Definition: Gravitation
"Every particle of matter in the universe attracts every other particle with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them. The direction of the force is along the line joining the particles."
or
The force by which the Earth attracts objects towards its centre is called gravitational force.
OR
The force of mutual attraction that any two objects in the universe exert on each other, which is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres, is called the gravitational force.
Definition: Universal Law of Gravitation
"Every particle of matter attracts every other particle of matter with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between them."
Definition: Universal Gravitational Constant
The constant of proportionality in Newton's Law of Gravitation, whose value is 6.67430 × 10−11 Nm²kg⁻², and which remains the same everywhere in the universe regardless of the medium or the nature of the bodies, is called the Universal Gravitational Constant.
Definition: Acceleration due to Gravity
The acceleration gained by a body due to the gravitational force of attraction of the Earth, whose value is constant at a given place but differs from place to place and is given by g = GM/R2, is called Acceleration due to Gravity.
Definition: Potential Energy
"Potential energy is the work done against conservative force (or forces) in achieving a certain position or configuration of a given system."
OR
The energy stored in an object because of its position or state is called potential energy.
OR
The amount of work done against conservative forces which causes a change in P.E. is called potential energy.
Definition: Gravitational Potential
The gravitational potential energy per unit mass at a point is called gravitational potential.
OR
The negative of the work done by the gravitational force in displacing a unit mass from that point to infinity (or equivalently, the work done in bringing unit mass from infinity to that point without acceleration), is called Gravitational Potential.
Definition: Gravitational Potential Energy
The amount of work done in bringing a given body from infinity to that point against the gravitational force is called gravitational potential energy.
OR
The energy possessed by a system of two or more bodies by virtue of their positions and mutual gravitational attraction, which equals the work done against the gravitational force in assembling the system from infinity, is called Gravitational Potential Energy.
U = −\[\frac {Gm_1m_2}{r}\]
Answer the following question.
Define the binding energy of a satellite.
The minimum energy required by a satellite to escape from Earth’s gravitational influence is the binding energy of the satellite.
Answer the following question in detail.
What is a critical velocity?
The exact horizontal velocity of projection that must be given to a satellite at a certain height so that it can revolve in a circular orbit round the Earth is called the critical velocity or orbital velocity (vc).
Definition: Satellite
The objects that revolve around the Earth are called Earth satellites.
Definition: Artificial Satellite
A man-made object that orbits a planet or other celestial body is called an artificial satellite.
Definition: Geosynchronous Satellite
A satellite that orbits the Earth at a height of approximately 36,000 km above the equator with a period of revolution of 24 hours is called a geosynchronous satellite.
Definition: Polar Satellite
A satellite that travels over Earth's poles, passing close to the Earth's surface, with a period of revolution of nearly 85 minutes is called a polar satellite.
Definition: Binding Energy of Satellite
"The minimum energy required by a satellite to escape from Earth’s gravitational influence is the binding energy of the satellite."
OR
The energy that must be given to an orbiting satellite to make it escape to infinity (equal to the negative of total energy of the satellite), given by BE=GMm/2rBE=GMm/2r, is called Binding Energy.
Definition: Escape velocity
"The minimum velocity with which a body should be thrown vertically upwards from the surface of the Earth so that it escapes the Earth’s gravitational field, is called the escape velocity (ve) of the body."
Definition: Time Period of a Satellite
The time taken by a satellite to complete one full revolution around the Earth is called the Time Period of the satellite.
Formula: Gravitation
Newton’s Universal Law of Gravitation:
F = \[G\frac{m_1m_2}{r^2}\]
where:
Formula: Kepler's Law
Kepler’s Third Law relates the time period T of a planet’s revolution to the semi-major axis a of its elliptical orbit:
T2 ∝ a3
where,
Formula: Kepler's Second Law
The area swept by the planet of mass m in a given interval Δt is:
\[\Delta\vec{A}=\frac{1}{2}(\vec{r}\times\vec{v}\Delta t)\]
- \[\vec r\]: Position vector of the planet (distance from Sun).
- \[\vec v\]: Velocity vector of the planet.
- Δt: Time interval.
