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Unit 8: MOTION IN ORBITS
Unit 8: MOTION IN ORBITS
Topic Area: MOTION IN FIELD
Sub-Topic Area: PLANETARY MOTION
Key unit competence: By the end of this unit, I should be able to evaluate Newton’s law of gravitation and apply Kepler’s laws of planetary motion.
Unit Objectives:
By the end of this unit learners will be able to;
◊ Explain the terms, concept and characteristics of waves properly.
◊ Explain the properties of waves.
◊ Explain the behavior of waves in vibrating strings and applications of waves properly.
8.1. INTRODUCTION
Gravity is the mysterious force that makes everything fall down towards the Earth. But after research it has turned out that all objects have gravity. It’s just that some objects, like the Earth and the Sun, have a stronger gravity than others. How much gravity an object has depends its mass. It also depends on how close you are to the object. The closer you are, the stronger the gravity.
Gravity is very important to our everyday lives. Without Earth’s gravity we would fly right off it. If you kicked a ball, it would fly off forever. While it might be fun to try for a few minutes, we certainly can’t live without gravity. Gravity also is important on a larger scale. It is the Sun’s gravity that keeps the Earth in orbit around the Sun. Life on Earth needs the Sun’s light and warmth to survive. Gravity helps the Earth to stay at just the right distance from the Sun, so it’s not too hot or too cold.
8.2. NEWTON’S LAW OF GRAVITATION
This is also called the universal law of gravitation or inverse square law. It states that “the gravitational force of attraction between two masses m1 and m2 is directly proportional to the product of masses and inversely proportional to the square of their mean distance apart.”
Remember two objects exert equal and opposite force of gravitation on each other.
where G = 6.7 × 10–11 Nm2/kg2 and is called the universal gravitational constant.
Notes:
• The value of G in the laboratory was first determined by Cavendish using the torsional balance.
• The value of G is 6.67 × 10–11 N–m2 kg–2 in S.I.
• Dimensional formula [M –1 L3 T –2].
• The value of G does not depend upon the nature and size of the bodies.
• It also does not depend upon the nature of the medium between the two bodies.
• As G is very small hence gravitational forces are very small, unless one (or both) of the masses is huge.
Properties of Gravitational Force
• It is always attractive in nature while electric and magnetic force can be attractive or repulsive.
• It is independent of the medium between the particles while electric and magnetic forces depend on the nature of the medium between the particles.
• It holds good over a wide range of distances. It is found true for interplanetary to interatomic distances.
• It is a central force, i.e. it acts along the line joining the centers of two interacting bodies.
• 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, i.e.
• On the contrary, nuclear force is a many-body interaction.
• It is the weakest force in nature : As Fnuclear > Felectromagnetic > Fgravitational.
• The ratio of gravitational force to electrostatic force between two electrons is of the order of 10–43.
• 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.
• 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.
8.3. KEPLER’S LAWS OF PLANETARY MOTION
Planets are large natural bodies rotating around a star in definite orbits. The planetary system of the star sun, called solar system, consists of nine planets, viz. mercury, venus, earth, mars, jupiter, saturn, uranus, neptune and pluto. Out of these planets mercury is the smallest, closest to the sun. jupiter is the largest and has the maximum number of moons. Venus is closest to the earth and the brightest planet. Kepler, after a life time study, worked out three empirical laws which govern the motion of these planets and are known as Kepler’s laws of planetary motion. These are stated below.
8.4. VERIFICATION OF KEPLER’S THIRD LAW OF PLANETARY MOTION
Assuming that a planet’s orbit is circular (which is not exactly correct but is a good approximation in most cases), then the mean distance from the sun is constant –radius. Suppose, a planet of mass m2 moving around the sun of mass m1. If the motion of the planet is circular, there are two types of forces:
(a) Gravitational force of attraction F1 between the sun and the planet,
(b) Centripetal force F2 responsible for keeping the planet moving in a circular motion around the sun.
