11Gravitation

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GRAVITATION
▢ Newton's Law of Gravitation:

Table 1: Universal Law of Gravitation

Quantity
Formula / Point
Gravitational force
\(F=\frac{Gm_1m_2}{r^2}\)
Direction
Along line joining centres of two masses
Nature
Always attractive
Universal gravitational constant
\(G=6.67\times10^{-11}\ Nm^2kg^{-2}\)
▢ Acceleration Due to Gravity:
❖ Formula: \(g=\frac{GM}{r^2}\)

Table 1: Variation of g

Condition
Formula / Point
Due to shape of Earth
\(g\) increases from equator to pole
At height \(h\)
\(g'=g\left(\frac{R}{R+h}\right)^2\)
At small height \(h\ll R\)
\(g'=g\left(1-\frac{2h}{R}\right)\)
At depth \(d\)
\(g'=g\left(1-\frac{d}{R}\right)\)
Due to rotation of Earth at latitude \(\lambda\)
\(g'=g-\omega^2R\cos^2\lambda\)
At equator
\(g\) minimum
At pole
\(g\) maximum
▢ Gravitational Field:

Table 1: Field, Potential and Potential Energy

Quantity
Formula
Meaning
Gravitational field strength / intensity
\(\vec E=\frac{GM}{r^2}\)
Force per unit mass
Gravitational potential
\(V=-\frac{GM}{r}\)
Work done per unit mass in bringing test mass from infinity
Gravitational potential energy
\(U=-\frac{GMm}{r}\)
Energy due to position in gravitational field
Potential energy difference from surface to height \(h\)
\(\Delta U=\frac{mghR}{R+h}\)
For \(h\ll R\), \(\Delta U=mgh\)
▢ Satellite:

Table 1: Satellite Formulae

Quantity
Formula / Point
Centripetal condition
\(\frac{mv^2}{r}=\frac{GMm}{r^2}\)
Orbital velocity
\(v=\sqrt{\frac{GM}{r}}\)
First cosmic velocity
\(\frac{v_0^2}{Rg}=1\Rightarrow v_0=\sqrt{Rg}=\sqrt{\frac{GM}{R}}\)
Escape velocity / second cosmic velocity
\(\frac{v_e^2}{Rg}=2\Rightarrow v_e=\sqrt{2Rg}=\sqrt{\frac{2GM}{R}}\)
Third cosmic velocity
Velocity required to escape from solar system
Kinetic energy of satellite
\(E_K=\frac{GMm}{2r}\)
Potential energy of satellite
\(E_P=-\frac{GMm}{r}\)
Total energy of satellite
\(E_T=-\frac{GMm}{2r}\)
Energy relation
\(-E_T=E_K=-\frac{1}{2}E_P\)
▢ Field and Potential of Spherical Bodies:
❖ Symbols:
  • •
    \(r\) = distance of point from centre
  • •
    \(R\) = radius of sphere
  • •
    \(M\) = mass of sphere

Table 1: Gravitational Field Strength

Position
Hollow sphere
Solid sphere
Outside \((r>R)\)
\(E=\frac{GM}{r^2}\)
\(E=\frac{GM}{r^2}\)
Surface \((r=R)\)
\(E=\frac{GM}{R^2}\)
\(E=\frac{GM}{R^2}\)
Inside \((r
\(E=0\)
\(E=\frac{GMr}{R^3}\)
Centre \((r=0)\)
\(E=0\)
\(E=0\)

Table 2: Gravitational Potential

Position
Hollow sphere
Solid sphere
Outside \((r>R)\)
\(V=-\frac{GM}{r}\)
\(V=-\frac{GM}{r}\)
Surface \((r=R)\)
\(V=-\frac{GM}{R}\)
\(V=-\frac{GM}{R}\)
Inside \((r
\(V=-\frac{GM}{R}\)
\(V=-\frac{GM(3R^2-r^2)}{2R^3}\)
Centre \((r=0)\)
\(V=-\frac{GM}{R}\)
\(V=-\frac{3GM}{2R}\)
❖ Electrostatic Analogy:

Table 1: Gravitation-Electrostatic Analogy

Gravitation
Electrostatic analogue
\(E=\frac{GM}{r^2}\)
\(E=\frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\)
\(V=-\frac{GM}{r}\)
\(V=\frac{1}{4\pi\epsilon_0}\frac{q}{r}\)
▢ Kepler's Laws of Planetary Motion:

