Significance of Gravitational Force
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Significance of the gravitational force =
Discovery of planets ==
Accurate measurements on the orbits of the plantes indicated that they did not precisely follow Kepler's laws. Slight deviations from perfectly elliptical orbits were observed. Newton was aware that this was to be expected from the Law of Universal Gravitation. The derivation of perfectly elliptical ignores the forces due to the other planets. These deviations called perturbations are observed and led to the discovery of Neptune and Pluto. Planets around distant stars were also inferred from the regular wobble of each star due to the gravitational attraction of the revolving plant.
Ocean tides ==
Ocean tides are caused by differences in the gravitational pull between the Moon and the Earth on the opposite sides of the Earth. Gravitational force is stronger on the side of the Earth nearer to the Moon and is weaker on the side of the Earth farther from the Moon. The bulge that is caused in the Earth's oceans due to this gravitational pull results in two sets of tides on the Earth.
Concept #1 - Projectile and Satellite Motion
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Satellite
An Earth satellite is simply a projectile that falls around the Earth rather than into it. That means the horizontal falling distance matches the Earths curvature. Geometrically, the curvature of the surface is that its surface drops a vertical distance of 5 metres for every 8000 metres tangent to the surface.
Therefore, if we throw a rock or a ball at a high enough speed (about 29000 km/s), it would follow the curvature of the Earth. But at this speed, atmospheric friction (due to air drag) would burn up everything. This is why satellites are launched at an altitude high enough for the air drag to be negligible.
Satellite motion was understood by Newton who reasoned that the Moon was simply a projectile that was circling the Earth.
Activity 4 - Demonstration of satellite motion using simulation
The following simulation can illustrate how satellite motion takes place.
Kepler's Laws
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- The key concept to understand here is that gravitational forces play an important role in planetary motion.
- Three laws of planetary motion that describe the motion of the planets have been postulated based on detailed astronomical observations
Laws of Planetary Motion
We now know that satellites are continually falling towards the Earth following a curved path whose curvature is greater than that of the curvature of the Earth. The Moon is just such a satellite that moves around the Earth. In a similar way, all the planets that move around the Sun are satellites of the Sun. The motion described in such a situation is not strictly circular - it is elliptical.
Johannes Kepler, working with data painstakingly collected by Tycho Brahe without the aid of a telescope, developed three laws which described the motion of the planets across the sky.
1. The Law of Orbits: All planets move in elliptical orbits, with the sun at one focus.
2. The Law of Areas: Each planet moves so that an imaginary line drawn from the Sun to the planet sweeps out equal areas in equal periods of time.
3. The Law of Periods: The square of the period of any planet is proportional to the cube of the semimajor axis of its orbit.
Kepler's laws were derived for orbits around the sun, but they apply to satellite orbits as well.
The Law of Orbits
All planets move in elliptical orbits, with the sun at one focus. An ellipse is a closed curve such that the sum of the distances from any point P on the curve to two fixed points (called the foci, F1 and F2) remains constant.
Orbit eccentricity
The semi major axis of the ellipse is a and represents the planet's average distance from the Sun. The eccentricity, “e” is defined so that “ea” is the distance from the centre to either focus. A circle is a special case of an ellipse where the two foci coincide. The Earth and most of the other planets have nearly circular orbits. For Earth, “e” = 0.017.
The Law of Equal Areas
Kepler's second law states that each planet moves so that an imaginary line drawn from the Sun to the planet sweeps out equal areas in equal periods of time.
This can be shown to be true using the law of conservation of angular momentum.
If “v” is the velocity of the planet, in time “dt” the planet moves a distance vdt and sweeps out an area equal to the area of a triangle of base “r” and altitude vdt sinα.
Hence dA = ½ (r) (“v” x “dt” x sinα)
dA/ dt = ½ rv sinα
The magnitude of the angular momentum of the planet about the Sun is L = mvr sinα.
dA/ dt = (½)L/m
Because the angular momentum is conserved, the rate of change of area covered is constant. This means that the planets move with different velocities depending upon their position in the orbits.
The Law of Periods
The ratio of the squares of the periods of any two planets revolving about the Sun is equal to the ratio of the cubes of their semi-major axes.
Can you derive this?
G m1 Ms / r12 = m1 (v12)/ r1
v1 = 2πr1/T1
Substituting and rearranging we get
T12/ r13 = 4π2 / G Ms
Deriving this for another planet, we can arrive at the third law.
5. www.hyperphysics.com - From Classical Mechanics to General Relativity - This is a good description of the geometry of Newtonian gravity and how to move from classical mechanics to relativity.
8. http://www.physicsclassroom.com/Class/circles/U6L4b.cfm -This website describes the mathematics of orbital motion.
9. http://spaceflight.nasa.gov/gallery/images/station/crew-9/html/iss008e21996.html
Keywords
Mass, Inertial, Gravitational, Force field, Universal law of gravitation, Acceleration due to gravity, “g”, weight, weightlessness