Download Understanding Gravitation: Key Concepts and Applications and more Schemes and Mind Maps Physics in PDF only on Docsity!
GRAVITATION
(i) The value of G in the laboratory was first determined by Cavendish using the torsional balance. (ii) The value of G is 6.67 × 10–11^ N – m^2 kg –2^ in S.I. and 6.67 × 10–8^ dyne- cm^2 - g –2^ in C.G.S. system. (iii) Dimensional formula. (iv) The value of G does not depend upon the nature and size of the bodies. (v) It also does not depend upon the nature of the medium between the two bodies. (vi) As G is very small, hence gravitational forces are very small, unless one (or both) of the mass is huge. Properties of Gravitational Force (1) It is always attractive in nature while electric and magnetic force can be attractive or repulsive. (2) It is independent of the medium between the particles while electric and magnetic force depend on the nature of the medium between the particles. (3) It holds well over a wide range of distances. It is found true for interplanetary to inter atomic distances. (4) It is a central force i.e. acts along the line joining the centres of two interacting bodies. (5) It is a two-body interaction i.e. gravitational force between two particles is independent of the presence or absence of other particles; so the principle of superposition is valid i.e. force on a particle due to number of particles is the resultant of forces due to individual particles. While nuclear force is many body interaction (6) It is the weakest force in nature: As F nuclear > F (^) electromagnetic > F (^) gravitational. (7) The ratio of gravitational force to electrostatic force between two electrons is of the order of . (8) It is a conservative force i.e. work done by it is path independent or work done in moving a particle round a closed path under the action of gravitational force is zero. (9) It is an action reaction pair i.e. the force with which one body (say earth) attracts the second body (say moon) is equal to the force with which moon attracts the earth. This is in accordance with Newton's third law of motion. Acceleration Due to Gravity (1) From the expression it is clear that its value depends upon the mass radius and density of planet and it is independent of mass, shape and density of the body placed on the surface of the planet. i.e. a given planet (reference body) produces same acceleration in a light as well as heavy body. (2) The greater the value of or greater will be value of g for that planet. (3) Acceleration due to gravity is a vector quantity and its direction is always towards the centre of the planet.
(4) Dimension [ g ] = [ LT –2] (5) it’s average value is taken to be 9.8 m / s^2 or 981 cm/sec^2 or 32 feet/sec^2 , on the surface of the earth at mean sea level. (6) The values of acceleration due to gravity vary due to the following factors: (a) Shape of the earth, (b) Height above the earth surface, (c) Depth below the earth surface and (d) Axial rotation of the earth. Variation in the value of g Shape of the Earth ● The earth is not a perfect sphere. It is somewhat flat at the poles. The equatorial radius is approximately 21 km more than the polar radius. Acceleration due to gravity at a height above the surface of Earth ● As we go above the surface of the earth, the value of g decreases because. ● If then , i.e., at infinite distance from the earth, the value of g becomes zero. ● If i.e., height is negligible in comparison to the radius then, we get [h << R] ● If then decrease in the value of g with height : Absolute decrease Fractional decrease Percentage decrease
● The effective value of g is not truly vertical i.e. it is not directed exactly towards the centre of earth. ● The effect of centrifugal force due to rotation of earth is to reduce the effective value of g. ● At equators, = 0º therefore, g’ = g – Rω^2 ● At poles, = 90º, therefore, g’ = g Thus, at equator g’ is minimum while at poles g’ is maximum. Substituting in the above expression we get ∴ there is no effect of rotational motion of the earth on the value of at the poles. Substituting in the above expression we get ∴ The effect of rotation of earth on the value of at the equator is maximum. When a body of mass m is moved from the equator to the poles, its weight increases by an amount Gravitational Field ● The space surrounding a material body in which gravitational force of attraction is experienced is called its gravitational field. Gravitational field intensity: ● The intensity of the gravitational field of a material body at any point in its field is defined as the force experienced by a unit mass (determined using a test mass placed at that point provided the test mass itself does not produce any change in the field of the body). ● If a test mass m at a point in a gravitational field experiences a force then ● It is a vector quantity and is always directed towards the centre of gravity of body whose gravitational field is considered. Units: Newton/kg or m/s^2 Dimension: [M^0 LT–2]