For the planet to move around the sun in orbit of constant radius: F1 = F2
EXAMPLE 8.1:
The distance of a planet from the sun is 5 times the distance between the earth and the sun. What is the time period of revolution of the planet?
8.5. ACCELERATION DUE TO GRAVITY AT THE SURFACE OF THE EARTH
where M = mass of the earth and R = radius of the earth. If g is the acceleration due to gravity, then the force on the body due to earth is given by
Notes:
• 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. • The greater the value of (M/R2) or rR, greater will be the value of g for that planet.
• Acceleration due to gravity is a vector quantity and its direction is always towards the centre of the planet.
• Dimensions of [g] = [LT –2]
• Average value of g is taken as 9.8 m/s2 or 981 cm/s2, on the surface of the earth at mean sea level.
• In general, the value 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.
8.6. VARIATION OF ACCELERATION DUE TO GRAVITY WITH HEIGHT
Consider a particle placed at a height h above the surface of the earth where acceleration due to gravity is g′ as shown on the figure below.
Acceleration due to gravity at the surface of the earth
Acceleration due to gravity at height h from the surface of the earth
From equations 8-8 and 8-9:
Notes:
• As we go above the surface of the earth, the value of g decreases because
. This expression can be plotted on the graph as:
• If r = ∞ then g′ = 0, i.e. at infinite distance from the earth, the value of g becomes zero.
• If h << R, i.e. height is negligible in comparison to the radius. Then from equation (iii), we get
• If h << R, then decrease in the value of g with height:
EXAMPLE 8.5:
The acceleration of a body due to the attraction of the earth (radius R) is g. Find the acceleration due to gravity at a distance 2R from the surface of the earth.
EXAMPLE 8.6:
Find the height of the point above the earth’s surface, at which acceleration due to gravity becomes 1% of its value at the surface is (Radius of the earth = R).
8.7. VARIATION OF GRAVITY WITH DEPTH
Notes:
• The value of g decreases on going below the surface of the earth. From equation 8-12, we get g′ ∝ (R – d). So it is clear that if d increases, the value of g decreases.
Combining the graphs for variation of acceleration due to gravity below and above the surface of the earth will give the graph as shown
• At the centre of earth d = R \ g′ = 0, i.e., the acceleration due to gravity at the centre of earth becomes zero.
• Decrease in the value of g with depth
• The rate of decrease of gravity outside the earth (h << R) is double to that of inside the earth.
EXAMPLE 8.7:
Weight of a body of mass m decreases by 1% when it is raised to height h above the earth’s surface. If the body is taken to a depth h in a mine, what is the change in its weight?
EXAMPLE 8.8:
What is the depth at which the effective value of acceleration due to gravity
8.8. VARIATION IN G DUE TO ROTATION OF EARTH
i.e., there is no effect of rotational motion of the earth on the value of g at the poles.
. Work done in planetary motion by gravity of sun is zero. This is because force acting on the planet is towards the center of sun while the direction of displacement is
forward. Since the two are perpendicular, work done = becomes zero. This is because, q = 90° ⇒ cos 90° = 0. EXAMPLE 8.9: What is the angular velocity of the earth with which it has to rotate so that acceleration due to gravity on 60° latitude becomes zero? (Radius of earth = 6400 km. At the poles g = 10 ms–2)
8.9. VARIATION OF ‘g’ DUE TO SHAPE OF EARTH
Earth is elliptical in shape. It is flattened at the poles and bulged out at the equator.
The equatorial radius is about 21 km longer than polar radius.
Therefore, the weight of body increases as it is taken from equator to the pole.
8.10. ROCKETS
Spacecraft Propulsion
Spacecraft Propulsion is characterized in general by its complete integration within the spacecraft (e.g. satellites). Its function is to provide forces and torques in (empty) space to:
• transfer the spacecraft: used for interplanetary travel
• position the spacecraft: used for orbit control
• orient the spacecraft: used for altitude control
The jet propulsion systems for launching rockets are also called primary propulsion systems. Spacecrafts, e.g. satellites, are operated by secondary propulsion systems.
