PHYSICS PROJECT GRAVITATION
GRAVITATION Gravitation, or gravity, is a natural phenomenon by which all physical bodies attract each other. It is most commonly experienced as the agent that gives weight to objects with mass and causes them to fall to the ground when dropped. Gravitation is one of the four fundamental interactions of nature, along with electromagnetism, and the nuclear strong force and weak force. In modern physics, the phenomenon of gravitation is most accurately described by the general theory of relativity by Einstein, in which the phenomenon itself is a consequence of the curvature of spacetime governing the motion of inertial objects. The simpler Newton's law of universal gravitation postulates the gravity force proportional to masses of interacting bodies and inversely proportional to the square of the distance between them. It provides an accurate approximation for most physical situations including calculations as critical as spacecraft trajectory.
FACTORS EFFECTING GRAVITATIONVariation in gravity and apparent gravity A perfect sphere of spherically uniform density (density varies solely with distance from centre) would produce a gravitational field of uniform magnitude at all points on its surface, always pointing directly towards the sphere's centre. However, the Earth deviates slightly from this ideal, and there are consequently slight
deviations in both the magnitude and direction of gravity across its surface. Furthermore, the net force exerted on an object due to the Earth, called "effective gravity" or "apparent gravity", varies due to the presence of other factors, such as inertial response to the Earth's rotation. A scale or plumb bob measures only this effective gravity. Parameters affecting the apparent or actual strength of Earth's gravity include latitude, altitude, and the local topography and geology. Apparent gravity on the earth's surface varies by around 0.7%, from 9.7639 m/s2 on the Nevado Huascarán Mountain in Peru to 9.8337 m/s2 at the surface of the Arctic Ocean. In large cities, it ranges from 9.766 in Kuala Lumpur,Mexico City, and Singapore to 9.825 in Oslo and Helsinki.
Latitude The differences of Earth's gravity around the Antarctic continent. The surface of the Earth is rotating, so it is not an inertial frame of reference. At latitudes nearer the Equator, the outward centrifugal force produced by Earth's rotation is larger than at polar latitudes. This counteracts the Earth's gravity to a
small degree – up to a maximum of 0.3% at the Equator – and reduces the apparent downward acceleration of falling objects. The second major reason for the difference in gravity at different latitudes is that the Earth's equatorial bulge (itself also caused by inertia) causes objects at the Equator to be farther from the planet's centre than objects at the poles. Because the force due to gravitational attraction between two bodies (the Earth and the object being weighed) varies inversely with the square of the distance between them, an object at the Equator experiences a weaker gravitational pull than an object at the poles. In combination, the equatorial bulge and the effects of the Earth's inertia mean that sea-level gravitational acceleration increases from about 9.780 m·s−2 at the Equator to about 9.832 m·s−2 at the poles, so an object will weigh about 0.5% more at the poles than at the Equator. The same two factors influence the direction of the effective gravity. Anywhere on Earth away from the Equator or poles, effective gravity points not exactly toward the centre of the Earth, but rather perpendicular to the surface of the geoid, which, due to the flattened shape of the Earth, is somewhat toward the opposite pole. About half of the deflection is due to inertia, and half because the extra mass around the Equator causes a change in the direction of the true gravitational force relative to what it would be on a spherical Earth.
Altitude The graph shows the variation in gravity relative to the height of an object Gravity decreases with altitude as one rises above the earth's surface because greater altitude means greater distance from the Earth's centre. All other things being equal, an increase in altitude from sea level to 9,000 metres (30,000 ft) causes a weight decrease of about 0.29%. (An additional factor affecting apparent weight is the decrease in air density at altitude, which lessens an object's buoyancy. This would increase a person's apparent weight at an altitude of 9,000 metres by about 0.08%) It is a common misconception that astronauts in orbit are weightless because they have flown high enough to "escape" the Earth's gravity. In fact, at an altitude of 400 kilometres (250 mi), equivalent to a typical orbit of the Space Shuttle, gravity is still nearly 90% as strong as at the Earth's surface. Weightlessness actually occurs because orbiting objects are in free-fall. The effect of ground elevation depends on the density of the ground (see "Slab correction" below). A person flying at 30 000 ft above sea level over mountains will feel more gravity than someone at the same elevation but over the sea. However, a
person standing on the earth's surface feels less gravity when the elevation is higher. The following formula approximates the Earth's gravity variation with altitude:
Where
gh is the gravitational acceleration at height above sea level.
re is the Earth's mean radius.
g0 is the standard gravitational acceleration.
