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Gravitation is an important topic from JEE Main / JEE Advanced Exam Point of view. Every year there are 1-2 questions asked from this topic. It is very interesting and easy topic, therefore, one can score good marks from this topic. This short notes on Gravitation will help you in revising the topic before the JEE Main & IIT JEE Advanced Exam.
Iit Jee Physics Notes Pdf
Notemonk is an exciting new way to experience books. An easy to use webtool helping students to prepare better by downloading books, taking notes, sharing questions, watching related videos and collaborating with their peers. Free eBook of chemistry for IIT JEE Main and UPTU exams are listed with download links. Readers can download these chemistry study notes PDF files from the download links given below.
You can also download Gravitation notes in the PDF format through the link shared at the end of this article.
Gravitation Notes
The Universal Law of Gravitation and Gravitational Constant
In the universe, everybody attracts every other body with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. This attraction force is known as the gravitational force.
where M1 and M2 are the masses of two bodies, R is the distance between and G is the Gravitational Constant. G = 6.67 × 10-11 Nm2(kg)-2
1. Gravitational force is always attractive in nature.
2. Gravitational force is independent of the nature of the intervening medium.
3. Gravitational force is conservative in nature.
4. It is a central force so it acts along the line joining the center of the two interacting bodies and it obeys inverse square law.
Acceleration due to Gravity of the Earth and its Variation
The gravitational pull exerted by the earth is called gravity. The acceleration produced in a body due to the force of gravity is called as acceleration due to gravity(g).
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, where Me is the mass of earth, and Re is the radius of the earth.
Let the density of the earth is ρ. Then the acceleration due to gravity of the earth is
Variation of Acceleration due to gravity
Due to altitude (h)
The acceleration due to gravity at a height h above the earth's surface is
Thus the value of acceleration due to gravity decreases with the increase in height h.
Due to depth (d)
The acceleration due to gravity at depth d below the earth's surface is
Thus the value of acceleration due to gravity decreases with the increase in depth d and becomes zero at the center of the earth.
Variation of acceleration due to gravity (g') with distance from the center of the earth (R) is as shown below
Due to rotation of the earth about its axis
The acceleration due to gravity at latitude λ is
where ω is the angular speed of rotation of the earth about its axis.
At the equator, λ = 0o, value acceleration due to gravity
At the pole, λ = 90o, value acceleration due to gravity
Thus, the value of acceleration due to gravity increases from the equator to the pole due to the rotation of the earth.
If the earth stops rotating about its axis, ω = 0, the value of g will increase everywhere, except at the poles. But if there is increase in the angular speed of the earth, then except at the poles the value of g will decrease at all places.
Kepler's Laws of Planetary Motion
To explain the motion of planets, Kepler formulated the following three laws.
1. Law of Orbits (First Law): The planets in the solar system revolve in elliptical orbits around the Sun in elliptical orbits with the Sun located at any one of the foci of the elliptical path set by the respective planet.
2. Law of areas (Second Law): The rate of the area swept by the position vector of the revolving planet with respect to the Sun per unit time remains same irrespective of the position of the planets on the set elliptical path. Kepler's second law follows the law of conservation of angular momentum.
According to the Kepler's second law, the areal velocity of the planet is constant. that means the planet is closer to the sun on the elliptical path, it moves faster, thus covering more path-area in the given time.
3. Law of periods (Third Law): The square of the period of revolution around the sun of a planet is proportional to the cube of the semimajor axis of its orbit-path around the sun.
Gravitational Field and Potential Energy
Gravitational Field (E) - It is the space around a material body in which its gravitational pull can be experienced by other bodies. The intensity of the gravitational field at a point due to a body of mass M, at a distance r from the center of the body, is
Gravitational Potential (V) - Gravitational Potential at a point in the gravitational field of a body is defined as the amount of work done in bringing body of unit mass from infinity to that point.
Gravitational potential (V) is related with gravitational Field (E) as
Gravitational Potential Energy- The Gravitational potential energy of a body at a point in a gravitational field of another body is defined as the amount of work done in bringing the given body from infinity to that point.
The gravitational potential energy of mass m in the gravitational field of mass M at a distance r from it is
Satellite and its Velocity
Satellite is a natural or artificial body describing the orbit around a planet under its gravitational attraction.
Escape velocity- The velocity of the object needed for it to escape the earth’s gravitational pull is known as the escape velocity of the earth. The escape velocity of the earth is,
Orbital velocity- The orbital velocity of the satellite revolving around the earth at a height h is
When the satellite is orbiting close to the earth's surface, h << Re, then the orbital velocity of the satellite is,
For a point close to earth's surface the escape velocity and orbital velocity related as,
Time period of a satellite
Time period is the time taken by satellite to complete one revolution around the earth,
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When the satellite is orbiting close to the earth's surface, h << Re, then
Energy of an orbiting satellite
The kinetic energy of a satellite is,
The potential energy of a satellite is,
The total energy of the satellite is,
Geostationary Satellite- A satellite which revolves around the earth in its equatorial plane with the same angular speed and in the same direction as the earth rotates about its own axis is called a geostationary satellite.
1. They have a fixed height of 3600 km from the earth’s surface.
2. They revolve in an orbit oriented in the equatorial plane of the earth.
3. They have their rotation same as that of earth about its own axis i.e., from west to east.
4. The period of revolution around the earth is the same as that of the earth about its own axis.
Polar satellite- A satellite revolves in a polar orbit is called a polar satellite.
1. These satellites have their orbit such that they pass the north and the south pole once every 24 hours. They revolve around the earth along the meridian lines.