Table 1: Kepler's Laws

Law
Statement
Important Point
Law of orbit / 1st law
Planets revolve around Sun in elliptical orbits
Sun lies at one focus
Law of area / 2nd law
Position vector of planet w.r.t. Sun sweeps equal areas in equal intervals of time
Areal velocity is constant
Law of time period / 3rd law
Square of time period is directly proportional to cube of semi-major axis
\(T^2\propto a^3\)
❖ Second Law:
  • •
    \(\frac{dA}{dt}=constant\)
  • •
    Based on conservation of angular momentum
  • •
    Planet speed maximum when closest to Sun
  • •
    Planet speed minimum when farthest from Sun
❖ Third Law:
  • •
    Larger distance from Sun → greater time period
  • •
    Nearest planet has least time period
  • •
    Farthest planet has greatest time period
▢ High-Yield Recall:

Table 1: Gravitation One-Liners

Fact
Answer
Newton's gravitational force
\(F=\frac{Gm_1m_2}{r^2}\)
Value of \(G\)
\(6.67\times10^{-11}\ Nm^2kg^{-2}\)
Acceleration due to gravity
\(g=\frac{GM}{r^2}\)
\(g\) from equator to pole
Increases
\(g\) at height \(h\)
\(g'=g\left(\frac{R}{R+h}\right)^2\)
For small height
\(g'=g\left(1-\frac{2h}{R}\right)\)
\(g\) at depth \(d\)
\(g'=g\left(1-\frac{d}{R}\right)\)
\(g\) due to rotation
\(g'=g-\omega^2R\cos^2\lambda\)
Gravitational field intensity
\(E=\frac{GM}{r^2}\)
Gravitational potential
\(V=-\frac{GM}{r}\)
Gravitational potential energy
\(U=-\frac{GMm}{r}\)
Potential difference from surface to height \(h\)
\(\Delta U=\frac{mghR}{R+h}\)
Orbital velocity
\(v=\sqrt{\frac{GM}{r}}\)
First cosmic velocity
\(v_0=\sqrt{Rg}\)
Escape velocity
\(v_e=\sqrt{2Rg}\)
Third cosmic velocity
Velocity to escape solar system
Satellite total energy
\(E_T=-\frac{GMm}{2r}\)
Satellite energy relation
\(-E_T=E_K=-\frac{1}{2}E_P\)
Field inside hollow sphere
Zero
Potential inside hollow sphere
Constant
Field inside solid sphere
\(E=\frac{GMr}{R^3}\)
Potential at centre of solid sphere
\(V=-\frac{3GM}{2R}\)
Kepler's 1st law
Elliptical orbit
Kepler's 2nd law
Equal areas in equal times
Kepler's 2nd law based on
Conservation of angular momentum
Kepler's 3rd law
\(T^2\propto a^3\)
Planet speed maximum
When closest to Sun
Planet speed minimum
When farthest from Sun
Q1.
At what depth from earth surface will value of g become g/4? Radius of earth = R.
📅CEE MBBS 2025
Q2.
Escape velocity on the surface of earth is:
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Q3.
The value of 'g' decreases in going from surface to deep down the Earth. What happens to value of 'g' on going to height 'h' from the Earth surface?
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Q4.
Escape velocity of Earth is 11.2 km/s. Escape velocity of a planet whose g is twice that of Earth and diameter is twice that of Earth is:
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Q5.
If the speed of rotation of Earth increases, the weight of a body:
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Q6.
The escape velocity on Earth for a satellite launched vertically is 11 km/s. If the satellite is launched at 45°, the escape velocity will be:
📅CEE MBBS 2023
Q7.
At what height from Earth's surface does acceleration due to gravity become \(g/4\), where \(g\) is its surface value and \(R\) is Earth's radius?
📅CEE MBBS 2023
Q8.
Escape velocity on the surface of Earth is:
📅CEE MBBS 2022
Q9.
If angular velocity of Earth increases, the weight of a body:
📅CEE MBBS 2022
Q10.
The value of g decreases on going from the surface toward the centre of Earth. What happens to g on going to a height h above the surface?
📅CEE MBBS 2022
Q11.
Which statement about g is correct?
📅CEE MBBS 2022
Q12.
Escape velocity for earth is 11 km/s. If satellite is launched at 45° with horizontal, then escape velocity would be
📅CEE MBBS 2021
Q13.
The orbital velocity depends upon:
📅CEE BDS 2023
Q14.
If the escape velocity of planet is 11.2 m/s and the radius of the planet gets doubled and mass increases by 8 times, then the new escape velocity will be:
📅CEE BDS 2023
Q15.
The ratio of escape velocity of an earth satellite to its orbital velocity is nearly equal to:
📅CEE BDS 2021
Q16.
If the earth suddenly looses quality to attract things:
📅CEE BDS 2021
Q17.
A planet has twice Earth’s radius but the same mean density. What is its escape velocity? (Earth: 11.2 km/s)
📅BP 2014
Q18.
A satellite moves in a circular orbit close to the surface of a planet at speed \(V\). If the planet’s radius doubles at unchanged density, what is the new near-surface orbital speed?
📅BP 2014
Q19.
When earth doesn't rotate about its axis then what effect can be seen in its gravity? (Expect at poles)
📅BP 2014•BP 2013
Q20.
Orbital velocity of satellite depends upon
📅BP 2013
Q21.
What is true for a satellite orbiting around the earth
📅BP 2013
Q22.
Which is true?
📅BP 2013