This formula treats the Earth as a perfect sphere with a radially symmetric distribution of mass; a more accurate mathematical treatment is discussed below. Depth An approximate depth dependence of density in the Earth can be obtained by assuming that the mass is spherically symmetric (it depends only on depth, not on latitude or longitude). In such a body, the gravitational acceleration is towards the center. The gravity at a radius r depends only on the mass inside the sphere of radius r; all the contributions from outside cancel out. This is a consequence of the inverse-square law of gravitation. Another
consequence is that the gravity is the same as if all the mass were concentrated at the center of the Earth. Thus, the gravitational acceleration at this radius is
where G is the gravitational constant and M(r) is the total mass enclosed within radius r. If the Earth had a constant density ρ, the mass would be M(r) = (4/3)πρr3 and the dependence of gravity on depth would be
If the density decreased linearly with increasing radius from a density ρ0 at the centre to ρ1 at the surface, then ρ(r) = ρ0 − (ρ0 − ρ1) r / re, and the dependence would be
KEPLER’S LAWS1)The Law of Orbits: The orbit of every planet is an ellipse with the Sun at one of the two focii.
An ellipse is a closed plane curve that resembles a stretched out circle (see the figure to the right). Note that the Sun is not at the center of the ellipse, but at one of its foci. This focal point is sometimes called the occupied focus. The other focal point, known as the empty or vacant focus, marked with a lighter dot, has no physical significance for the orbit. The center of an ellipse is the midpoint of the line segment joining its focal points. A circle is a special case of an ellipse where both focal points coincide.
2)Law of Area: A line joining a planet and the Sun sweeps out equal areas during equal intervals of time. - To satisfy law of area, a planet will have to revolve faster when it is closer to the sun. PROVE: Let P1 and P2 be the positions of the planet in the orbit in a small interval of time of time dt . Where v is the linear velocity Angular momentum is the moment of the linear momentum (L) L= mvR2 =m (v=
R2
R)
L/m=
R2
The area swept is equal to
the area of the triangular shaded region dA =
½ (P1 P2)R
=
½ vdtR
=
½
dA = ½
R dt R R2
dA/dT= ½
R2
= ½ L/m dA/dT= constant Hence Kepler’s Law of Area
3)Law of Time Period: The square of the time period of a revolution of a planet is directly proportional to the cube of the mean distance between the planet and the sun. The third law, published by Kepler in 1619 captures the relationship between the distance of planets from the Sun, and their orbital periods. Symbolically, the law can be expressed as
where
is the orbital period of the planet and is the semi-major
axis of the orbit. The constant of proportionality is
for a sidereal year (yr), and astronomical unit (AU). (For numeric values see List of gravitationally rounded objects of the Solar System). Kepler enunciated this third law in a laborious attempt to determine what he viewed as the "music of the spheres" according to precise laws, and express it in terms of musical notation.] So it used to be known as the harmonic law. For circular orbits, Kepler's 3rd Law is also commonly represented as
Where
is the period,
is the Gravitational constant,
mass of the larger body, and
is the
is the distance between the centers
of mass of the two bodies.
NEWTONS LAW OF UNIVERSAL GRAVITATION-
Newton's law of universal gravitation states that every point mass in the universe attracts every other point mass with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. Every point mass attracts every single other point mass by a forcE pointing along the line intersecting both points. The force is proportional to the product of the two masses and inversely proportional to the square of the distance between them:
, where:
F is the force between the masses,
G is the gravitational constant,
m1 is the first mass,
m2 is the second mass, and
r is the distance between the centres of the masses.