Characteristics of Spacecraft Propulsion Systems
In order to fulfill altitude and orbit operational requirements of spacecraft, spacecraft propulsion systems are characterized by:
• Very high velocity increment capability (many km/s)
• Low thrust levels (1 mN to 500 N) with low acceleration levels
• Continuous operation mode for orbit control
• Pulsed operation mode for altitude control
• Predictable, accurate and repeatable performance (impulse bits)
• Reliable, leak-free long time operation (storable propellants)
• Minimum and predictable thrust exhaust impingement effects
Classification of Propulsion Systems
Spacecraft propulsion can be classified according to the source of energy utilized for the ejection of propellant:
• Chemical propulsion use heat energy produced by a chemical reaction to generate gases at high temperature and pressure in a combustion chamber. These hot gases are accelerated through a nozzle and ejected from the system at a high exit velocity to produce thrust force.
• Electric propulsion uses electric or electromagnetic energy to eject matter at high velocity to produce thrust force.
• Nuclear propulsion uses energy from a nuclear reactor to heat gases which are then accelerated through a nozzle and ejected from the system at a high exit velocity to produce thrust force.
Notes:
• While chemical and electric systems are used for the propulsion of today’s spacecrafts, nuclear propulsion is still under study. Therefore, only chemical and electric propulsion will be dealt with in this book.
8.12. SATELLITES
A satellite is an artificial or a natural body placed in orbit round the earth or another planet in order to collect information or for communication. Communication satellites are satellites that are used specifically to communicate. Part of that communication will be the usual commands and signals we get from any satellite. The payload of the satellite consists of huge collection of powerful radio transmitters and a big dish or something like that, to enable it to talk to things on the ground. And we’ll use them to transmit TV signals, to transmit radio signals, and in some cases, it might be to be transmit internet signals. So, all of that gets turned into radio somehow and transmitted up into space and then bounced back down somewhere else.
There is only one main force acting on a satellite when it is in orbit, and that is the gravitational force exerted on the satellite by the Earth. This force is constantly pulling the satellite towards the center of the Earth.
A satellite doesn’t fall straight down to the Earth because of its velocity. Throughout a satellite’s orbit there is a perfect balance between the gravitational force due to the Earth, and the centripetal force necessary to maintain the orbit of the satellite.
8.12.1. Orbital Velocity of Satellite.
Orbital velocity of a satellite is the velocity required to put the satellite into its orbit around the earth. For revolution of satellite around the earth, the gravitational pull provides the required centripetal force.
Notes:
• Orbital velocity is independent of the mass of the orbiting body and is always along the tangent of the orbit, i.e. satellites of different masses have the same orbital velocity, if they are in the same orbit.
• Orbital velocity depends on the mass of central body and radius of orbit.
• For a given planet, greater the radius of orbit, lesser will be the orbital velocity of the satellite
• Orbital velocity of the satellite when it revolves very close to the surface of the planet:
• Close to the surface of planet,
It means that if the speed of a satellite orbiting close to the earth is made
times (or increased by 41%) then it will escape from the gravitational field. • If the gravitational force of attraction of the sun on the planet varies as
then the orbital velocity varies as: 
EXAMPLE 8.10:
Two satellites A and B go round a planet P in circular orbits having radii 4R and R respectively. If the speed of the satellite A is 3V, what is the speed of the satellite B?
EXAMPLE 8.11:
A satellite is moving around the earth with speed v in a circular orbit of radius r. If the orbit radius is decreased by 1%, what is its speed?
8.12.2. Time Period of Satellite
It is the time taken by satellite to go once around the earth.
Notes:
• From
, it is clear that time period is independent of the mass of orbiting body and depends on the mass of central body and radius of the orbit.
