THEORY OF PLANETARY MOTION-

Geocentric Theory of Planetary Motion- The Earth is the center of the Universe and all planets, Sun, Stars revolve around its orbit in circular path.

After some time Heliocentric Theory were proposed by the great Indian mathematician and astronomer, Aryabhata. Heliocentric Theory says that the sun was center and all the planets revolve around of Sun.

KEPLER ‘S LAW OF PLANETARY MOTION-

There are three laws-

  1. Kepler ‘s First Law of Orbits- There are all planets are revolved in elliptical orbits with Sun is situated at 1 of the foci of ellipse.

This law find that the distance between the Earth and the Sun are continuously changing. The closest point ‘P’ called perihelion and the farthest point ‘A‘ called aphelion.

2. Kepler ‘s Second Law of Areas- The Planet sweep out equal areas in equal times as it orbit the sun. That means A1 = A2 = A3, where a is area. The area velocity of the Planet around the Sun is constant.

3. Kepler ‘s Third Law of Period- The square of time period of revolution of Planet around the Sun is directly proportional to cube of semi-major axis of its elliptical orbit. That means T2 ∝ a3.

They all Kepler laws are applicable in Solar System, Artificial Satellites etc.

UNIVERSAL LAW OF GRAVITATION-

The every body attract to each other with a force whose magnitude is directly proportional to the product of their masses and inversely proportional to the square of distance between their centers.

Gravitational force, F=G(m1 m2)/r2 , dimension is [M-1L3T-2].

Gravitational force always attractive force(not like electric or magnetic which also repel each other). This is conservative force. The value of G does not depend on nature and size of masses.

After derivation(derivation in question section), we get F12=-F21, so gravitational force is central force.

G=6.67 * 10-11 N-m/kg

Vector form of Newton ‘s Law of Gravitation– After small derivation (derivation in question section), we get F12=-F21.

The value of G (gravitational constant) is find by the English Scientist Henry Cavendish in 1798. After the cavendish ‘s experiment(experiment in question section), we get G=6.67* 10-11 N-m2/kg2. The G is universal constant, so its value does not effect by change in temperature, pressure, heat etc.

PRINCIPAL OF SUPERPOSITION-

Principal of Superposition- If many gravitational forces(F) are acting on a same object, then the total gravitational forces are equal to the vector sum of these forces. After derivation(derivation in question section ), we get F=-Gm1[m2/r212 r21 + m3/r213 r31+…. mn/r21n  rn1], where G is gravitational constant, m is masses, r is vector form, F is force.

ACCELERATION DUE TO GRAVITY OF EARTH-

ACCELERATION DUE TO GRAVITY(g)– It is a velocity of a body which gain when a body fall freely toward the Earth. The value g is 9.8 m/sec2 on the Earth surface.

This value is differ place to place on Earth and also varies when we increase height form earth surface or when we increase depth from the surface of the Earth.

Acceleration due to gravity on the Earth surface-After derivation(derivation in question section), we get the value of acceleration due to gravity on earth surface is g=GM/r2 , where M is mass of Earth(6.4*1024 kg), R is radius of earth(6.4*106 m) and G is gravitation constant. The value of g on earth surface is equal to 1/6 time of value of g on moon ‘s surface. The value of g only depend on mass and radius of Earth.

Acceleration due to gravity above the Earth surface- After derivation(derivation in question section), we get the value of acceleration due to gravity above the Earth surface is gh= g(1-2h/R) ,where h is height form earth surface, R is radius of Earth(6.4*106 m). The value of g is inversely proportional to the h.

Acceleration due to gravity below the Earth surface-After derivation(derivation in question section), we get the value of acceleration due to gravity below the Earth surface is gd= g(1-d/R), where d is depth, R is radius of Earth(6.4*106 m).

INTENSITY OF GRAVITATIONAL FIELD(I) ON POINT-

It is the force which experienced by a unit mass placed under a particular point. I=GM/r2.

Gravitational Field- It is the region where gravitational force can affect(just like magnetic or electric field).

GRAVITATIONAL POTENTIAL-

Gravitational Potential- It is a work done per unit mass to bring an object from infinity to a specific point in a gravitational field. It SI unit is J/kg, C.G.S unit is erg/g. I=-dV/dr, where negative sign indicates direction. V=W/m , where W is work done, m is mass of infinity to point and its dimension formula is [M0L2T-2].

Expression of Gravitational Potential(V)-

After derivation(derivation in question section), we get Gravitational Potential, V=-GM/r.

GRAVITATIONAL POTENTIAL ENERGY(U)-

It is amount of work done in bringing the given body form infinity to that point against the gravitational force.

After derivation(derivation in question section), we get U=m*(-GM/r) or Gravitational potential energy = gravitational potential * Mass of body.

ESCAPE SPEED-

Escape Speed- It is a minimum speed required for a body to break free form gravitational field without propulsion.

Different expressions of Escape Speed- After derivation(derivation in question section), we get ve=2g R1/2 , where ve is escape speed or velocity, R is radius of Earth(6.4*106 m) and also ve =R under root 8/3 πGp, where p is density of earth.

