Introduction to Gravitation and Newton's Law
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Questions and Answers

What does the gravitational force depend on?

  • The distance divided by the masses of the objects
  • The product of the masses of the two objects and their distance apart (correct)
  • The sum of the masses of the two objects
  • The velocity of the objects

The universal gravitational constant G is constant regardless of the masses of the objects.

True (A)

Who published the Principia Mathematica?

Sir Isaac Newton

According to Newton's law of gravitation, the force of attraction F is given by F = G * ____.

<p>m1m2/r^2</p> Signup and view all the answers

Match the following terms with their correct descriptions:

<p>Gravitational force = Attraction between two masses Universal gravitational constant = G, constant for gravitation Principia Mathematica = Published by Sir Isaac Newton Inverse-square law = Force varies inversely with the square of distance</p> Signup and view all the answers

In vector form, the force exerted on particle m2 by particle m1 is represented as:

<p>F21 = -G(m1m2/r^2) (A)</p> Signup and view all the answers

The distance used in the gravitational force formula is always squared.

<p>True (A)</p> Signup and view all the answers

If two unit masses are placed at a unit distance apart, the gravitational force between them is equal to the value of ____.

<p>G</p> Signup and view all the answers

What is the value of G?

<p>6.67 x 10^-11 Nm2/kg2 (D)</p> Signup and view all the answers

The gravitational force is dependent on the medium between two masses.

<p>False (B)</p> Signup and view all the answers

State the dimensions of the gravitational constant G.

<p>M^-1L3T^-2</p> Signup and view all the answers

The gravitational force between two particles acts along the line that connects them and they are part of an action-reaction pair known as Newton's _______ law.

<p>third</p> Signup and view all the answers

What happens to the acceleration due to gravity g at a height h above the Earth's surface?

<p>It decreases. (A)</p> Signup and view all the answers

The weight of a body at height h can be expressed as mg h = GMm/(R + h)^2.

<p>True (A)</p> Signup and view all the answers

What is the relationship for g at a very small height h above the Earth's surface?

<p>gh = g (1 - h/2R + 2h^2/R^2...)</p> Signup and view all the answers

Match the values with their corresponding meanings:

<p>G = Gravitational constant g = Acceleration due to gravity m = Mass of the object R = Radius of the Earth</p> Signup and view all the answers

What is the formula for escape velocity (Ve) in terms of gravitational acceleration (g) and the radius of the Earth (R)?

<p>$2gR$ (A)</p> Signup and view all the answers

What height above the Earth's surface do geostationary satellites typically orbit?

<p>36,000 km (A)</p> Signup and view all the answers

The escape velocity of a body from the surface of the Earth is equal to its critical velocity when revolving close to the Earth's surface.

<p>False (B)</p> Signup and view all the answers

What is the mean density of a planet represented by?

<p>ρ (rho)</p> Signup and view all the answers

Astronauts experience weightlessness because there is no gravitational force acting on them.

<p>False (B)</p> Signup and view all the answers

The velocity at which a satellite is in a state of free fall towards Earth without falling is called its __________.

<p>critical velocity</p> Signup and view all the answers

What is one primary use of communication satellites?

<p>Sending TV signals over large distances</p> Signup and view all the answers

The weight of a body is defined as the gravitational force that pulls it towards the ________.

<p>Earth's center</p> Signup and view all the answers

Match the following terms with their definitions:

<p>Escape Velocity = Speed required to break free from gravitational attraction Geostationary Satellite = Satellite that remains fixed over one point on the Earth's surface Earth's Mass (M) = Approximately $5.97 imes 10^{24} kg$ Gravitational Force = Attractive force between two masses</p> Signup and view all the answers

What results from equating centripetal force to gravitational force for a body in circular orbit near Earth's surface?

<p>$v_c = rac{GM}{R}$ (A)</p> Signup and view all the answers

Match the following terms with their descriptions:

<p>Geostationary Satellite = Appears stationary from Earth Weightlessness = Result of equal acceleration Kepler's Law of Orbit = Planets move in elliptical orbits Space Adaptation Syndrome = Initial hours of weightlessness discomfort</p> Signup and view all the answers

The escape velocity from Earth is derived from the formula $v_e = rac{2GM}{R}$.