Intensity due to uniform solid sphere Outside the surface r > R On the surface r = R Inside the surface r < R Intensity due to spherical shell Outside the surface r > R On the surface r = R Inside the surface r < R = 0 Intensity due to uniform circular ring At a point on its axis At the centre of the ring E = 0 Intensity due to uniform disc At a point on its axis At the centre of the disc
Outside the surface r > R On the surface r = R Inside the surface r < R at the centre ( r = 0) ( max .) V centre = Potential due to a Uniform Thin Spherical Shell Outside the surface r > R On the surface r = R Inside the surface r < R Potential due to a Uniform Ring at a Point on its Axis
At a point on its axis At the centre Gravitational Potential Energy Important Points ● Potential energy is a scalar quantity. ● Unit : Joule ● Dimension: [ML^2 T–2] ● Gravitational potential energy is always negative in the gravitational field because the force is always attractive in nature. ● As the distance increases, the gravitational potential energy becomes less negative i.e., it increases. ● If then it becomes zero (maximum) ● In case of discrete distribution of masses ● Gravitational potential energy ● If the body of mass is moved from a point at a distance to a point at distance then change in potential energy
Orbital Speed Period of Revolution Energy of Satellite ● This energy is constant and negative, i.e., the system is closed. The farther the satellite from the earth the greater its total energy. Kepler’s Laws of Planetary Motion First Law (Law of Orbits) ● Every planet moves around the sun in an elliptical orbit with sun at one of the foci. Second Law (Law of Area) ● The line joining the sun to the planet sweeps out equal areas in equal interval of time. i.e. areal velocity is constant.
Third Law (Law of Periods) ● The square of period of revolution of any planet around sun is directly proportional to the cube of the semi-major axis of the orbit. or Velocity of a Planet in Terms of Eccentricity , Geostationary Satellite ● The satellite which appears stationary relative to earth is called geostationary or geosynchronous satellite, communication satellite. (i) It should revolve in an orbit concentric and coplanar with the equatorial plane. (ii) Its sense of rotation should be same as that of earth about its own axis i.e., in anti- clockwise direction (from west to east). (iii) Its period of revolution around the earth should be same as that of earth about its own axis.
(ii) Condition of weightlessness can be experienced only when the mass of satellite is negligible so that it does not produce its own gravity.
Points to Remember (1) When two equal spheres each of radius r are placed in contact with each other, then the force of attraction between the two sphere is inversely proportional to square of radius, i.e.,. It is to be noted that by taking and showing , is not correct because it is against the basic concept which is more important than a numerical value. As inverse square law is fundamental law here and gravitational force is for masses and not for densities. (2) If the value of G becomes 10 times its present value, then we would be crushed to the floor by earth’s attraction. If the value of G becomes 1\10th^ of its present value then the earth’s attraction becomes very weak and in that case we can take jump over a building. (3) When a particle of mass m is placed at the centre of an arc of wire off mass M, radius r, it experiences a gravitational force due to wire given by. (4) If a mass M is split into two parts m and (M – m), which are then separated by a certain distance the gravitational force between two parts is maximum of m/M = 1/2. (5) The acceleration due to gravity is independent of mass, shape, size etc. of the falling body, i.e., there is equal acceleration in lighter and heavier body while falling freely (neglecting the air friction). (6) If the radius of a planet decreases by n%, keeping its mass unchanged, the acceleration due to gravity on its surface increases by 2n%. (7) If the mass of a planet increases by n%, keeping its radius unchanged, acceleration due to gravity on its surface increase by n%. (8) If the density of a planet decreases by x%, keeping its radius unchanged, the acceleration due to gravity decreases by x%. (9) If a hydrogen balloon is released from a height on the surface of moon, it will fall with an acceleration nearly towards the surface of moon. (10) If the same body is dropped from the same height on two different places on the earth where acceleration due to gravity are then (i) velocity on reaching the ground, (ii) Time of fall (11) If a man can jump to a maximum height of on the surface of one planet and at a height on another planet, then. (12) If a man can have a long jump through a maximum horizontal on one planet where acceleration due to gravity is and through on the second planet where acceleration to