Escape Speed according to principal of conservation of energy, is Minimum speed (vi)min= (2gR)1/2

EARTH SATELLITES-

Satellite is a body which continuously revolving around the Earth. eg-Moon is natural satellite and INSAT-1B is artificial satellite of Earth.

Equation of time period of Earth ‘s Satellite- After derivation(derivation in question section), we get T=2π(R/g)1/2 , where T is time period, R is radius of Earth(6.4*106 m), g is acceleration due to gravity, 9.8 m/s2 .

ENERGY OF AN ORBITING SATELLITE-

When satellite revolve, then both potential energy and kinetic energy possess by satellite.

The expression of Kinetic Energy(K) ,Potential Energy(U), and Total Energy of satellite is-

  1. Kinetic Energy(due to motion on orbit)- K= GMm/2(R+h),
  2. Potential Energy(due to against the gravitational pull on Earth)- U= -GMm/R+h,
  3. Total Energy of satellite- E= -GMm/2(R+h),

Where G is gravitational constant(6.67 * 10-11 N-m/kg), M is mass of Earth(6.4*1024 kg), m is mass of satellite, R is radius of earth(6.4*106 m) and h is height of satellite. The satellite are always Finite distance of Earth, so it energy always be negative.

These all are the notes of chapter 1 in physics. And after some time you get important questions and derivation HERE. *#THANKS FOR VISITING, VISIT AGAIN#* 😊

Important Questions And Answers Which Strong Your Concept Are-

  1. What is the Geocentric Theory of Planetary Motion?
    • Answer- The Earth is the center, and all planets, the Sun, and stars revolve around it in circular paths.
  2. According to the Geocentric Theory, which celestial body is at the center of the Universe?
    • Answer- The Earth.
  3. Who proposed the Heliocentric Theory?
    • Answer- Aryabhata.
  4. What does the Heliocentric Theory state?
    • Answer- Heliocentric Theory: The Sun is the center, and all planets revolve around it.
  5. How many laws are there in Kepler’s theory of planetary motion?
    • Answer- Three.
  6. What does Kepler’s First Law state about planetary orbits?
    • Answer- First Law: Planets move in elliptical orbits with the Sun at one focus.
  7. Define perihelion and aphelion.
    • Answer- Perihelion and Aphelion: Closest and farthest points of a planet from the Sun, respectively.
  8. What is the key concept of Kepler’s Second Law of Areas?
    • Answer- Second Law: Planets sweep out equal areas in equal times; area velocity is constant.
  9. According to Kepler’s Third Law, how is the time period of a planet’s orbit related to the semi-major axis of its orbit?
    • Answer- Third Law: The square of a planet’s orbital period is proportional to the cube of the semi-major axis.
  10. What does the Universal Law of Gravitation state?
    • Answer- Law: Every object attracts others with a force proportional to their masses and inversely proportional to the square of their distance.
  11. Write the formula for gravitational force and its dimensions.
    • Answer- Formula= F = G (m1m2)/r2, dimension= [M1L3T-2]
  12. What is the value of the gravitational constant G?
    • Answer- Value of G= 6.67* 10-11 N-m2/kg2
  13. Who determined the value of G experimentally, and in which year?
    • Answer- Henry Cavendish in the year 1798.
  14. What is the Principle of Superposition in gravitation?
    • Answer- Definition: Total gravitational force on an object equals the vector sum of individual forces.
  15. What is the acceleration due to gravity on the Earth’s surface?
    • Answer- Value of g: 9.8 m/s2 on Earth’s surface.
  16. How does the value of g vary with height above the Earth’s surface?
    • Answer- Variation with Height: gh= g(1−2h/R) decreases with height.
  17. What is the formula for g at a depth below the Earth’s surface?
    • Answer- ariation with Depth: gd= g(1−d/R), decreases with depth.
  18. Define gravitational field intensity.
    • Answer- Gravitational Field Intensity: Force experienced by a unit mass; I= GM​/r2.            
  19. What is gravitational potential? What is its SI unit?
    • Answer- ravitational Potential: Work done per unit mass to bring an object from infinity to a point. SI unit: J/kg.
  20. Write the formula for gravitational potential energy.
    • Answer- Formula for Potential Energy: U= -GMm/r
  21. What is escape speed?
    • Answer- Definition: Minimum speed needed to escape a gravitational field without propulsion.
  22. Write the formula for escape speed in terms of g and R.
    • Answer- Formula: ve​= (2gR)1/2
  23. What is a satellite? Give one example of a natural and an artificial satellite.
    • Answer- Definition and Examples: Bodies revolving around Earth. Natural: Moon, Artificial: INSAT-1B.
  24. Write the formula for the time period of an Earth satellite.
    • Answer- Time Period Formula: T=2π (R/g)1/2
  25. What are the expressions for kinetic energy, potential energy, and total energy of an orbiting satellite?
    • Answer- Energy Expressions:
      • Kinetic Energy: K=GMm/2(R+h)​.
      • Potential Energy: U=−GMm/R+h​.
      • Total Energy: E=−GMm/2(R+h)​.