6.6 Satellites and Kepler's Laws: An Argument for Simplicity

Cavendish used an apparatus like this to measure the attraction between the two suspended spheres and the two on the stand by observing the amount of twisting in the fiber.

The force on distance can be checked.
Modern experiments of gravity continue.

There are many examples of gravitational orbits.

There are hundreds of artificial satellites and thousands of pieces of debris.
Humans have been interested in the Moon's position about Earth for a long time.
There are planets, asteroids, and comets around the Sun.
Almost unimaginable numbers of stars, galaxies, and other objects are interacting through gravity if we look further.

It is possible to describe all of these motions to different degrees of precision.

Large computers are needed to make precise descriptions of complex systems.
Without the use of computers, we can describe an important class of orbits.

A mass is larger than a small one.

The motion can be viewed as if it were stationary, without significant error.
The mass is the satellite if it is bound.

The system is not part of any other mass.

We can ignore any small effects due to outside mass.

To a good approximation, the conditions are satisfied by Earth's satellites, as well as by the satellites of other planets.

Historically, planets were studied first, and there is a classical set of three laws, called Kepler's laws of planetary motion, that describe the orbits of all bodies satisfying the two previous conditions.
The laws are named after Johannes Kepler, the German astronomer who created them after studying a large amount of planetary motion recorded by Tycho Brahe.
Good science involves careful collection and recording of methods.
New interpretations and meanings can be constructed from data.

Each planet has an ellipse with the Sun at one focus.

You can draw an ellipse if you put a pin at each focus, place a string around a pencil and the pins, and trace a line on paper.

Any point on the circle is the same distance from the center as the two foci.
This is a fact for planets around the Sun.

Each planet moves so that an imaginary line drawn from the Sun to the planet sweeps out equal areas in equal times.

The ratio of the squares of the periods of any two planets about the Sun is the same as the ratio of the cubes of their average distances from the Sun.

This equation can only be used to compare two small mass to large mass.

This is a descriptive equation and no information is given as to the cause of the equality.

There are equal areas in the shaded regions.

It takes the same amount of time to go from A to B, from C to D, and from E to F. The second law was created for planets around the Sun.

For historical reasons, the laws for planets around the Sun are valid for all bodies, even if they don't meet the two previously stated conditions.

Given that the Moon is an average distance from the center of Earth and that it is an altitude of 1500 km above the surface, calculate the period of an artificial satellite.

The period, or time for one, is related to the circle by the third law.

The subscript 1 is for the moon and the subscript 2 is for the satellite.
The period of the Moon is given by the information given.

The height of the artificial satellite above Earth's surface is given, and so we must add the radius of Earth to get.

All quantities can be found.

This is a good period for a satellite.

It is interesting to know that any satellite at this altitude will stay in the same position for the same amount of time.
The condition that the satellite's mass is small is related to this fact.

When broadly applicable laws are discovered, people immediately search for deeper meaning.

The law of universal gravitation was proposed byNewton.
The cause of what was happening was discovered byNewton.

We will derive the third law from the laws of motion and gravitation.

The purpose is to show that the force of gravity is the cause of the laws.

The two conditions stated at the beginning of this section should be satisfied by a circular circle of a small mass around a large mass.

The centripetal force is supplied by gravity.

The fact that cancels out is a part of the fact that everyone falls with the same speed at a given location.

All of the mass are at the same speed.
To get at the third law, we need the period into the equation.
Period is the time for one complete circle.

This is the third law.

The mass of the parent body can only be canceled by comparing the satellites of the same parent body.

If we solve for the ratio, what will we get?

The mass of the parent can be calculated if they are known for a satellite.

This principle has been used to find the mass of heavenly bodies.
The ratio should be constant for all satellites of the same parent body.

It is clear from Table 6.2 that the ratio of is constant, at least to the third digit, for all listed satellites of the Sun and Jupiter.

There are two causes of small variations in that ratio.
The location of new planets and moons can be predicted by those perturbations.
This is a verification of the universal law of gravitation.

Einstein's general theory of relativity modified the universal law of gravitation, which is still an excellent approximation for most situations.

Near black holes are where Einstein's modification is most noticeable.
Small but long-known deviations of the planet Mercury from classical predictions are explained by general relativity.

The history of ideas was influenced by the development of the universal law of gravitation.

We note some important points even though it is beyond the scope of this text.
The definition of a planet in the solar system was set in 2006 by the IAU.

After scientists revised their definition of a "true" planet, Pluto was demoted to a "dwarf planet".

A good example of a physical principle that is applicable is the universal law of gravitation.

All situations in which gravity acts are described by that single equation.
It causes a lot of effects, such as the moons in the solar system.
It shows the underlying unity and simplicity of physics.

The Greek philosopher who lived in the second century AD had a view called the Ptolemaic view.

The model is characterized by a list of facts for the motions of planets.
There was a lack of simplicity and a different rule for each heavenly body.

The modern or Copernican model is represented in Figure 6.31(b).

A small set of rules and a single underlying force explain all the motions in the solar system.
The laws of physics are easy to understand.
The simplicity of nature's laws has become more evident as our knowledge of nature has grown.

The model that can be made more accurate by adding more circles contains no clues as to the causes of the motions.

It is explained by a small number of laws of physics.

