experimentally as 6.672 x

10-11 . - The distance r between m1 and m2 is measured from

the center of m1 to the center of m2.

Newtons Universal Law of Gravity

- By Newtons third law, the magnitude of the force

exerted by m1 on m2 is equal to the force exerted

by m2 on m1, but opposite in direction. These

gravitational forces form an action-reaction pair.

Properties of the Gravitational Force

- The gravitational force acts as an

action-at-a-distance force, which also exists

between two particles, regardless of the medium

that separates them. - The force varies as the inverse square of the

distance between the particles and therefore

decreases rapidly with increasing distance

between the particles. - The gravitational force is proportional to the

mass of each particle.

Properties of the Gravitational Force

- The force on a particle of mass m at the Earths

surface has the magnitude - ME is the Earths mass (5.98 x 1024 kg) RE is

the radius of the Earth (6.37 x 106 m). - The net force is directed toward the center of

the Earth both masses accelerate, but the

Earths acceleration is not noticeable due to its

extremely large mass. The smaller mass

accelerates towards the Earth.

Weight and Gravitational Force

- Weight was previously defined as FW mg, where

g is the magnitude of the acceleration due to

gravity. With the new perspective related to the

attractive forces existing between any two

objects in the universe, - The mass m cancels out, giving us (also called

surface gravity)

Bodies Above the Surface of a Mass

- Consider a body of mass m at a distance h above

the Earths surface, or a distance r from the

Earths center, where r Re h. The magnitude

of the gravitational force acting on the mass is

given by

Gravity/Radius Ratio

- If the body is in free fall, then the

acceleration of gravity at the altitude h is

given by - Thus, it follows that g decreases with increasing

altitude.

Gravity/Radius Ratio

- The value of g at any given location can be

determined using the following proportional

relationship - This proportional relationship can also be

applied to the weight of an object

Keplers Laws

- Kepler formulated three kinematic laws to

describe the motion of planets about the Sun - Keplers First Law All planets move in

elliptical orbits with the sun at one of the

focal points. - Keplers Second Law The radius vector drawn

from the sun to any planet sweeps out equal areas

in equal time intervals. - Keplers Third Law The square of the orbital

period of any planet is proportional to the cube

of the semi-major axis of the elliptical orbit.

Keplers Laws

- First law

- Second law

Keplers Laws

- Third law equation
- where k is a constant 3.35 x 1018 m3/s2
- r is the radius of rotation
- T is the period of rotation (the time necessary

to complete one revolution) - Keplers laws apply to any body that orbits the

Sun, manmade spaceship as well as planets,

comets, and other natural objects. The mass of

the orbiting body does not enter into the

calculation.

Keplers Laws

- The ratio of the squares of the periods (T) of

any two planets revolving about the Sun is equal

to the ratio of the cubes of their average

distances r from the Sun

Period of a Satellite

- The period of a satellite or planet orbiting

about a central body is given by - Mbody is the mass of the central body being

orbited.

The Gravitational Field

- The gravitational field concept revolves around

the general idea that an object modifies the

space surrounding it by establishing a

gravitational field which extends outward in all

directions, falling to zero at infinity. - Any other mass located within this field

experiences a force because of its location. So,

it is the strength of the gravitational field at

that location that produces the force - not the

distant object. - The situation is symmetrical - each object

experiences a gravitational force because of the

field set up by any other object.

The Gravitational Field

- The gravitational field is a vector quantity

equal to the gravitational force acting on a

particle divided by the mass of the particle - The gravitational field equation can be used to

determine the value of g at any location by

Escape Velocity

- Suppose you want to launch a rocket vertically

upward and give it just enough kinetic energy

(energy of motion) to escape the Earths

gravitational pull. - The minimum initial velocity of an object at the

Earths surface that would allow the object to

escape the Earth, never to return, is the escape

velocity. - Escape velocity from Earth

Satellite Orbits

- A satellite is held in a circular orbit because

the force of gravity supplies the necessary

centripetal force to keep the object moving in a

circular path about the central body. - In order for a satellite to orbit around a

central body, such as the Earth, there must be a

net force on the object directed toward the

center of the circular orbit, a centripetal

force. - For a satellite in orbit around Earth, the

centripetal force is equal to the gravitational

force exerted by the Earth on the satellite.

Satellite Orbits

- A satellite does not fall because it is moving,

being given a tangential velocity by the rocket

that launched it. It does not travel off in a

straight line because Earths gravity pulls it

toward the Earth. - The tangential speed of an object in a circular

orbit is given by - If the period of the orbit is known, the velocity

may be determined using

Satellite Orbits

- The period of a satellite can be determined by

Satellite Orbits

- The Goldilocks principle can be used to explain

the relationship between the speed of a satellite

and its orbit. The velocity of the satellite is

critical, and the velocity described by the

equation - Vmin describes the minimum velocity necessary for

the satellite to maintain its proper circular

orbit (JUST RIGHT). If the satellite velocity is

TOO HOT (greater than the vmin), it will not

maintain the proper circular orbit and fly into

space. If the satellite velocity is TOO COLD

(less than vmin), it will be pulled into the

Earths atmosphere by the Earths gravitational

force, where it will either burn up in the

atmosphere or slam into the Earths surface.

Gravitational Potential Energy Revisited

- Gravitational potential energy near the surface

of the Earth is given by the equation Ug mgh,

where h is the height of the object above or

below a reference level. This equation is only

valid for an object near the Earths surface. - For objects high above the Earths surface, the

equation for potential energy is

Gravitational Potential Energy

- The negative sign comes from the work done

against the gravity force in bringing a mass in

from infinity where the potential energy is

assigned the value zero, towards the Earth. This

work is stored in the mass as potential energy. - As r gets larger, the potential energy gets

smaller the gravitational force approaches zero

as r approaches infinity.

Gravitational Potential Energy

- Only changes in gravitational potential energy

are important. - For an object that moves from point B to point A,

the expression for the change in potential energy

is

Web Sites

- Kepler's Laws (with animations)
- Kepler's Three Laws
- Kepler's Three Laws of Planetary Motion
- Kepler's Laws of Planetary Motion