Assuming SI units, F is measured in newtons (N), M1 M2 in kilograms (kg), r in meters (m), and the constant G is
approximately equal to 6.674×10−11 N m2 kg−2.[4] The value of the constant G was first accurately determined from the results of the Cavendish experiment conducted by the British scientist Henry Cavendish in 1798, although Cavendish did not himself calculate a numerical value for G.[5] This experiment was also the first test of Newton's theory of gravitation between masses in the laboratory. It took place 111 years after the publication of Newton's Principia and 71 years after Newton's death, so none of Newton's calculations could use the value of G; instead he could only calculate a force relative to another force
GRAVITATIONAL CONSTANTThe gravitational constant denoted by letter G, is an empirical physical constant involved in the calculation(s) of gravitational force between two bodies. It usually appears in Sir Isaac Newton's law of universal gravitation, and in Albert Einstein's theory of general relativity. It is also known as the universal gravitational constant, Newton's constant, and colloquially as Big G. It should not be confused with "little g" (g), which is the local gravitational field (equivalent to the free-fall acceleration), especially that at the Earth's surface. According to the law of universal gravitation, the attractive force (F) between two bodies is proportional to the
product of their masses (m1 and m2), and inversely proportional to the square of the distance, r, (inverse square law) between them:
The constant of proportionality, G, is the gravitational constant.
ACCELERATION DUE TO GRAVITYthat a free-falling object is an object that is falling under the sole influence of gravity. A free-falling object has an acceleration of 9.8 m/s/s, downward (on Earth). This numerical value for the acceleration of a free-falling object is such an important value that it is given a special name. It is known as the acceleration of gravity - the acceleration for any object moving under the sole influence of gravity. A matter of fact, this quantity known as the acceleration of gravity is such an important quantity that physicists have a special symbol to denote it - the symbol g. The numerical value for the acceleration of gravity is most accurately known as 9.8 m/s/s. There are slight variations in this numerical value (to the second
decimal place) that are dependent primarily upon on altitude. We will occasionally use the approximated value of 10 m/s/s in The Physics Classroom Tutorial in order to reduce the complexity of the many mathematical tasks that we will perform with this number. By so doing, we will be able to better focus on the conceptual nature of physics without too much of a sacrifice in numerical accuracy.
g = 9.8 m/s/s, downward ( ~ 10 m/s/s, downward)
ORBITAL VELOCITY OF A SATELLITE The motion of objects is governed by Newton's laws. The same simple laws that govern the motion of objects on earth also extend to the heavens to govern the motion of planets, moons, and other satellites. Consider a satellite with mass Msat orbiting a central body with a mass of mass MCentral. The central body could be a planet, the sun or some other large mass capable of causing sufficient acceleration on a less massive nearby object. If the satellite moves in circular motion, then the net centripetal force acting upon this orbiting satellite is given by the relationship
Fnet = ( Msat • v2 ) / R
This net centripetal force is the result of the gravitational force that attracts the satellite towards the central body and can be represented as Fgrav = ( G • Msat • MCentral ) / R2 Since Fgrav = Fnet, the above expressions for centripetal force and gravitational force can be set equal to each other. Thus, (Msat • v2) / R = (G • Msat • MCentral ) / R2 Observe that the mass of the satellite is present on both sides of the equation; thus it can be canceled by dividing through by Msat. Then both sides of the equation can be multiplied by R, leaving the following equation. v2 = (G • MCentral ) / R Taking the square root of each side, leaves the following equation for the velocity of a satellite moving about a central body in circular motion
where G is 6.673 x 10-11 N•m2/kg2, Mcentral is the mass of the central body about which the satellite orbits, and R is the radius of orbit for the satellite.