Q23.
The cause of day and night is due to
📅BP 2011
Q24.
Work done in raising a body of mass 'm' from surface of earth height equal to radius of the earth is given by
📅BP 2011
Q25.
The time period of communication satellite
📅BP 2011
Q26.
Two bodies of mass is 20 kg and 30 kg a 30 m apart. Then the gravitational for between the bodies is: IMOE 201
📅MOE 201
Q27.
Earth's escape velocity for a satellite launched vertically is 11km/s. If the satellite is launched at 60 with verticle, th escape velocity would be.
Q28.
An earth satellite revolves round a circular orbit at a height 300 km above the earth surface. If the radius of earth is 6400 k the velocity of the satellite will be nearly MOE 2011
📅MOE 2011
Q29.
The escape velocity of moon of mass 7.2 10" kg and radius 1.7 x 10' m
Q30.
If 'A' is the areal velocity of the planet of mass 'M' then angular momentum is:
📅TE 2010
Q31.
Two solid spheres of radii \(r\) and \(2r\), of the same density, touch each other. How does the gravitational force vary with \(r\)?
📅TE 2010
Q32.
The escape velocity of the planet is V. It the mass of the planet is increased four times keeping the radius same. The escape velocity becomes: [1.E 2011)
📅IE 2011
Q33.
The escape velocity from earth surface is 11 km/sec then what is the escape velocity from another planet of double radius having mean density same
📅IOM 2012
Q34.
X of 10 kg, Y of 20 kg and acceleration due to gravity on the surface of moon(gm) is 1.6 m/s then gravitation field intensity of:
📅KU 2014
Q35.
The planet of mass and diameter is three times than the earth. Then find the acceleration due to gravity on planet is:
📅KU 2013•KU 2011
Q36.
. A body is projected with a velocity equal to twice the escape velocity from the earth. The velocity of the body in free space will be
📅IOM 04
Q37.
At what height from earth, g becomes g/2
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Q38.
A satellite is moving round the earth in a circular orbit of radius 8000Km with a velocity of 800m/s. The acceleration due to gravity on the satellite is nearly.
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Q39.
Two planets have radii r, and r, and densities d, and d, respectively. The ratio of th acceleration due to gravity on them will be
📅MOE 2010
Q40.
The orbital velocity of an artificial satellite in a circular orbit just above the earth's surface is v. For a satellite orbiting at an altitude of half the earth's radius, the orbital velocity is
📅MOE 2010
Q41.
g is the acceleration due to gravity at the equator. Its value at the pole is
📅MOE 2000
Q42.
Let G be the gravitational constant and R be the radius of the earth. If the radius of the earth becomes half, the new value of the gravitational constant will be
📅MOE
Q43.
An astronaut feels weightlessness when
📅TE-02
Q44.
The satellite of mass M and 9M are orbiting a planet in a circular orbit or radius R. Their time period of revolution will be in the ratio of
📅TE-08
Q45.
A mass \(m\) falls from height \(h=R/5\) to Earth’s surface. Find the decrease in gravitational potential energy, given surface gravity \(g\).
📅IE-08
Q46.
A man has weight 80 kg on earth's surface. The height above ground where he will have weight 40 kg is, (radius of earth 'R'= 6400 km)
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Q47.
If a planet had mass and radius half of the earth, the acceleration due to gravity of that planet is:
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Q48.
A satellite is revolving around a planet of radius 'r' with speed 'V.'. If the satellite is made to revolve in another planet of twice the mass than original and constant radius then ratio of initial to final velocity is
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Q49.
At what height above the earth's surface will the weight of a body be half as that on earth's surface?
Q50.
At what altitude above Earth is the gravitational acceleration 1% of its surface value? Take \(R=6400\ \mathrm{km}\).
Q51.
If the value of 'g' is same at depth 'd' inside earth and height 'h' above carth then
Q52.
A mass 'M' is broken into two parts one of which has mass 'm'. To have maximum gravitational force of attraction between the broken masses
Q53.
If a body weighs \(x\ \mathrm N\) at Earth’s surface, what does it weigh at depth \(R/2\) inside a uniformly dense Earth?
Q54.
The gravitational force between two identical steel balls each of radius R touching their surfaces is F. Then gravitational force between two identical steel balls each of radius 2R touching their surfaces is
Q55.
Two identical touching copper spheres of radius \(R\) attract with force \(F\). What is the force between touching copper spheres of radii \(R\) and \(2R\)?
Q56.