A uniform circular motion is motion in a circle at constant speed.

Centripetal acceleration is the ratio of the arcs length to the radius of curvature experienced while in uniform circular motion.
It always points to the center of rotation.
The magnitude is the distance traveled along a circular path and the radius of the path's curve.
The quantity is measured in radians.

Centripetal force is any force causing uniform.

The center-seeking force is the rate of change of an angle and always points toward the center of rotation.

It has a magnitude where a rotation takes place in a time.
The radians per second are the units which can be expressed.

Each planet has an ellipse with the Sun at one focus.

The Coriolis force is needed from the Sun to the planet to explain motion in such frames.

Every particle in planets about the Sun is equal to the ratio of the cubes the universe attracts every other particle with a force of their average distances from the Sun.

The force is proportional to the product of their mass and the square of the distance between them.
This is the average of the circles.

The period and radius of a satellite is related to the magnitude of the gravitational force.

There is an analogy between the two physical quantities.

You can use a body diagram in your answer.

A child is riding on a merry-go-round at a distance between its center and edge.

She put her lunch box on wax paper so that it wouldn't touch the merry-go-round.
There is a trail on the merry-go-round.

There are two paths around the curve.

The inside path is called cutting the corner and is used by race car drivers to take the curve at the highest speed.

The cars are attached to the rails in a way that they cannot fall off.

The centripetal force will be supplied if the car goes over the top at the right speed.

A child on a merry-go-round releases her lunch box at point P.

Amusement rides with a vertical loop are an example of curved motion.

In one amusement park ride, riders enter a large vertical table.

The case for gravity was once thought to be illogical and therefore untrue.

Two friends are talking.

Anna says that a satellite is in freefall because it keeps falling.

A frame of reference at the center of the Sun is very close to an insturment.

The case for gravity was once thought to be illogical and therefore untrue.

Two friends are talking.

When a toilet is flushed or a sink is drained, the water satellite is in freefall because the satellite keeps rotating about the drain on falling toward Earth.

Tom says the satellite is not going down.
Explain what if there is no initial rotation and a flow in freefall because the gravity is not directly towards the drain.

There is a satellite in the northern hemisphere.

There is an odometer on one of the walls.

The trailer wheel has the full value.
The hub is weighted so that it doesn't fall away and it has gears to count the number of wheel riders so they don't slide down.
Then it calculates the distance traveled.
If the wheel has a 1.15 m diameter and goes through velocity that assures the riders will not slide down the 200,000 rotation, how many kilometers should the wall?
There is a free body diagram of a single rider.

Microwave ovens are rotating at a rate of 6 revolutions per minute.

What is the riders' clothing?

They were wearing them out before the Centripetal Acceleration km.

A fairground ride spins its passengers inside a flying tires, neglecting any backing up and any change saucer-shaped container.

A baseball pitcher throws a pitch with his arm in the air and his forearm in the air.

Taking the age of Earth to be about years and stick by rotating the stick and forearm.

What is the angle of the tires?

The helicopter blades are strong.

A large centripetal ship hangs from a large pivot as riders in an amusement park ride shaped like a viking are spun at rapid rates.

The ship swings under the rotates at 300 revolutions per minute.

Skaters can spin at about 5 revolutions per minute.

The lowest point of his path is the percentage of the acceleration at Earth's surface.

The swing is suspended above the child's center of mass due to gravity.

It is rotating at 50,000rpm from its center.

The force approximates it as being circular.

An artificial round is created by a rotating space station.

If she is 1.25 m from its acceleration, she must exert a force to stay on.

A commercial jet has a takeoff speed of 60.0 m/s.

A 105 km/h speed limit highway has a diameter of 0.850 m.

To be used to expose aspiring astronauts to stable accelerations, the ground must be on a line similar to those experienced in rocket launches and going through the center of gravity.

The ability to lean at the correct angle becomes instinctive as a bicyclist negotiates a turn on level ground.

The center of gravity is where the force of the ground on the wheel needs to be.
The centripetal force is the net external force on the system.
The weight of the system is canceled by the vertical component of the force on the wheel.
The relationship between the angle, speed, and radius of the turn are similar to that of the ideal banking of the roads.

The rider's total force is allowed to be along the cage's axis at all times.

The Moon has a smaller gravity on the surface of Earth.

The tides are mostly due to the roller coaster at the top of the loop if the number of miles from the Moon is not taken into account.

Mars has a mass and a radius.

The common center of mass of the Moon and Earth is 4700 km away.

Comment on whether or not they are equal.

At the top, astrology makes decreases to a minimum.

The position of the planets at the moment of centripetal acceleration builds from zero to a maximum at one's birth.
The only force a planet has is on top.
Earth is not spherical.

The centripetal acceleration can be away from birth with a small exerted on a 4.20 kilogram baby by a 100 kilogram father.

How does Jupiter keep them in place?

The mass of Jupiter can be found by looking at the data for Neptune's orbit.

Compare your result with the discovery of the moon near its predicted position.

The ratio of Jupiter's mass to that of Earth is found by looking at the data in Table 6.2.

Neptune's position was not well known.

The mass of the dwarf planet is.

The existence of "dark matter" in the universe and the existence of Uranus are implied by these calculations.

The mass of the black holes at the Uranus is.