EXAMPLE 8.12:
A satellite is launched into a circular orbit of radius ‘R’ around earth while a second satellite is launched into an orbit of radius 1.02 R. What is the percentage difference in the time periods of the two satellites?
EXAMPLE 8.13:
What is the periodic time of a satellite revolving above Earth’s surface at a height equal to R, where R is the radius of Earth?
8.12.3. Height of Satellite
As we know, time period of satellite
By squaring and rearranging both sides,
By knowing the value of time period we can calculate the height of satellite the surface of the earth.
EXAMPLE 8.14:
Given radius of earth ‘R’ and length of a day ‘T’, what is the height of a geostationary satellite?
From the expression
EXAMPLE 8.15:
A satellite is revolving round the earth in circular orbit at some height above surface of the earth. It takes 5.26 × 103 seconds to complete a revolution while its centripetal acceleration is 9.32 m/s2. What is the height of satellite above the surface of earth? (Radius of the earth 6.37 × 106 m)
8.12.4. Geostationary Satellite
The satellite which appears stationary relative to earth is called geostationary or geosynchronous satellite, e.g. communication satellite.
A geostationary satellite always stays over the same place above the earth. Such a satellite is never at rest. It appears stationary due to its zero relative velocity w.r.t. that place on earth.
The orbit of a geostationary satellite is known as the parking orbit.
Notes:
• It should revolve in an orbit concentric and coplanar with the equatorial plane.
• 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).
• Its period of revolution around the earth should be the same as that of earth about its own axis.
T = 24 hr = 86400 s• Height of geostationary satellite
Substituting the value of G and M we get R + h = r = 42000 km = 7R
Height of geostationary satellite from the surface of earth, h = 6R = 36000 km • Orbital velocity of geostationary satellite can be calculated by
• Substituting the value of G and M, we get v = 3.08 km/s
8.12.5. Energy of Satellite
When a satellite revolves around a planet in its orbit, it possesses both potential energy (due to its position against gravitational pull of earth) and kinetic energy (due to orbital motion).
Notes
• Kinetic energy, potential energy or total energy of a satellite depends on the mass of the satellite and the central body and also on the radius of the orbit.
• From the above expressions we can say that
Kinetic energy (K) = – (Total energy)
Potential energy (U) = 2 (Total energy)
Potential energy (K) = – 2 (Kinetic energy)
• If the orbit of a satellite is elliptical, then
(a) Total energy
constant; where a is semi-major axis. (b) Kinetic energy (K) will be maximum when the satellite is closest to the central body (at perigee) and maximum when it is farthest from the central body (at apogee).
(c) Potential energy (U) will be minimum when kinetic energy is maximum, i.e. when satellite is closest to the central body (at perigee). Potential energy is maximum when kinetic energy is minimum, i.e. the satellite is farthest from the central body (at apogee).
• Binding Energy: Total energy of a satellite in its orbit is negative. Negative energy means that the satellite is bound to the central body by an attractive force and energy must be supplied to remove it from the orbit to infinity. The energy required to remove the satellite from its orbit to infinity is called Binding Energy of the system, i.e.
EXAMPLE 8.17:
What is the Potential energy of a satellite having mass ‘m’ and rotating at a height of 6.4 × 106 m from the earth’s center?
EXAMPLE 8.18:
Two satellites are moving around the earth in circular orbits at height R and 3R respectively, R being the radius of the earth. What is the ratio of their kinetic energies?
EXERCISE 8-1
1. The distance of Neptune and Saturn from sun are nearly 1013 and 1012 meters respectively. Assuming that they move in circular orbits, what will be their periodic times in the ratio?
2. A spherical planet far out in space has a mass M0 and diameter D0. A particle of mass m falling freely near the surface of this planet will experience an acceleration due to gravity which is equal to g. Derive the expression of g in terms of D.
3. At surface of earth, weight of a person is 72 N. What is his weight at height R/2 from surface of earth (R = radius of earth)?