A concentric spherical cavity of radius \(R/2\) is removed from an otherwise uniform spherical Earth. What is the gravitational force on a test mass at its surface, if it was initially \(F\)?
Q57.
If the removed cavity of radius \(R/2\) is tangent internally to Earth’s surface at the test mass, what is the remaining gravitational force? (Original solid-Earth force \(F\).)
Q58.
A small particle is held inside an isolated hollow sphere at distance x from centre of the sphere. The gravitational force on that particle by the sphere is
📅IOM
Q59.
The gravitational potential at the surface of an isolated thin spherical soap bubble is \(V_0\). What is the potential at its centre?
Q60.
The gravitational potential at the surface of a uniform solid sphere is \(V_s\). What is its potential at the centre?
Q61.
A hollow sphere shrinks maintaining its shape. Then gravitational potential at its centre
Q62.
$1. Two planets of equal density having radii R and 2R. Then ratio of acceleration due to gravity on their surface will be
Q63.
If earth shrinks so that its radius decreases by 50%, then the value of acceleration due to gravity at its surface
Q64.
At what depth below earth surface, value of acceleration due to gravity becomes (If R = radius of the earth)
Q65.
Find the ratio of gravitational accelerations at height \(R/2\) and depth \(R/2\) from the surface of a uniformly dense Earth.
Q66.
What angular speed must Earth attain for objects at the equator to become weightless? Take \(R=6.4\times10^6\ \mathrm m\).
Q67.
If g be the acceleration due to gravity at earth's surface and R be radius of earth then escape velocity of a particle at height R will be:
Q68.
Compare the escape speed at height \(R\) above Earth with the orbital speed just above its surface.
Q69.
Two planets of equal densities have radii R and 2R. Then ratio of escape velocities of particles from their surface is
Q70.
In a missile launched with velocity less than escape velocity, the sum of its KE and PE is always
📅IOM
Q71.
Two satellites of masses m and 9m are orbiting a planet in a circular orbit of radius R. Their time periods of revolution will be in the ratio of
Q72.
A satellite with kinetic energy E is revolving round the earth in a circular orbit. The minimum additional K.E. required for it to escape into outer space is
Q73.
By what percentage is escape speed greater than the speed of satellite close to earth surface
Q74.
A rocket is launched vertically from Earth at 90% of the escape speed. What maximum altitude does it reach (ignore air resistance)?
Q75.
A projectile starts vertically upwards with half the kinetic energy required for escape. What maximum altitude above Earth does it attain?
Q76.
What initial vertical speed is needed for a body to rise from Earth’s surface to a maximum height of \(10R\)?
Q77.
Mass of moon is 1/81 times that of earth and its radius is 1/4 times the earth's radius. If escape velocity on earth's surfac is 11.2 km/s, its value at the surface of the moon is
Q78.
Approximately how long does a radio signal take to travel from Earth to a geostationary satellite and back?
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Q79.
A satellite A of mass 'm' is at a distance 'r' from earth's surface. Another satellite of mass 2m is at a distance 2r from earth's surface. Their time periods are in the ratio of
Q80.
Earth's escape velocity for a satellite launched vertically is 11 km/s. If the satellite is launched at 60" with vertical, the escape velocity would be
Q81.
If the earth were to suddenly contract to half the present radius (without any external torque acting on it), what would be the time of rotation of earth?
Q82.
A body is projected vertically upwards from the surface of a planet of radius R with a velocity equal to half the escape velocity for that planet. The maximum height attained by the body is
Q83.
Two particles of equal masses (each of mass M) go round a circle of radius R under the action of their mutual attraction. The speed of each particle is:
Q84.
Which expression is constant at different heights in the exterior gravitational field of the same spherical planet?
Q85.
A planet has nine times Earth’s gravitational acceleration. A person jumps 2 m high on that planet. How high could they jump on Earth with the same takeoff speed?
Q86.
Which one of the following statement is correct?
📅KU 2015
Q87.
Escape velocity of a body projected from earth:
📅KU 2016
Q88.
Which statement correctly describes a geostationary satellite?
📅KU 2016
Q89.
If a body is projected with velocity V less. than the escape velocity of earth, then the total energy of the projected body is:
📅KU 2016
Q90.
At particular point, the acceleration due to gravity is always constant for same or different mass due to :
📅KU 2016
Q91.
Star that appear stationary form the earth
📅KU 2017