Gravitational Field

Historical facts Geocentric Theory Heliocentric Theory – Nicholas Copernicus (1473 – 1543) Nicholas Copernicus – All planets, including Earth, move in orbits around the sun

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

The gravitational force is given by: Gravitational field (G-field) (2) Where G is a constant called the Universal Gravitational Constant

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

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

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

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

Radial field The field lines are directed radially inwards towards a centreradially

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

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)

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

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:

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)

Celestial evidence of inverse square law (1) The speed v of the moon along its orbit is = 1.02  10 3 m s -1

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

Assume that: Celestial evidence of inverse square law (3) i.e. n  2 Therefore, gravitational force 

Gravitational constant G Unit: N m 2 kg -2 Value: 6.67  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

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

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)

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)

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,

Variation of V with r (1) Slope = field intensity g Gravitational potential energy, U = mV = is valid for r > R

Gravitational potential energy is a scalar Variation of V with r (2) The p.e. of the system =

Relationship with the gravitational field strength g: Variation of V with r (3)

Artificial satellite (1)

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

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)

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 km from the centre of the earth Artificial satellite (4)

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)

Velocity of escape Sufficient energy is required to escape completely from the influence of the earth Escape velocity = 11.2 km s -1

Potential, kinetic & total energy of a satellite in orbit At a orbit of radius r figure

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

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

– 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)

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

Deriving the 3 rd law from inverse square law (1) Also

 Deriving the 3 rd law from inverse square law (2)

Explanation of 2nd law (1) Angular momentum L = mvh = …. (1) Area swept out …. (2) 

 (2) in (1) Explanation of 2nd law (2) Hence, constant areal velocity  the radius vector sweeps out equal areas in equal times

Mass of the sun Consider the motion of the earth round the sun: T = days, r = 1.5  m kg

Mass of the earth Consider the motion of the moon round the earth : T = 27.3 days, r = 3.84  10 8 m  6  kg

Nicholas Copernicus Details

Field lines Radial fieldUniform field

Equipotential surface: uniform field

Equipotential surface: radial field

Effect of Earth ’ s rotation

K.E., P.E. & Total Energy

Johannes Kepler (1571 – 1630) A German mathematician Details

Kepler ’ s 1 st law

Kepler ’ s 2 nd law

Kepler ’ s 3 rd law