4. Assuming earth to be a sphere of a uniform density, what is the value of gravitational acceleration in a mine 100 km below the earth’s surface (Given R = 6400 km)?
5. If the gravitational force between two objects was proportional to 1/R; where R is separation between them, then a particle in circular orbit under such a force would have its orbital speed v proportional to which value?
6. An earth satellite S has an orbital radius which is 4 times that of a communication satellite C. What is its period of revolution?
8.13 TYPES AND APPLICATIONS OF SATELLITE SYSTEMS
Four different types of satellite orbits have been identified depending on the shape and diameter of each orbit:
• GEO (Geo-stationary earth orbit)
• MEO (medium earth orbit)
• LEO (Low earth orbit) and
• HEO (Highly elliptical orbit)
GEO (geostationary orbit)
A geostationary orbit or geosynchronous equatorial orbit (GEO) has a circular orbit 35,786 kilometers above the Earth’s equator and following the direction of the Earth’s rotation. An object in such an orbit has an orbital period equal to the Earth’s rotational period (one sidereal day) and thus appears motionless, at a fixed position in the sky, to ground observers.
Most common geostationary satellites are either weather satellites or communication satellites relaying signals between two or more ground stations and satellites that broadcast signals to a large area on the planet. All radio and TV, whether satellite etc. are launched in this orbit.
Advantages of Geo-Stationary Earth Orbit
1. It is possible to cover almost all parts of the earth with just 3 geo satellites.
2. Antennas need not be adjusted every now and then, but can be fixed permanently.
3. The life-time of a GEO satellite is quite high usually around 15 years.
Disadvantages of Geo-Stationary Earth Orbit
1. Larger antennas are required for northern/southern regions of the earth.
2. High buildings in a city limit the transmission quality.
3. High transmission power is required.
4. These satellites cannot be used for small mobile phones.
5. Fixing a satellite at Geo stationary orbit is very expensive.
LEO (Low Earth Orbit)
Satellites in low Earth orbits are normally military reconnaissance satellites that can locate out tanks from 160 km above the
1. The antennas can have low transmission power of about 1 watt.
2. The delay of packets is relatively low.
3. Useful for smaller foot prints
Disadvantages of Low Earth Orbit
1. If global coverage is required, it requires at least 50-200 satellites in this orbit.
2. Special handover mechanisms are required.
3. These satellites involve complex design.
4. Very short life: Time of 5-8 years. Assuming 48 satellites with a life-time of 8 years each, a new satellite is needed every 2 months.
5. Data packets should be routed from satellite to satellite. MEO (Medium Earth Orbit) or ICO (Intermediate Circular Orbit) Medium Earth Orbit satellites move around the earth at a height of 600020000 km above earth’s surface. Their signal takes 50 to 150 milliseconds to make the round trip. MEO satellites cover more earth area than LEOs but have a higher latency. MEOS are often used in conjunction with GEO satellite systems.
Advantages of Medium Earth Orbit
1. Compared to LEO system, MEO requires only a dozen satellites.
2. Simple in design.
3. Requires very few handovers.
Disadvantages of Medium Earth Orbit
1. Satellites require higher transmission power.
2. Special antennas are required.
HEO (Highly Elliptical Orbit)
A satellite in elliptical orbit follows an oval-shaped path. One part of the orbit is closest to the centre of Earth (perigee) and another part is farthest away (apogee). A satellite in this type of orbit generally has an inclination angle of 64 degrees and takes about 12 hours to circle the planet. This type of orbit covers regions of high latitude for a large fraction of its orbital period.
8.14. COSMIC VELOCITY (FIRST, SECOND AND THIRD)
The cosmic velocity is the initial velocity which a body must have to be able to overcome the gravity of another object.
We have:
1. The first cosmic velocity
2. Second cosmic velocity (escape velocity)
3. The third cosmic velocity
4. The fourth cosmic velocity
8.14.1. The first cosmic velocity
As you know the satellites which were sent by a human are orbiting around the Earth. They had to be launched with a very high velocity, namely, with the first cosmic velocity.
