Newtonian Gravitation and Orbits

SPH4U – Grade 12 Physics

Unit 1

Recall:

The force of Gravity, is one of the fundamental forces of nature. It is incredibly important in understanding the universe. It is a vector, and it has magnitude and direction.

is NOT the same as ,which is the acceleration due to gravity. (also called the gravitational field intensity).

GF

GF

g

Recall:

The force of gravity exists between any two objects in the Universe. It is not just about planets and satellites and other things in space!

Because the force of gravity is calculated between two objects, both objects exert a force. This means that if planet Earth is exerting a force of gravity down on you, you in turn are exerting a force of gravity up on the Earth.

Universal Law of Gravitation

For any two objects of mass m1 and m2, whose centers are separated by a distance of r, the force of gravity will be determined by this formula, which is the Universal Law of Gravitation.

In order for the force of gravity to be noticeable, at least one of the objects must have a large mass relative to the distance between the object centers.

Force of Gravity The force of gravity is always attractive. Every

mass attracts every other mass. Therefore the direction will always be towards the center of the other mass.

The magnitude of m1 on m2 is equal in strength to the magnitude of m2 on m1.

The Acceleration due to Gravity: .

As already stated, and are not always the same thing.

In grade 11 Physics, we said that was equal to 9.81m/s2.

In truth, this is just an approximation, and the value of varies at different points on the earth depending on how far one is from the earth’s surface. This is because of the inverse-square relationship of the Universal Gravitational Law.

g

GF

g

g

g

The Acceleration due to Gravity: .

In grade 11 Physics, we calculated the force of gravity using the equation

In truth, this is only valid for the force of gravity that is near the surface of the Earth. It represents just a particular instance of the Universal Law of Gravitation.

g

gmFG

Also see table 1 on page 289 of your textbook.

Acceleration due to gravity - g

Gravitational Fields

Gravitational Fields

Recall from grade 11 physics, that a field is a region in space where a force can be felt.

In this course, we will describe a field as a collection of vectors, one at each point in space, that determines the magnitude and direction of the force.

Notice that the arrows are drawn bigger near the center. That is no accident! Vectors should always be drawn to represent the strength of the force. The force is stronger in this region, so the arrows are bigger.

Gravitational Field Strength/ Intensity The gravitational field strength is the force of

attraction per the unit mass of an object that is placed in the field.

It is equal to the gravitational force on the object divided by the object’s mass.

Near the surface of the Earth, this is of course

g

Gravitational Field Strength/ Intensity To calculate the gravitational field strength as a

function of a spherical mass we combine the universal law of gravitation and Newton’s second law to get this equation:

G = 6.67x10-11 Nm2/kg2

r = radius of the planet

mplanet = mass of the planet

g = magnitude of the gravitational field strength on the surface

2r

Gmg planet

Example 1

Calculate the magnitude of the gravitational field strength of a white dwarf with a radius of 7.0x106m and a mass of 1.2x1030kg.

Satellites and Orbits

A satellite is an object or a body that revolves around another body due to gravitational attraction.

An artificial satellite is an object that has been intentionally placed by humans into orbit around Earth or another body. (Natural satellites, like the moon, are not placed there by humans).

Satellites and Orbits

Orbits are an example of centripetal motion, which we will study later in the course.

When an object is orbiting something like a

planet there is a force of attraction between the object and the planet. The orbital velocity however, keeps the object from falling into the planet. Both of these components, the orbital velocity and the gravitational force, are necessary for an object to orbit something.

Satellites and Orbits

Orbits are typically elliptical, not circular, but we approximate the orbits by assuming they are circular.

Satellites and Orbits

You can solve for a satellites orbiting speed using the following equation:

v = orbiting speedG = 6.67x10-11 N·m2/kg2

m = mass of the planet or large bodyr = orbital radius from center of satellite to the center of the planet

r

Gmv

Example 2

An asteroid has a mean radius of orbit around the Sun of 4.8x1011m. What is its orbital velocity?

Satellites and Orbits

A geosynchronous orbit is when a satellite orbits the Earth with a speed that matches the Earth’s rotational speed. A geosynchronous satellite will appear to travel through the same point in the sky every 24 hours.

A geostationary orbit is an orbit is a type of geosynchronous orbit where the satellite orbits directly over the equator. The satellite will appear to observers on the earth to remain fixed in the sky at all times.

General Relativity and Gravity

The Universal Law of Gravitation that we have been studying is based on Newtonian physics. It accurately describes a lot of phenomenon.

It is important to note however that General relativity, proposed by Einstein, has a different way of describing the force of gravity and explains gravity in terms of geometry and space. General relativity is able to describe all of the phenomena that the Newtonian model could, but it can also explain things like black holes, which the Newtonian model is limited in.

General Relativity and Gravity

This tends to be true for a lot of Newtonian physics. Newtonian physics describes some phenomena quite well, but not everything, and it does not work for other things at all. It is still good for us to learn, but as you advance in physics you will learn more about how General Relativity is used to describe gravity because it tends to explain things we see in the universe more accurately.

General Relativity and Gravity

General Relativity is the main way that scientists view gravity today, but it also does not answer all the questions we have about the Universe. Perhaps in the future it will be replaced in turn by another theory…

Homework

Read Sections 6.1, 6.2, and 6.4Make additional notes to supplement the

lesson notes. Complete the following questions:

Pg. 296 # 1, 2, 4, 5, 9, 11 Pg. 303 # 1, 4, 5, 6, 7, 8

Don’t forget to finish your Tribal Coat of Arms!