<p>True (A)</p> Signup and view all the answers

What phenomenon is experienced by astronauts during the initial hours of weightlessness?

<p>Space sickness (A)</p> Signup and view all the answers

The weight of an object (mg) can be expressed as the product of its mass (m) and the acceleration due to gravity (g), which is derived from the equation __________.

<p>GMm/R^2</p> Signup and view all the answers

Kepler's Law states that each planet orbits the sun in a circular path.

<p>False (B)</p> Signup and view all the answers

In an orbiting satellite, the astronaut and the satellite share the same ________ towards the center of the Earth.

<p>acceleration</p> Signup and view all the answers

What is the formula for calculating eccentricity (e) of an ellipse?

<p>e = SO / AO (D)</p> Signup and view all the answers

The distance of the closest approach to the sun is called apogee.

<p>False (B)</p> Signup and view all the answers

What does the law of areas state regarding the movement of planets?

<p>A planet sweeps out equal areas in equal times.</p> Signup and view all the answers

The apogee of a planet can be calculated using the formula ______.

<p>a(1 + e)</p> Signup and view all the answers

Match the terms related to orbits with their definitions:

<p>Perigee = Closest point to the sun Apogee = Farthest point from the sun Eccentricity = Measure of the deviation from a circle Law of Periods = T^2 is proportional to R^3</p> Signup and view all the answers

According to the law of periods, which of the following statements is true?

<p>T^2 is proportional to the cube of the semi-major axis. (A)</p> Signup and view all the answers

Gravity is a type of force that always acts outward from the Earth's center.

<p>False (B)</p> Signup and view all the answers

What is the relationship between the gravitational pull and weight of a body?

<p>The gravitational pull on a body is equal to its weight.</p> Signup and view all the answers

What is the mass of the Earth?

<p>$5.96 \times 10^{24} kg$ (A)</p> Signup and view all the answers

The critical minimum velocity needed for a satellite to maintain a circular orbit around the Earth is 8 km/sec.

<p>True (A)</p> Signup and view all the answers

What celestial body has a mean radius of $1.74 \times 10^{6}$ m?

<p>Moon</p> Signup and view all the answers

Newton's second law of motion states that F = _____, where F is the external force, mi is the inertial mass, and a is the acceleration.

<p>mi * a</p> Signup and view all the answers

Match the following celestial bodies to their mean densities (10 kg/m^3):

<p>Sun = 1.41 Earth = 5.52 Moon = 3.30</p> Signup and view all the answers

What happens if the energy of a satellite in orbit is too low?

<p>It will intersect the Earth and fall back. (C)</p> Signup and view all the answers

Gravitational mass is defined in the same manner as inertial mass.

<p>False (B)</p> Signup and view all the answers

Which body has the mean radius of $6.37 \times 10^{6}$ m?

<p>Earth</p> Signup and view all the answers

Flashcards

Newton's Law of Gravitation

Every object in the universe attracts every other object with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.

Universal Gravitational Constant (G)

A constant of proportionality in Newton's law of gravitation.

Gravitational Force

The attractive force between any two objects with mass.

Inverse Square Law

The force of gravity weakens rapidly as the distance between objects increases. This is how far apart matters. (Distance matters squared)

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Vector Form (Newton's Law)

The gravitational force is a vector. It acts along the line joining the centers of the objects.

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Gravitational Constant (numerical value)

The force of attraction between two unit masses put at a unit distance apart.

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Masses attraction

Masses with greater values attract each other more intensely

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Inverse Relationship

The force of attraction decreases as the distance between objects increases.

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SI unit of Gravitational constant

Newton metre squared per kilogram squared (Nm^2/kg^2)

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Value of Gravitational constant (G)

6.67 x 10^-11 Nm^2/kg^2

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Dimensions of G

M^-1 L^3 T^-2

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Gravitational Force - Intervening Medium

Independent of the substance between the masses.

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Acceleration due to gravity (at surface)

g = GM/R^2 (Where G is gravitational constant, M is earth's mass and R is earth's radius).