This velocity can be calculated using the gravitational force and the centripetal force of the satellite:
Satellites must have extremely high velocity to orbit around the Earth. In fact, satellites go around the Earth at the height h = 160 km in order not to break into the atmosphere.
8.15.2. Second cosmic velocity (escape velocity)
In the previous section we calculated the velocity which a body has to have to go around the Earth, which means that we calculated the value of the first cosmic velocity. Now it is time to give attention to calculating the second cosmic velocity -it is the speed needed to “break free” from the gravitational attraction of the Earth.
In order to understand this issue we should know something about kinetic and potential energy.
This value is calculated using the fact that as the body moves away from the Earth, the kinetic energy decreases and the potential energy increases. At infinity, both the energies are equal to zero, because, when the distance between the body and the Earth increases, the kinetic energy decreases and at infinity, it has the value of 0.
The potential energy at infinity has got the highest value but if we put infinity in the previous formula, we will obtain zero (or an extremely small fraction).
The value of the second cosmic velocity is calculated as follows;
And finally, the practical curiosity.
We can also obtain the value of the second cosmic velocity by multiply the value of the first cosmic velocity by the square root of two.
8.14.3. The third cosmic velocity
The third cosmic velocity is the initial velocity which a body has to have to leave the Solar System and its value is:
At the surface of the Earth, this velocity is about 42 km/s. But due to its revolution, it is enough to launch the body with velocity 16.7 km/s in the direction of this movement.
8.14.4. The fourth cosmic velocity
It is the initial velocity which a body should have to leave the Milky Way.
This velocity is about 350 km/s but since Sun is going around the galaxy center, so it is enough to launch the body with the velocity of 130 km/s in the direction of the Sun’s movement.
1. A satellite A of mass m is at a distance of r from the center of the earth. Another satellite B of mass 2m is at distance of 2r from the earth’s center. What is the ratio of their time periods?
2. Mass of moon is 7.34 × 1022 kg. If the acceleration due to gravity on the moon is 1.4 m/s2, find the radius of moon. Use (G = 6.67 × 10–11 Nm2/kg2).
3. A planet has mass 1/10 of that of earth, while radius is 1/3 that of earth. If a person can throw a stone on earth surface to a height of 90 m, to what height will he be able to throw the stone on that planet?
4. If the distance between centers of earth and moon is D and the mass of earth is 81 times the mass of moon, then at what distance from center of earth the gravitational force will be zero?
5. What is the depth d at which the value of acceleration due to gravity becomes
times the value at the surface? [R = radius of the earth]6. The distance between center of the earth and moon is 384000 km. If the mass of the earth is 6 × 1024 kg and G = 6.67 × 10–11 Nm2/kg2, what is the speed of the moon?
7. One project after deviation from its path, starts moving round the earth in a circular path at radius equal to nine times the radius at earth R, what is its time?
8. A satellite A of mass m is revolving round the earth at a height ‘r’ from the center. Another satellite B of mass 2m is revolving at a height 2r. What is the ratio of their time periods?
Newton’s law of gravitation
This is also called the universal law of gravitation or inverse square law. And sates that “the gravitational force of attraction between two masses m1 and m2 is directly proportional to the product of masses and inversely proportional to the square of their mean distance apart.”
Kepler’s laws of planetary motion
1st Law: This law is called the law of orbits and states that planets move in ellipses with the sun as one of their foci. It can also be stated that planets describe ellipses about the sun as one focus.
2nd Law: This is called the law of areas and states that the line joining the sun and the planet sweeps out equal areas in equal periods of time.
3rd Law: The law of periods states that the square of the periods T of revolution of planets are proportional to the cubes of their mean distances R from the sun.
Verification of Kepler’s third law of planetary motion
Gravitational force of attraction of the sun and the planet
Centripetal force responsible for keeping the planet moving in a circular motion around the sun.