2. They are situated at an altitude much lower than the geostationary satellites(850 km).
3. They are therefore capable of providing more detailed info about the clouds and storms.
Weightlessness- When the body is unsupported and no force is working on your body, it is the experience of weightlessness.
When an object is in free fall with an acceleration equal to the acceleration of earth, the object is said to be weightless as there is no force acting on it.
Short Notes on Gravitation, Download PDF!
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Vector is one of the fundamentals for the study in other areas of mathematics and of vital importance in physics. Thus, it becomes one of the most important topics in JEE Main, JEE Advanced and other engineering entrance examinations. Every year one can expect 4-5 questions from the topic along with some questions asked in coherence with other topics. Download the Vector short notes pdf from the link given at the end of the article.
1. Vector Quantities
Vectors are those quantities which are described by the magnitude of the quantity and its direction.
2. Linearly Independent and Dependent Vectors
A set of vectors is said to be linearly independent if
A set of vectors is said to be linearly dependent if there exist scalars x1 , x2 , …, xn, not all zero such that
3. Collinearity
(i) Two vectors and are collinear ⇔ for some scalar λ.
(ii) Three vectors, and are collinear, if there exists scalars x, y, z such that where x+y+z=0
Also the points A, B, C are collinear if for some scalar λ.
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4. Coplanarity
(i) Three vectors s, and are coplanar if one of them is a linear combination of the other two if there exist scalars x and y such that
(ii) Four vectors , and are coplanar if scalars x, y, z, w not all zero simultaneously such that
where x + y + z + w = 0.
5. Scalar or Dot product
The scalar product of two vectors is given by (0≤θ≤π) where θ is the angle between
5.1 Properties of the Scalar Product
(i)
(ii) Two vectors make an acute angle with each other , an obtuse angle if and are inclined at a right angle if .
(iii) Projection of
(iv) Projection of
(v) Components of a vector r in the direction of a vector a and perpendicular to vector a are respectively.
(vi) If are three unit vectors along three mutually perpendicular lines, then
(vii) and
6. Vector or Cross Product
The vector product of two vectors is given by where θ is the angle between the vectors and is the unit vector perpendicular to .
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6.1 Properties of the Vector Product
(i)
(ii)if the vectors are either along the direction or opposite in the direction.
(iii)
(iv) then
(v) The vector perpendicular to both is given by
(vi) The unit vector perpendicular to the plane of is
(vii) If three unit vectors are along three mutually perpendicular lines, then they follow the circular rule of cross product
(viii) If are collinear and non-zero vectors then
(ix)(a) The area of a triangle if adjacent sides are is given by
(b) The Area of a parallelogram if adjacent sides are is given by
(c) The Area of a parallelogram if diagonals are is given by
7. Scalar Triple Product
If be there vectors, there called the scalar triple product of these three vectors.
Note: The scalar triple product is usually written as and termed as the box a,b,c
7.1 Properties of the scalar triple product
(i)
(ii)
(iii) If λ is a scalar then
(iv) If
(v) The value of the scalar triple product, if two of its vectors are equal, is zero i.e.,
(vi)
(vii) The volume of the parallelepiped whose adjacent sides are represented by the vectors
(viii) The volume of the tetrahedrane whose adjacent sides are represented by the vectors
(ix) The volume of the triangular prism whose adjacent sides are represented by the vectors
(x) If then are coplanar.
(xi) If then are coplanar
(xii) Three vectors form a right handed or left handed system according to as
(xiii)
(xiv)
8. Vector Triple Product
The vector triple product of three vectors is the vector and
Also,
Clearly
Equality holds if either of the vectors is zero or all the three vectors are collinear or all three vectors are mutually perpendicular to each other.
9. The scalar product of four vectors
If are four vectors then is called the scalar product of four vectors.
This relation is known as Lagrange’s Identity.
10. Vector product of four Vectors
If are four vectors, the products are called vector products four vectors.
i.e., Also,
An expression for any vector, in space, as a linear combination of three non-coplanar vectors
11. Reciprocal System of Vectors
If be three non-coplanar vectors, then the three vectors are defined by the equations are called reciprocal system of vectors to the vectors
11.1 Properties of Reciprocal system of Vectors
(i)
(ii)
(iii)
(iv)
(v) The system of three mutually perpendicular unit vectors is its own reciprocal.
12. APPLICATION IN GEOMETRY
(1) The bisectors of the angles between the lines
‘+’ sign for internal bisector and ‘-‘ sign for external bisector.
(2)Section Formula: If are the position vectors of A and B and be the position vector of the point X which divides to join of A and B in the ratio m:n then
‘+’ sign takes for internal
‘-‘ sign takes for external.
(3) If be the position vectors of ΔABC and be the position vector of the centroid of ΔABC. Then
(4)The equation of a straight line
(i) Vector equation of the straight line passing through origin and parallel to is given by where t is scalar.
(ii) Vector equation of the straight line passing through and parallel to is given by where t is scalar.
(iii) Vector equation of the plane passing through isWhere s & t are scalars.
(5) Perpendicular distance of the line from the position vector of a point C represented by is given as
(6) Perpendicular distance of the plane i.e., from the point P represented by vector a is
(7) The condition that two lines and (where t & t1 are scalars) are coplanar and non-parallel is given by
(8) The shortest distance between two non-intersecting lines (skew lines ( where t & t1 are scalars ) is given by
(9) Vector equation of the sphere with the position vector of center and radius p is
(10) Vector equation of sphere when extremities of diameter being is given by
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