Similar reasoning can be used to determine an equation for the acceleration of our satellite that is expressed in terms of masses and radius of orbit. The acceleration value of a satellite is equal to the acceleration of gravity of the satellite at whatever location that it is orbiting. The equation for the acceleration of gravity was given as g = (G • Mcentral)/R2 Thus, the acceleration of a satellite in circular motion about some central body is given by the following equation
where G is 6.673 x 10-11 N•m2/kg2, Mcentral is the mass of the central body about which the satellite orbits, and R is the average radius of orbit for the satellite. The final equation that is useful in describing the motion of satellites is Newton's form of Kepler's third law. Since the logic behind the development of the equation has been presented elsewhere, only the equation will be presented here. The period of a satellite (T) and the mean distance from the central body (R) are related by the following equation:
where T is the period of the satellite, R is the average radius of orbit for the satellite (distance from center of central planet), and G is 6.673 x 10-11 N•m2/kg2.
There is an important concept evident in all three of these equations - the period, speed and the acceleration of an orbiting satellite are not dependent upon the mass of the satellite.
None of these three equations has the variable Msatellite in them. The period, speed and acceleration of a satellite are only dependent upon the radius of orbit and the mass of the central body that the satellite is orbiting. Just as in the case of the motion of projectiles on earth, the mass of the projectile has no affect upon the acceleration towards the earth and the speed at any instant. When air resistance is negligible and only gravity is present, the mass of the moving object becomes a non-factor. Such is the case of orbiting satellites.
Gravitational Potential Energy
Gravitational potential energy is energy an object possesses because of its position in a gravitational field. The most common use of gravitational
potential energy is for an object near the surface of the Earth where the gravitational acceleration can be assumed to be constant at about 9.8 m/s2. Since the zero of gravitational potential energy can be chosen at any point (like the choice of the zero of a coordinate system), the potential energy at a height h above that point is equal to the work which would be required to lift the object to that height with no net change in kinetic energy. Since the force required to lift it is equal to its weight, it follows that the gravitational potential energy is equal to its weight times the height to which it is lifted.
The general expression for gravitational potential energy arises from the law of gravity and is equal to the work done against gravity to bring a mass to a given point in space. Because of the inverse square nature of the gravity force, the force approaches zero for large distances, and it makes sense to choose
the zero of gravitational potential energy at an infinite distance away. The gravitational potential energy near a planet is then negative, since gravity does positive work as the mass approaches. This negative potential is indicative of a "bound state"; once a mass is near a large body, it is trapped until something can provide enough energy to allow it to escape. The general form of the gravitational potential energy of mass m is:
where G is the gravitation constant, M is the mass of the attracting body, and r is the distance between their centers. This is the form for the gravitational potential energy which is most useful for calculating the escape velocity from the earth's gravity.
GEOSTATIONARY SATELLITE A satellite that appears to be stationary with respect to a particular place on the surface of the earth is known as a geostationary satellite. The time period of a revolution of a geostationary satellite should be equal to the time period of the rotation of the earth i.e, 24hrs.
Both should move in the same direction also. Geostationary satellites are used for live telecast and telecommunication. For covering the entire globe a minimum of 3 geostationary satellites placed 120o apart are required.
GRAVITATIONAL FIELD OF A BODY It is the region around a body where the gravitational force of attraction is experienced.
GRAVITATIONAL FIELD IN DENSITY Gravitational field in density at a point inside the gravitational field of the body is defined as force acting on unit mass placed at that point. Gravitational field in density= GM/x2 Where x is the distance unit mass from the body producing the gravitational field. It is a vector quantity and its direction is along the direction of the force acting on that unit mass.
Energy in orbit of satellites around the earth lost? If the total mechanical energy in a satellite's orbit (assuming circular) is greater when it is closer to the earth, and hence smaller when it is farther from the earth, then we can say that as the moon drifts from the earth, the moon loses energy in translational speed and gravitational potential energy. If only those two are taken into consideration, then there is a net energy loss from the moon.
I had first thought that the energy a satellite has increases as it goes on a larger orbit, but I ran some numbers and it didn't appear so. If I went wrong somewhere, please someone, correct me. Here are my numbers:
For a geostationary satellite (r = 42 164 km, v = 11 068 km/s, m = 1 kg), its total energy is PE + KE. PE = mgh, but g = 0.22416 m/s^2. The result is PE = 9 451.650 kJ, KE = 4 726.582 kJ
For a satellite at r = 45 000 km , m = 1kg, then v = sqrt(GM/r) = 2 976.06 km/s. g at that height is g = 0.19680 The result is PE = 8 856.094 kJ, KE = 4 428.047 kJ
At the larger orbit, both PE and KE are lower than if it was at a lower orbit. Is this right?
