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Gravitational Field
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Historical facts Geocentric Theory Heliocentric Theory – Nicholas Copernicus (1473 – 1543) Nicholas Copernicus – All planets, including Earth, move in orbits around the sun
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Gravitational field (G-field) (1) A region where a force is experienced by a unit mass A gravitational field is set up in the neighbourhood of a mass and through the interaction of this field with another mass, a force is experienced by the second body A gravitational field is represented by field linesfield lines
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The gravitational force is given by: Gravitational field (G-field) (2) Where G is a constant called the Universal Gravitational Constant
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Gravitational field strength It is also called the field intensity The gravitational field strength at a point is the gravitational force acting on a unit mass placing at the point Unit: N kg -1 On the earth’s surface, g 10 N kg -1 On the moon’s surface, g 1.7 N kg -1
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Gravitational potential Due to the existence of the field, a net amount of work has to be done to move a unit mass from one point to another. We say that different points in the field have different gravitational potential It represents the work done in taking a unit mass from one point in a g-field to another Unit: J kg -1
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Equipotential surfaces Surfaces containing points having the same gravitational potential (figure)figure The spacing of the equipotential surface is an indication of the field strength The g-field is pointing in the direction of decreasing gravitational potential
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Uniform field Near the Earth’s surface, the pull of the Earth on a body is constant in size and in direction. The gravitational field g is thus uniform within these limits uniform
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Radial field The field lines are directed radially inwards towards a centreradially
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Important points (1) The reference zero level for the gravitational potential is defined to be a point at infinity Work must be done in moving a mass from the earth’s surface to the infinity. At a closer distance from the earth the gravitational potential takes a negative value
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The equipotential surfaces have a non- uniform spacing, being closer to each other nearer the earth’s surface – Explanation: Near the earth’s surface, the g- field is stronger. The same amount of work can only cause a small displacement from the earth’s surface The density of the field lines is a measure of the field strength Important points (2)
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For radial field, field strength Important points (3) where r is the distance from the centre In a radial gravitational field, the path of an object is a conic
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Law of universal gravitation (1) Every particle of matter in the universe attracts every other particle with a force which is directly proportional to the product of their masses and inversely proportional to the square of their distances apart Mathematical form:
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As the force is inversely proportional to the square of the separation of the masses, it is called an inverse square law The law applies to: – particles – the attraction exerted at an external point by a sphere of uniform density (the whole mass should be treated as if concentrated at centre of the sphere) Law of universal gravitation (2)
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Celestial evidence of inverse square law (1) The speed v of the moon along its orbit is = 1.02 10 3 m s -1
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The moon’s centripetal acceleration a is Celestial evidence of inverse square law (2) On the earth’s surface, acceleration = g = 9.8 m s -2, radius of earth = R = 6.38 10 6 m
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Assume that: Celestial evidence of inverse square law (3) i.e. n 2 Therefore, gravitational force
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Gravitational constant G Unit: N m 2 kg -2 Value: 6.67 10 -11 N m 2 kg -2 Relation between g and G Note: g is independent of m Mass of earth can be estimated from the above equation
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Variation of g (1) A. With height h above the Earth’s surface If g’ = acceleration due to gravity at a distance a from the centre of the earth where a>R
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B. With depth beneath the Earth’s surface – If g 1 = acceleration due to gravity at a distance b from the centre of the Earth where b < R Variation of g (2)
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With latitude – Factors Oblateness of the earth: the equatorial radius exceeds its polar radius by about 21 km and thereby making g greater at the poles than at the equator where a body is farther from the centre of the earth Earth’s rotation: at latitude , the gravitational force mg has two componentstwo components –m 2 r cos , providing centripetal force –mg o, which is the observed gravitational force Variation of g (3)
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Gravitational potential The gravitational potential V at a point in the field is defined as the work done in taking a unit mass from infinity to that point The zero level is defined to be a point at infinity At a finite distance r from a planet of mass M,
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Variation of V with r (1) Slope = field intensity g Gravitational potential energy, U = mV = is valid for r > R
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Gravitational potential energy is a scalar Variation of V with r (2) The p.e. of the system =
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Relationship with the gravitational field strength g: Variation of V with r (3)
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Artificial satellite (1)
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Artificial satellite (2) Satellite orbit – Centripetal force comes from the gravitational attraction of the earth – If R = radius of Earth, for a satellite orbiting with speed v and radius r
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each orbit requires a certain speed of flight of the satellite the greater the orbit radius r the smaller the speed v Artificial satellite (3)
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Parking orbit – an equatorial orbit and has a period of exactly one day rotating in the same direction as the earth – the satellite will always stay above the same spot on earth – it is at 42300 km from the centre of the earth Artificial satellite (4)
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Launching a satellite – The satellite is brought by a two-stage rocket to its desired orbital height – The second-stage rocket then fires and increases the speed to that required for a circular orbit at this height – By firing small rockets, the satellite is separated from the second stage rocket and travels in the orbit Artificial satellite (5)
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Velocity of escape Sufficient energy is required to escape completely from the influence of the earth Escape velocity = 11.2 km s -1
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Potential, kinetic & total energy of a satellite in orbit At a orbit of radius r figure
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Weightlessness True weightlessness occurs only when an object is not subjected to any gravitational force an object may appear to be `weightless’ if there is no reaction force giving the sensation of weight. e.g. free fall, projectile motion and satellite motion
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Kepler ’ s laws (1) First law (The law of orbits) – Each planet moves in an ellipse which has the sun at one focus – Figure Figure Second law (The law of areas) – The line joining the sun to the moving planet sweeps out equal areas in equal times – Figure Figure
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– According to the 2nd law: The motion of a planet is faster when it is closer to the sun It is a consequence of the law of conservation of angular momentum Kepler ’ s laws (2)
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Third law (The harmonic law) – The squares of the periods of revolution of the planets about the sun are proportional to the cubes of their mean distance from it (i.e. T 2 r 3 ) – Figure Figure Kepler ’ s laws (3) Further reading
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Deriving the 3 rd law from inverse square law (1) Also
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Deriving the 3 rd law from inverse square law (2)
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Explanation of 2nd law (1) Angular momentum L = mvh = …. (1) Area swept out …. (2)
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(2) in (1) Explanation of 2nd law (2) Hence, constant areal velocity the radius vector sweeps out equal areas in equal times
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Mass of the sun Consider the motion of the earth round the sun: T = 365.25 days, r = 1.5 10 11 m kg
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Mass of the earth Consider the motion of the moon round the earth : T = 27.3 days, r = 3.84 10 8 m 6 10 24 kg
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Nicholas Copernicus Details
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Field lines Radial fieldUniform field
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Equipotential surface: uniform field
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Equipotential surface: radial field
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Effect of Earth ’ s rotation
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K.E., P.E. & Total Energy
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Johannes Kepler (1571 – 1630) A German mathematician Details
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Kepler ’ s 1 st law
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Kepler ’ s 2 nd law
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Kepler ’ s 3 rd law
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