Gravity & Motion

The textbook web resources for Chapter 2 are here.

 

Newton's First Law: Inertia

Inertia is the tendency of a body...

 ...at rest to remain at rest.

 ...in motion to remain in motion at a constant speed in a straight line.

Inertia is described in Newton's First Law of Motion (the Law of Inertia):

A body continues in a state of rest or motion in a straight line at a constant speed unless made to change that state by forces acting on it.

A force can be defined as any push or pull on an object.  In terms of Newton's Law, we are really talking about a net force: the sum of all forces acting on an object.  For example, you sit on a chair.  Gravity pulls you down, but the chair holds you up.  The forces are balanced, so you remain at rest.  The net force is zero.

In terms of Astronomy, we must consider that any object NOT moving in a straight line at a constant speed must have a net force acting on it.  For example, the Moon goes around the Earth in a curved path.  There is a force causing its path to be curved.

 

Orbital Motion and Gravity

Newton was not the first to recognize that gravity was the force holding the Moon in orbit around the Earth.  He did, however, figure out how this worked, by conducting the following thought experiment.

If you place a cannon on a mountain top and fire a cannonball horizontally, the cannonball will be pulled downward by gravity until it hits the ground.  Making the ball go faster will make it travel farther before it hits the ground.  Eventually, if you keep increasing its speed, the curve of the Earth's surface will drop away as the ball falls.  At some point, the curve or the Earth will match the curve of the ball's path as it falls.  The cannonball will now be in orbit.

So, there are two forces acting on the Moon...

  • inertia carries it forward.
  • gravity causes its path to be curved around the Earth.

 

Newton's Second Law: Acceleration

Speed measures how far something travels in a given amount of time.

Velocity measures speed in a given direction.

Acceleration is any change in velocity (change in speed or direction).

Given Newton's First Law, an object can only accelerate if a net force is acting on it.  However, the amount of acceleration is determined by the amount of force put on an object and the amount of matter the object contains.

Mass is a measure of how much matter an object contains.  Since more mass means more inertia, mass is also a measure of inertia.  (Note: Mass and weight are not the same!  Weight measures the force of gravity on an object.)

Newton's Second Law of Motion states that the amount of acceleration (a) that a force (F) can produce depends on the mass (m) of the object being accelerated.  The formula for this is...

                                              F = ma

 

The Law of Gravity

Newton's Law of Gravity, based on his study of the Moon, is...

Every mass exerts a force of attraction on every other mass.  The strength of the force is directly proportional to the product of the masses divided by the square of their separation.

The formula for this is... 

                             GMm

                   F = -------------

                                r2

    F = force of attraction between the two bodies

    M and m = the masses of the two bodies

    r = the distance between the centers of the two masses

    G = the Gravitational Constant 6.67x10-11 m3/kg•s2 

Just looking at the formula, we can see a few relationships...

  • If the mass of either object is increased, the force of gravity between them will increase.
  • If the distance between the two objects is increased, the force of gravity will decrease.
    • If the distance is doubled (2x), the force is quartered (x¼).
  • Although the force of gravity weakens with increased distance, it never disappears.

Newton's Third Law

Newton's Third Law of Motion (the Law of Action-Reaction) states:

When two objects interact, they create equal and opposite forces on each other.

The Sun pulls on the Earth (gravity).  The Earth pulls on the Sun (gravity).  The force of the Sun pulling on the Earth is the same as the force of the Earth pulling on the Sun.  Remember, though, that Force = mass x acceleration.  Since the Sun's mass is 300,000x larger than the Earth, it will accelerate 300,000x less than the Earth.

However, the Earth does cause the Sun to move slightly.

 

Measuring Mass Using Orbital Motion

Using Kepler's Third Law and Newton's Laws of Motion and Gravity, it is possible to calculate an object's mass by its orbital properties.  This is based on two concepts...

  • The masses or orbiting objects determine the force of their gravitational attraction.
  • The gravitational force determines the properties of the orbit.

If a mass (m) moves with velocity (V) around an orbit or radius (r), then we can calculate the force (F) needed to hold it in this circle with the formula:

                            mV2

                  F =  ---------

                               r

If the F in this equation is the same as the F in the equation for Newton's Law of Gravity, we can solve for V (which gives us the orbital velocity.

                              V2 = GM / r

Once we have the orbital velocity, we can go back to solve for mass.

 

Surface Gravity

Surface gravity measures the force of gravity at the surface of a planet or star.  This is an acceleration of mass caused by the gravitational force.  The symbol for surface gravity is g.

Surface gravity determines...

  • How much a given mass will weigh on a planet/star.
  • The shape of the object.
    • Asteroids are often not round because their surface gravity is too weak to pull them into a round shape.
  • Whether the planet has an atmosphere.

Surface gravity depends on the object's mass (M) and radius (R).  The formula is...

                                 GM

                        g = ---------

                                   R2

From this, we can see that...

  • Two planets with the same radius, but different masses, will have different surface gravities.  (The more massive one will have greater surface gravity.)
  • Two planets with the same mass, but different radii, will have different surface gravities.  (The one with the smaller radius will have greater surface gravity.)

Escape Velocity

Escape velocity is the speed at which an object needs to move away from another body, so that it is not pulled back by the body's gravity.

Escape velocity determines...

  • How fast a rocket has to move to go places other than Earth.
  • Whether a planet has an atmosphere.
  • The nature of black holes.  (The escape velocity of a back hole is equal to the speed of light.)

The formula for escape velocity is...

                                              Vesc = √2GM/R

Vesc = escape velocity

G = gravitational constant

M = mass of the body to be escaped from

R = radius of the body to be escaped from

 

Escape velocity from Earth is  11 km/s.

escape velocity from the Moon is  2.4 km/s.


Web Resources 

 

Homework from the Text:

  • Read Chapter 2 (pg 75-88).
  • Review Questions #1-8 (pg 88).