Acceleration due to gravity at the surface of the earth
At the surface of the earth acceleration due to gravity is given by;
This value is constant and it’s average value is taken to be 9.8 m/s2
Variation of acceleration due to gravity with height
The acceleration due to gravity at a point above the surface of the earth is given by;
This value decreases as you move further from the surface of the earth.
Variation of gravity with depth
At a point below the surface of the earth, acceleration due to gravity is given by;
The depth d is measured from the surface of the earth. The value of acceleration due to gravity increases as we move towards the surface. At center of earth g = 0.
Variation in g Due to Rotation of Earth
As the earth rotates, a body placed on its surface moves along the circular path and hence experiences centrifugal force, due to which the apparent weight of the body decreases.
By solving, the acceleration due to gravity is given by;
Variation of gravity g Due to Shape of Earth
The value of acceleration due to gravity will vary depending on someone’s position at the surface of the earth as;
Rockets and spacecraft
A rocket is a device that produces thrust by ejecting stored matter. Spacecraft Propulsion is characterized in general by its complete integration within the spacecraft (e.g. satellites).
Satellites
A satellite is an artificial body placed in orbit round the earth or another planet in order to collect information or for communication.
Orbital Velocity of Satellite
Time Period of Satellite
The period of a satellite is given by;
Height of Satellite
The height atIt is seen that angular momentum of satellite depends on both the mass of orbiting and central body as well as the radius of orbit. Energy of Satellite When a satellite revolves around a planet in its orbit, it possesses both potential energy (due to its position against gravitational pull of earth) and kinetic energy (due to orbital motion). which a satellite is launched is given by;
Geostationary Satellite
The satellite which appears stationary relative to earth is called geostationary or geosynchronous satellite e.g. communication satellite.
Angular Momentum of Satellite
The angular momentum of a satellite is given by;
It is seen that angular momentum of satellite depends on both the mass of orbiting and central body as well as the radius of orbit.
Energy of Satellite
When a satellite revolves around a planet in its orbit, it possesses both potential energy (due to its position against gravitational pull of earth) and kinetic energy (due to orbital motion).
Types and applications of Satellite Systems
• GEO (Geo-stationary earth orbit)
• MEO (medium earth orbit)
• LEO (Low earth orbit) and
• HEO (Highly elliptical orbit)
Cosmic velocity
The first cosmic velocity
Second cosmic velocity
This is also called the escape velocity,
Third cosmic velocity
The third cosmic velocity is the initial velocity which a body has to have to escape the Solar System and its value is given by;
At the surface of the Earth, this velocity is about 42 km/s. But due to its revolution, it is enough to launch the body with velocity 16.7 km/s in the direction of this movement.
8.14.4. The fourth cosmic velocity
It is the initial velocity which a body should have to leave the Milky Way.
v4 = 130 km/s
This velocity is about 350 km/s but since Sun is going around the galaxy center, so it is enough to launch the body with the velocity of 130 km/s in the direction of the Sun’s movement.
Using the Across and Down clues, write the correct words in the numbered grid below.
ACROSS
1.The only natural satellite of Earth.
5. An object in orbit around a planet.
6. The smallest planet and farthest from the Sun.
7. This planet probably got this name due to its red color and is sometimes referred to as the Red Planet.
9. This planet’s blue color is the result of absorption of red light by methane in the upper atmosphere.
10. It is the brightest object in the sky except for the Sun and the Moon.
DOWN
2. Named after the Roman god of the sea.
3. The closest planet to the Sun and the eighth largest.
4. A large cloud of dust and gas which escapes from the nucleus of an active comet.
8. The largest object in the solar system.
1. A satellite A of mass m is at a distance of r from the center of the earth. Another satellite B of mass 2m is at distance of 2r from the earth’s center. What is the ratio of their time periods?
2. Mass of moon is 7.34 × 1022 kg. If the acceleration due to gravity on the moon is 1.4 m/s2, find the radius of moon. Use (G = 6.67 × 10–11 Nm2/kg2).