- \[\vec p\]: Linear momentum (\[\vec p\] = m\[\vec v\])
- \[\vec L\]: Angular momentum (\[\vec L\] = \[\vec r\] × \[\vec p\])
Formula: Kepler's Third Law
Formula: Universal Law of Gravitation
The gravitational force of attraction (F) between two bodies of mass m1 and m2 separated by a distance r is:
\[\mathbf{F} = \mathbf{G}\frac{m_1 m_2}{r^2}\]
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F: Gravitational Force of attraction (in Newtons, N).
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\[m_1, m_2\]: Masses of the two objects (in kilograms, kg).
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r (or d in the first part): Distance between the two objects (in meters, m).
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G: The constant of proportionality, called the Universal gravitational constant.
Formula: Variation of g with Depth
gd = g(1 - \[\frac {d}{R}\])
Formula: Variation of g with Altitude
gh = g\[\left(\frac{R^2}{(R+h)^2}\right)\] = g(1 - \[\frac {2h}{R}\])(for h < < R)
Formula: Variation of g with Latitude
gλ = g − ω2 R cos2 λ
Formula: Potential Energy
Based on the relationship between work and energy, the change in potential energy is given by:
\[\vec F\] · d\[\vec x\] = dU
- \[\vec{F}\]: The force acting on the object (external force applied against the conservative force).
- \[d\vec{x}\]: The small displacement of the object.
- dU: The change (increase) in the potential energy of the system.
Formula: Binding Energy
Binding Energy = +\[\frac {1}{2}\frac {GMm}{r}\]
Where:
- G = Universal Gravitational Constant
- M = Mass of the Earth
- m = Mass of the satellite
- r = Radius of the orbit (Distance from the center of the Earth)
Formula: Escape velocity
\[v_e=\sqrt{\frac{2GM}{R}}\]
- ve = Escape velocity (minimum speed needed to escape Earth’s gravity)
- G = Universal gravitational constant (6.674 × 10−11 Nm2/kg2)
- M = Mass of the Earth (or celestial body)
- R = Radius of the Earth (or distance from the centre of the mass to the object)
Formula: Time Period of Satellite
T = \[2\pi\sqrt{\frac{(R+h)^3}{GM}}\]
Where:
- T = Time period of the satellite (in seconds)
- R = Radius of the Earth
- h = Height of the satellite above Earth's surface
- G = Universal gravitational constant
- M = Mass of the Earth
- (R + h) = r = Radius of the satellite's orbit
Law: Kepler's First Law
Kepler's First Law (Law of Ellipses)
- Each planet moves in an elliptical orbit with the Sun at one focus.
- This means planetary orbits are stretched circles, not perfect circles.
- The ellipse has two foci; the Sun occupies one of these.
Law: Kepler's Second Law
Kepler's Second Law (Law of Equal Areas)
- A line joining the planet and the Sun sweeps out equal areas in equal time intervals.
- When the planet is nearer the Sun (perihelion), it moves faster.
- When the planet is farther from the Sun (aphelion), it moves more slowly.
- This law reflects conservation of angular momentum.
Law: Kepler's Third Law
Kepler's Third Law (Law of Periods)
- The square of the time period of revolution of a planet is proportional to the cube of the semi-major axis of its orbit.
- This means a planet farther from the Sun takes a longer time to complete an orbit.
Law: Universal Law of Gravitation
Statement:
The law which states that every particle of matter attracts every other particle in the universe with a force whose magnitude is directly proportional to the product of masses and inversely proportional to the square of distance between them is called Newton's Law of Gravitation.
Derivation:
Newton's Universal Law of Gravitation states that every particle of matter attracts every other particle of matter with a force which is:
- Directly proportional to the product of their masses: F ∝ m1 ⋅ m2
- Inversely proportional to the square of the distance between them: F ∝ \[\frac {1}{r^2}\]
Combining both, the gravitational force is expressed as:
F = G\[\frac{m_1m_2}{r^2}\]
where G is the Universal Gravitational Constant, measured by Henry Cavendish using the Cavendish balance, with the value:
G = 6.67 × 10−11Nm2/kg2