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Acceleration due to gravity at height 'h'

g'h = g(R/(R+h))^2

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Acceleration due to gravity at small height (approximation)

g'h = g*(1-2h/R)

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Gravitational Force - Action-Reaction Pair

Forces are equal and opposite, acting along the line connecting the objects.

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Escape Velocity (ve)

The minimum speed required for an object to escape a planet's gravitational pull.

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Escape velocity expression in terms of g (gravity)

Escape velocity, ve, is calculated as ve = √(2gR).

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Escape velocity expression in terms of planet´s mean density

Escape velocity is also ve = √(2 * (4/3) * π * G * ρ * R)

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Critical Velocity (vc)

The velocity needed for a body to maintain a stable orbit around a planet or celestial body

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Relationship between escape and critical velocity

Escape velocity (ve) is twice the critical velocity (vc) – ve = 2vc

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Geo-stationary Satellite

A satellite that orbits Earth at a fixed point relative to the Earth’s surface, appearing stationary.

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Derivation of ve=2gR

Starts with the relationship between gravitational force and weight (mg) and mass then substitutes to get the expression for escape velocity.

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Derivation of ve in terms of density

Uses the mass of the planet, related to its density and radius, to express escape velocity.

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Height of a geosynchronous satellite

Approximately 36,000 km above Earth's surface.

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Weightlessness in Space

The feeling of weightlessness experienced by astronauts in orbit is due to the astronaut and spacecraft experiencing the same acceleration towards Earth.

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Weightlessness vs. Zero Gravity

Weightlessness is not the absence of gravity, but the absence of a reaction force (like the ground pushing back) when the acceleration is the same for both the astronaut and the spacecraft (falling).

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Space Adaptation Syndrome (space sickness)

A common problem faced by astronauts in initial weightlessness.

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Kepler's First Law

Planets orbit the sun in elliptical paths, with the sun at one focus.

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Earth's gravitational force

The force pulling objects towards the Earth's center.

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Satellite uses

Communication (TV signals, telecommunications), weather forecasting, astronomical photography, and solar/cosmic radiation studies.

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Eccentricity of an ellipse

The ratio of the distance from a point on the ellipse to the focus to the distance from the same point to the closest point on the major axis.

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Perigee

The point in the orbit of an object around the Sun that is closest to the Sun.

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Apogee

The point in the orbit of an object around the Sun that is farthest from the Sun.

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Kepler's Second Law

A line joining a planet and the Sun sweeps out equal areas during equal intervals of time.

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Area velocity

The rate at which the area is swept out by the line joining a planet and the Sun.

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Kepler's Third Law

The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit.

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Centripetal force

The force that keeps an object moving in a circular path.

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Earth's Radius

The average distance from the Earth's center to its surface, approximately 6.37 × 10⁶ meters.

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Sun's Mass

The total amount of matter in the Sun, approximately 1.97 × 10³⁰ kilograms.

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Moon's Density

The Moon's mass packed into its volume, approximately 3.30 × 10³ kg/m³.

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Satellite Launch Velocity

The minimum speed a spacecraft needs to achieve to enter orbit around Earth, approximately 8 km/s.

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Inertial Mass

A measure of an object's resistance to changes in motion. It's defined by how much force is needed to produce a given acceleration.

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Gravitational Mass

A measure of an object's gravitational pull. It's determined by how much force it exerts on other objects.

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What is the formula for calculating gravitational force?

The force of gravity between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. This is represented by the formula F = Gm₁m₂/r².

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What are the differences between inertial mass and gravitational mass?

Inertial mass is a measure of an object's resistance to changes in motion. It's determined by how much force is needed to produce a given acceleration. Gravitational mass is a measure of an object's gravitational pull. It's determined by how much force it exerts on other objects. Despite different measuring methods, they are equal.

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Study Notes

Introduction to Gravitation

  • The universe consists of galaxies, stars, planets, comets, asteroids, and meteoroids.
  • Gravitation is a natural force that causes objects to attract each other.
  • Sir Isaac Newton described the inverse-square law of gravitation in his Principia Mathematica.