Now, the earth slows down its rotation, which allows the moon to go into a larger orbit by conservation of angular momentum. Since the moon goes into a larger orbit, it loses energy. But, since the spin of the earth has slowed down, it also loses energy. Moreover, the moon is still tidally locked with the earth, so its rotational speed isn't increasing.
All in all, there seems to be an energy loss that's going on. How is this being compensated? Is it in the translational speed of the moon (so that the moon is actually moving faster than it should be to maintain a stable orbit)? That seems reasonable - there could be an increase in translational and rotational speed to compensate for the energy loss, maintaining the moon to be tidally locked.
GEOSTATIONARY SATELLITE
A geostationary orbit (GEO) is a geosynchronous orbit directly above the Earth's equator (0° latitude), with orbital eccentricity of zero. From the ground, a geostationary object appears motionless in the sky and is therefore the orbit of most interest to operators of artificial satellites (including communication and television satellites). Due to the constant 0° latitude, satellite locations may differ by longitude only.
The idea of a geosynchronous satellite for communication purposes was first published in 1928 by Herman
Potočnik. The geostationary orbit was first popularised by science fiction author Arthur C. Clarke in 1945 as a useful orbit for communications satellites. As a result this is sometimes referred to as the Clarke orbit. Similarly, the Clarke Belt is the part of space approximately 35,786 km above mean sea level in the plane of the equator where near-geostationary orbits may be achieved.
Geostationary orbits are useful because they cause a satellite to appear stationary with respect to a fixed point on the rotating Earth. As a result, an antenna can point in a fixed direction and maintain a link with the satellite. The satellite orbits in the direction of the Earth's rotation, at an altitude of approximately 35,786 km (22,240 statute miles) above ground. This altitude is significant because it produces an orbital period equal to the Earth's period of rotation, known as the sidereal day.
POLAR SATELLITE
A polar orbit is an orbit in which a satellite passes above or nearly above both poles of the planet (or other celestial body) it is orbiting on each revolution. It therefore has an inclination of (or very close to) 90 degrees to the equator. Except in the special case of a polar geosynchronous orbit, a satellite in a polar orbit will pass over the equator at a different longitude on each of its orbits.
Polar orbits are often used for earth-mapping-, earth observation- and reconnaissance satellites, as well as some weather satellites. The disadvantage to this orbit is that no one spot on the Earth's surface can be sensed continuously from a satellite in a polar orbit.
A satellite can hover over one polar area a large part of the time, albeit at a large distance, using a polar highly elliptical orbit with its apogee above that area. This is the principle behind a Molniya orbit.
WEIGHTLESSNESS Weightlessness, or an absence of 'weight', is in fact an absence of stress and strain resulting from externally applied forces, typically contact forces from floors, seats, beds, scales, and the like. Counterintuitively, a uniform gravitational field does not by itself cause stress or strain, and a body in free fall in such an environment experiences no g-force acceleration and feels weightless. This is also termed zero-g.
When bodies are acted upon by non-gravitational forces, as in a centrifuge, a rotating space station, or within a space ship with rockets firing, a sensation of weight is produced, as the forces overcome the body's inertia. In such cases, a sensation of weight, in
the sense of a state of stress can occur, even if the gravitational field were zero. In such cases, g-forces are felt, and bodies are not weightless.
When the gravitational field is non-uniform, a body in free fall suffers tidal effects and is not stress-free. Near a black hole, such tidal effects can be very strong. In the case of the Earth, the effects are minor, especially on objects of relatively small dimension (such as the human body or a spacecraft) and the overall sensation of weightlessness in these cases is preserved. This condition is known as microgravity and it prevails in orbiting spacecraft.