3. A planet has mass 1/10 of that of earth, while radius is 1/3 that of earth. If a person can throw a stone on earth surface to a height of 90 m, to what height will he be able to throw the stone on that planet?
5. What is the depth d at which the value of acceleration due to gravity becomes
times the value at the surface? [R = radius of the earth]6. The distance between center of the earth and moon is 384000 km. If the mass of the earth is 6 × 1024 kg and G = 6.67 × 10–11 Nm2/kg2, what is the speed of the moon?
7. One project after deviation from its path, starts moving round the earth in a circular path at radius equal to nine times the radius at earth R, what is its time?
8. A satellite A of mass m is revolving round the earth at a height ‘r’ from the center. Another satellite B of mass 2m is revolving at a height 2r. What is the ratio of their time periods?
Newton’s law of gravitation
This is also called the universal law of gravitation or inverse square law. And sates that “the gravitational force of attraction between two masses m1 and m2 is directly proportional to the product of masses and inversely proportional to the square of their mean distance apart.”
Kepler’s laws of planetary motion
1st Law: This law is called the law of orbits and states that planets move in ellipses with the sun as one of their foci. It can also be stated that planets describe ellipses about the sun as one focus.
2nd Law: This is called the law of areas and states that the line joining the sun and the planet sweeps out equal areas in equal periods of time.
3rd Law: The law of periods states that the square of the periods T of revolution of planets are proportional to the cubes of their mean distances R from the sun.
Verification of Kepler’s third law of planetary motion
Gravitational force of attraction of the sun and the planet
Centripetal force responsible for keeping the planet moving in a circular motion around the sun.
Acceleration due to gravity at the surface of the earth
At the surface of the earth acceleration due to gravity is given by;
This value is constant and it’s average value is taken to be 9.8 m/s2
Variation of acceleration due to gravity with height
The acceleration due to gravity at a point above the surface of the earth is given by;
This value decreases as you move further from the surface of the earth.
Variation of gravity with depth
At a point below the surface of the earth, acceleration due to gravity is given by;
The depth d is measured from the surface of the earth. The value of acceleration due to gravity increases as we move towards the surface. At center of earth g = 0
Variation in g Due to Rotation of Earth
As the earth rotates, a body placed on its surface moves along the circular path and hence experiences centrifugal force, due to which the apparent weight of the body decreases. By solving, the acceleration due to gravity is given by; .
Variation of gravity g Due to Shape of Earth
The value of acceleration due to gravity will vary depending on someone’s position at the surface of the earth as;
Rockets and spacecraft
A rocket is a device that produces thrust by ejecting stored matter. Spacecraft Propulsion is characterized in general by its complete integration within the spacecraft (e.g. satellites).
Satellites
A satellite is an artificial body placed in orbit round the earth or another planet in order to collect information or for communication.
Orbital Velocity of Satellite
Time Period of Satellite
The period of a satellite is given by;
Height of Satellite
The height at which a satellite is launched is given by;
Geostationary Satellite
The satellite which appears stationary relative to earth is called geostationary or geosynchronous satellite e.g. communication satellite.
Angular Momentum of Satellite
The angular momentum of a satellite is given by;
It is seen that angular momentum of satellite depends on both the mass of orbiting and central body as well as the radius of orbit.
Energy of Satellite
When a satellite revolves around a planet in its orbit, it possesses both potential energy (due to its position against gravitational pull of earth) and kinetic energy (due to orbital motion).
Types and applications of Satellite Systems
• GEO (Geo-stationary earth orbit)
• MEO (medium earth orbit)
• LEO (Low earth orbit) and
• HEO (Highly elliptical orbit)
Cosmic velocity
The first cosmic velocity
Second cosmic velocity
This is also called the escape velocity,
Third cosmic velocity
The third cosmic velocity is the initial velocity which a body has to have to escape the Solar System and its value is given by;