(27) At the centre of a uniform circular ring, the gravitational intensity, I = 0 gravitational potential, (28) Due to a uniform circular ring, the gravitational intensity is maximum at a point on the axis of ring whose distance from the centre of ring. (29) The work done to keep the satellite in the given orbit is zero. (30) The centripetal acceleration of the satellite is equal to acceleration due to gravity. (31) The orbital velocity of a satellite is independent of mass of the satellite but depends upon the mass and radius of the planet around which the rotation is taking place. (32) If a body is released from an artificial satellite it does not fall to the earth but will continue orbiting along with the satellite. (33) An astronaut in a satellite cannot use a pendulum clock. However he can use a spring clock or digital clock. (34) If the radius of the orbit of a satellite is made n times the radius of the earth, then its orbital velocity will be times the orbital velocity near the surface of earth. (35) If the altitude of a satellite is made n times the radius of the earth, then its orbital velocity will be times the orbital velocity near the surface of the earth. (36) It two satellites are orbiting in circular orbits of radii. Their orbital speeds will be in the ratio, which is independent of their masses. (37) The orbital velocity of a satellite decreases with an increase in the radius of orbit or increase in height from the surface of earth/planet. For a stationary satellite the orbital velocity increases with the increase in radius orbit or height from the surface of earth. (38) For a satellite, orbiting close to the surface of earth (h << R), the time period of revolution is (39) The angular velocity of a satellite orbiting close to the surface of earth is (40) The orbital velocity of geostationary satellite is. Its height above the surface of earth is about 36000 km. The relative angular velocity of geostationary satellite w.r.t. earth is zero. Geostationary satellites are launched in the equatorial plane. (41) If the gravitational force is inversely proportional to the nth power of distance r, then the orbital velocity of a satellite, and time period is. (42) When a body is projected horizontally with velocity υ, from any height from the surface of earth, then the following possibilities are there:
(i) If the body fails to revolve around earth and finally falls to the surface of earth. (ii) If the body will revolve around the earth in circular orbit. (iii) If the body will revolve around the earth in elliptical orbit. (iv) If the body will escape from the gravitational field of earth. (v) If the body will escape, following a hyperbola path. (43) If a body is released from a height equal to n-times the radius of earth, then its striking velocity on the surface of earth is. (44) (a) When the height of a satellite is increased, its potential energy will increase and kinetic energy will decreases. (b) When the velocity of a satellite is increased, its total energy will increase and it will start orbiting in a circular path of larger radius. (c) For a satellite orbiting in a circular orbit, the value of potential energy is always greater than its kinetic energy. (45) Total energy of a satellite in an orbit (46) Increase in gravitational potential energy of the particle of mass m at height h is . (47) When a particle of mass m is projected vertically upwards with a speed u, then the maximum height attained by the particle is given by. (48) The value of escape velocity on the surface of earth = 11.2 km/s and that of moon = 2.31 km/s. (49) The value of escape velocity is independent of (i) the mass of body (ii) the angle and direction of projection. (50) If a body falls freely from infinity, then it will be striking the earth with velocity 11.2 km/s. (51) The minimum energy required for the escape of a body from the surface of earth (52) The escape velocity/ (orbital velocity close to earth). (53) When a body is projected from the surface of earth with a velocity less than the escape velocity, the sum of its gravitational potential energy and kinetic energy is negative. (54) If the total energy of the satellite becomes positive, the satellite will escape from the gravitational pull of the earth. (55) For the orbiting satellite, the kinetic energy is less than its potential energy. When kinetic energy of a satellite becomes equal to its potential energy, the satellite escapes out of the gravitational field of earth. (56) If the velocity of a satellite orbiting close to the earth is increased by 41.4% (i.e. made times)
