Circular Motion 2

Circular Motion 2: The Law of Gravity

A. Newton’s Law of Gravity

In this section, we will discuss the concept of gravitational force with a more general treatment. The one force we have discussed associated with gravity is Weight.
The acceleration due to gravity 9.80 m/s² is the rate at which every object falls toward the Earth. Therefore, if two objects, one heavy and one light, are dropped from the same height, they will both accelerate at the same rate, g = 9.80 m/s²
This acceleration is specific to planet Earth. As we shall soon see, every planet and celestial body has it's own acceleration due to gravity.

We begin with a discussion of Newton's Law of Universal Gravitation. Basically, this law states that every object is attracted to every other object in the universe.
This attractive force is inversely proportional to the square of the distance between the objects, r. This force acts along the vector that goes from one object to the other.
In general, the universal law of gravitation is written as:

In the equation above: G = 6.67 x 10^-11 Nm²/kg² = universal constant of gravitation, r₁₂ is the distance between the two masses, and the unit vector tells us that the force acts along the radius vector between the two masses.
For example, consider the 2-body problem below. In the diagram, we see that the force acts along the radius vector between the 2 masses, and each force exerts a force of equal magnitude on the other object but in opposite directions.

So, what is the gravitational force that the Earth exerts on any object on or near Earth? Let r = R_E + h, where h = object’s distance above the surface of the Earth, and R_E and M_E are the radius and mass of the Earth respectively, and m is the mass of the object, then:

B. Weight and Gravitational Force

What if the object is close to the surface of the Earth or right at the surface of the Earth? Then h = 0, so that r = R_E . So the gravitational force between the Earth and an object of mass m becomes

Let us now use Newton’s 2^nd Law (F = ma) to obtain the acceleration of an object on or very near the surface of the Earth:

Therefore, weight is a special case of Newton’s law of gravitation in which we assume that g is constant for objects near the surface of the Earth.
With this in mind, we can find the acceleration due to gravity on any planet or celestial object in the universe using this universal law of gravitation. Therefore, for any object near a planet’s surface, the acceleration due to gravity of that planet is given by:

So as a precaution, remember that Weight is a more specific term for gravitational force or gravitational attraction.

C. Motion of Satellites around Planets

In this section, we perform a general analysis of the motion of satellites around planets under the following assumptions: that the motion is very nearly circular, and that this treatment applies to any object in circular motion about another object (planet around sun, moon around the planet, or man-made spacecraft around any planet).
First we establish that the "satellite" is orbiting a distance r = R_P + h above the center of the planet, where R_P is the radius of the planet, and h is the height above the surface of the planet.
Unless otherwise stated, the system is isolated, therefore the force that supplies the centripetal force for the object undergoing circular motion is the gravitational attraction between the planet and the object. Therefore:

From this equation, we will be able to solve for variables like the velocity of the object, the radius of the object, or even the mass of the planet.
Solving for the velocity v:

Knowing the velocity allows us to calculate the period of the object as it undergoes CIRCULAR motion around the planet. In previous chapters, we discussed that an object undergoing circular motion travels a distance of 2pr in one revolution, therefore the speed of the object is given by: v = (2pr)/T, where T is the period of revolution. Therefore:

Geosynchronous satellites are satellites that have periods of 24 hrs. They are satellites that hover above the same point of the Earth as they orbit (examples: communication satellites). The name geo-synchronous means it is synchronized with the earth’s motion.
LEO’s (Low Earth Orbiters) are satellites close to the atmosphere of the Earth, in fact many times they scrape above the upper atmosphere and experience frictional forces which degrade their orbits. Many maneuvers are needed to keep them in orbit.
Where a satellite is launched has much to do with the character of the orbit. The launching points in the US usually allow equatorial surveillance, while launching points used in Russia usually allowed for polar type orbits.

D. Kepler’s Laws

Over a period of about 20 years Tycho Brahe made detailed and meticulous measurements of the motion of many celestial objects. This was during a time when the geocentric model of the solar system was the current philosophical norm: the Earth is the center of the solar system.
But observation told scientists that something was wrong with that model. For instance the retrograde motion of the planets was hard to reconcile with the geocentric model.
Eventually, the heliocentric model was adapted: the sun was at the center of our solar system. The measurements made by Brahe were critical to the verification of these current theories.
Johannes Kepler was Brahe’s assistant, and upon Brahe’s death, he inherited all of the data. He studied and analyzed the data and came up with observations that are now known as Kepler’s Laws.
Before we get to the Laws, let us discuss the properties of planetary motion which is also called CENTRAL FORCE MOTION.

Central force motion is called that because the force acts along the radius vector drawn from one object to the other.
Central motion is confined to a plane.
The angular momentum and total energy of the central force system are constants, i.e., both Energy and Angular momentum are conserved, and the angular momentum is perpendicular to the plane of the motion.

This brings us to Kepler’s Laws of Motion:

1^st Law: All planets move in elliptical orbits with the sun at one of the focal points.

2^nd Law: The radius vector drawn from the sun to the planet sweeps out equal areas in equal dime intervals.

3^rd Law: The square of the orbital period of any planet is proportional to the cube of the semi-major axis of the elliptical orbit

Before we go any further, we will look at the properties of ellipses and elliptical orbits.

                                           a = semi-major axis                     e = eccentricity of the orbit
                                           b = semi-minor axis                     f₁ and f₂ = foci
                                           r = radius vector from sun to planet
                                           R_P = perihelion (nearest distance to sun)
                                           R_a = aphelion (farthest distance to sun)

A word on eccentricity of orbits. Eccentricity is a number that depends on the Energy of the orbital system. As a result, the shape of the orbit depends on the Energy of the system and therefore on the eccentricity.

If e = 0, then the orbit will be circular.
If 0 < e < 1, then the orbit is elliptical
If e = 1, the orbit is parabolic
If e > 1, then the orbit is hyperbolic.

Most planets have eccentricities close to zero, hence our approximations of circular motion are quite valid. Some comets have open orbits (parabolic and hyperbolic) thus once they pass the solar system, they will never be seen again.

E. More details on Kepler’s Laws
1) First Law: The law of orbits

All planets move in elliptical orbits with the sun at one focus.
For most planetary systems, the CM is virtually located inside the sun sin the mass of the sun is much larger than the mass of any planet.
The eccentricity of the Earth’s orbit is 0.0167, which is quite small, hence we usually just depict the orbit as circular without much loss generality.

2) Second Law: The law of areas

The radius vector drawn from the sun to the planet sweeps out equal areas in equal times.
Qualitatively: the planet moves faster when it is near the sun and slower when it is away from the sun.
This law is totally equivalent to the law of conservation of angular momentum.
Mathematically, the instantaneous rate that area is being swept up is given by

We already said that L is conserved for orbital motion, thus since the time rate of change of Area depends on L, then it is also constant.

3) Third Law: Law of periods

The square of the period of any planet is proportional to the cube of the semi-major axis of the orbit.