Evans SPH 4U1

Physics Grade 12

Unit 3: Circular Motion

Note 4: Universal Gravitation, Satellites

Reference Chapter 3.3/3.4

Newton's Law of Universal Gravitation:

The force of gravitational attraction between any two objects is directly proportional to the product of the masses and inversely proportional to the square of the distance between their centres.

Notes on Universal Gravitation:

~ Two equal opposite forces (i.e. the earth pulls on the moon, the moon pulls on the earth) ~ Newton's 3rd Law

~ One of the objects must be very large for the force to be noticeable.

~ Discovery of Neptune was made this way.

~ Theories suggest the attraction could be a result of the masses causing an indentation in space (bending of space).

~ Einstein said "Where light bends, space bends".

Planetary Motion:

Planets orbit around the sun because the sun and the planet are 2 masses that are attracted by universal gravitation.  Planets have an elliptical motion with the sun at one focus.  Elliptical orbits are more difficult to analyze than circular orbits because the instantaneous velocity is not constant.

Black Holes:

~ Imagine a star with a mass 10 times the sun.  Stars normally generate heat by converting hydrogen into helium.  The energy released creates pressure to support the star against its own gravity, giving an object with a radius about 5 times the sun's radius.  Once the star has used all its nuclear fuel, there is nothing to keep the outward pressure and the star collapses because of its own gravity.  As the star shrinks, the gravitational field at the surface becomes stronger and the escape velocity increases to greater than the speed of light. 

~ The gravitational force of a black hole is so great that nothing can escape it (not even light).

~ The boundary where it takes the speed of light to escape the gravity of the black hole is called the event horizon.

Satellites and Orbital Velocity:

Recall:

~ Satellites and space stations are really just falling down at the same rate as the curvature of the earth.  Basically they have a velocity (called the orbital velocity) that is just the velocity to overcome gravity that is pulling it down. We need alot of energy to launch the satellites up to this velocity but once they reach this orbital velocity, it is maintained (Newton's first law of inertia).  The only force acting on the satellite is the force of gravity from the planet which is equal to its centripetal force.

~ Any mass can have an object in orbit around it.

Example 1:

The magnitude of the force of gravity between 2 spherical masses is 30.0 N when their centres are 10.0m apart.  When the distance between the masses is changed the force becomes 90.0 N.  How far apart are the centres of the masses?

Answer: 5.8 m

Example 2:

Earth's gravitational pull on a satellite is 2.4 x 10² N in magnitude.  What will the magnitude of the force of gravity be on a second spacecraft with twice the mass of the first satellite and at a distance from the earth's centre that is 0.5 times as great?

Answer: 9.6 N

Example 3:

You want to launch a satellite into orbit around the earth.  Its altitude will be 700 km above the surface of the earth. What orbital velocity (speed required to maintain its orbit) is required? Use radius of the earth as 6.38 x 10⁶ m and the mass of the earth as 5.98 x 10²⁴ kg.

Answer: 7.51 x 10³ m/s

Note how this solution ends up being independent of the mass of the satellite.

Example 4:

Astronomers have identified a black hole at the centre of the galaxy M87 .  From the properties of the light observed, they have measured material at a distance of from the centre of the black hole, travelling at a speed of . Determine:

a) the mass of this black hole (assume the observed material is in circular  orbit). Answer: 4.8 x10³⁹ kg

b) What is the  ratio of the mass of the black hole to the mass of the sun ()? Answer: 2.4 x 10⁹

Example 5:

Question #5 on page 144 in Text.

a) Answer: 6.8 x 10 ^-8 N [W]

b) Answer: 7.9 x 10 ^-7 N [S42degreesW]