Newton's Law of Gravitation

  • Definition: Every particle of matter attracts every other particle of matter with a force that is proportional to the product of their masses and inversely proportional to the square of their separation.
  • Mathematical form: F ∝ (m₁m₂)/r² where F is the force of attraction, m₁ and m₂ are the masses of the particles, and r is the distance between them.
  • Vector form: The force exerted on particle m₂ by particle m₁ is given by F₂₁ = -G m₁m₂ î₂₁ / r²₁₂, where î₂₁ is a unit vector from m₁ to m₂.

Universal Gravitational Constant (G)

  • G = F r²/m₁m₂
  • The force of attraction between two unit masses at a unit distance is equal to G numerically.
  • G = 6.67 × 10⁻¹¹ Nm²/kg²
  • G has dimensions of M⁻¹L³T⁻².

Variation in 'g'

  • Acceleration due to gravity at a height (h): g' = g ( R² / (R + h)²), where g is the acceleration due to gravity at the surface of the earth and R is the radius of the earth.
  • Acceleration due to gravity at a depth (d): g' = g (1 - d/R), where d is the depth and R is the radius of the earth.

Effect of altitude

  • Acceleration due to gravity decreases with altitude.

Effect of depth

  • Acceleration due to gravity decreases with depth.

Variation of 'g' with latitude

  • g' = g - Rω² cos²λ, where g' is the acceleration due to gravity at a point on the earth's surface, g is the acceleration due to gravity at the equator, λ is the latitude, and ω is the angular velocity of the earth.
  • At the poles, g' = g, and at the equator, g' = g - Rω², which results in a difference in g, showing the effect of earth's rotation.

Satellite

  • Any smaller body revolving around a larger body due to gravity is known as a satellite.
  • Satellites can be natural (like the moon) or artificial.
  • Artificial satellites are launched by humans into circular orbits.
  • The minimum two-stage rocket is required to launch a satellite in a circular orbit around a planet.

Satellite Projection

  • Depending on the horizontal velocity, the satellite can follow an elliptical, parabolic, or hyperbolic path.
  • Critical velocity: Projections with critical velocity result in a circular orbit.
  • Escape velocity: Projections with an escape velocity result in a hyperbolic path, allowing the satellite to escape the earth's gravitational field.

Orbital Velocity

  • Definition: The minimum horizontal velocity required for a satellite to orbit the earth in a circular path is called orbital velocity.
  • Expression: v = √(GM/(R + h)), where G is the gravitational constant, M is the mass of the earth, and (R + h) is the radius of the orbit.

Gravitational Potential

  • Work done per unit mass to move a mass from infinity to a specific point in a gravitational field defines the gravitational potential at that point.
    • The formula for gravitational potential is V = -GM/r.

Gravitational Potential Energy

  • The work done in taking a body from infinity to a specific point defines the gravitational potential energy. This energy increases with distance and is negative for the earth-bound bodies.

Escape Velocity

  • The minimum velocity required for a body to escape from the gravitational pull of a planet is called the escape velocity. Its value depends on the mass and radius of the planet.
    • Formula: vₑ = √(2GM/R).

Communication Satellites

  • Artificial satellites, designed for communication purposes, are placed in geostationary or geosynchronous orbits with the same period as the Earth's rotation.
  • This ensures that they appear stationary relative to Earth's surface.

Weightlessness

  • Weightlessness is a phenomenon where the apparent weight is zero for a reason other than being no weight at all.
  • It occurs when the acceleration of a body equals that of the environment, making both forces equal and opposite resulting in no apparent weight.

Kepler's Laws

  • Kepler's First Law: Planets orbit the sun in elliptical paths with the sun at one focus.
  • Kepler's Second Law: A line joining a planet and the sun sweeps out equal areas during equal intervals of time.
  • Kepler's Third Law: The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit.

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Description

Explore the fundamentals of gravitation, including Sir Isaac Newton's laws and the universal gravitational constant. This quiz covers essential concepts such as the inverse-square law and mathematical representations of gravitational force. Test your understanding of how celestial bodies interact through gravitational forces.

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