11Gravitation

📚
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.
The escape velocity of a planet, double the size of the earth is (is p is same).
📅BP 2014
Q2.
A satellite is orbiting a planet of radius 'R' and mass 'M' with a velocity 'V. If the radius of the planet is double keeping the density unchanged the final velocity of satellite, he would be?
📅BP 2014
Q3.
When earth doesn't rotate about its axis then what effect can be seen in its gravity? (Expect at poles) [BP 2014, 20131
📅BP 2014•BP 2013
Q4.
Orbital velocity of satellite depends upon [BP 2013]
📅BP 2013
Q5.
What is true for a satellite orbiting around the earth [BP 2013]
📅BP 2013
Q6.
Which is true? [BP 2013]
📅BP 2013
Q7.
A plane is moving with velocity 200 m/s and an observer is just below it. If a shell is fired with 400 m/s by observer. What is minimum height for plane so would escape? [BP 2012]
📅BP 2012
Q8.
The earth satellite is 4 times higher than the communication satellite from surface of the earth. Then time period of earth [BP 2012] satellite is:
📅BP 2012
Q9.
The cause of day and night is due to [BP 2011]
📅BP 2011
Q10.
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]
📅BP 2011
Q11.
The time period of communication satellite [BP 2011]
📅BP 2011
Q12.
Two bodies of mass is 20 kg and 30 kg a 30 m apart. Then the gravitational for between the bodies is: IMOE 201
📅IMOE 201
Q13.
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.
Q14.
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
Q15.
The mass of the earth is 80 times that o the moon and their diameters are 1280 km and 3200 km respectively. The value of ion due to gravity on earth is 9.8 m/s', the value on the moon will be: [MOE 2011
📅MOE 2011
Q16.
The escape velocity of moon of mass 7.2 10" kg and radius 1.7 x 10' m
Q17.
If 'A' is the areal velocity of the planet of mass 'M' then angular momentum is: [TE 2010]
📅TE 2010
Q18.
Two solid spheres of radii 'r' and '2r made of same material are kept in contact mutual gravitational force of attraction between them is proportional to [TE 2010]
📅TE 2010
Q19.
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)
📅1.E 2011)
Q20.
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 [TOM 2012)
📅TOM 2012)
Q21.
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]
📅KU 2014
Q22.
The planet of mass and diameter is three times than the earth. Then find the acceleration due to gravity on planet is: [KU 2013, 2011]
📅KU 2013•KU 2011
Q23.
. 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 [TOM 04]
📅TOM 04
Q24.
At what height from earth, g becomes g/2 [IOM 2000]
📅IOM 2000
Q25.
What will be the time period of satellite moving around the earth if the radius of revolution is increases by 1 and half time
📅MOE 2008
Q26.
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.[MOE 2065
📅MOE 2065
Q27.
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]
📅MOE 2010
Q28.
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]
📅MOE 2010
Q29.
g is the acceleration due to gravity at the equator. Its value at the pole is [MOE 2000]
📅MOE 2000
Q30.
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]
📅MOE
Q31.
An astronaut feels weightlessness when [TE-02
📅TE-02
Q32.
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]
📅TE-08
Q33.
The earth radius is 'R', acceleration due to gravity at its surface is 'g'. If the body of mass 'M' falls from height h = R/5 from earth surface, its potential energy decreases by [IE-08]
📅IE-08
Q34.
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) [IE-08]
📅IE-08
Q35.
If a planet had mass and radius half of the earth, the acceleration due to gravity of [BPKIHS 05] that planet is:
📅BPKIHS 05
Q36.
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 [BPKIHS-06]
📅BPKIHS-06
Q37.
At what height above the earth's surface will the weight of a body be half as that on earth's surface?
Q38.
The value of g will 1% of its value at the surface of earth at a height (R. = 6400 km)
Q39.
If the value of 'g' is same at depth 'd' inside earth and height 'h' above carth then
Q40.
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
Q41.
A body weighs
Q42.
If a body weighs xN on the surface of earth, its weight half way down to the centre of the earth, assuming earth to be of uniform density is
Q43.
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
Q44.
The gravitational force between ty identical copper spheres each of radius R touch their surface is F. Then gravitational force between two copper spheres of radius R and 2R touching their surface is now
Q45.
The gravitational force on a small particle lying on earth's surface is F. If a concentric spherical cavity of radius R/2 is made inside earth, then force on the particle is
Q46.
In the above question, if the cavity of radius ? is made touching the particle then the force on the particle will
Q47.
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]
📅IOM
Q48.
The gravitational potential at surface of an isolated soap bubble is Vo. Thus potential at its centre will be
Q49.
The gravitational potential at the surface of earth (or mercury drop) is V.. The potential at its centre is
Q50.
A hollow sphere shrinks maintaining its shape. Then gravitational potential at its centre
Q51.
$1. Two planets of equal density having radii R and 2R. Then ratio of acceleration due to gravity on their surface will be
Q52.
2, If earth shrinks so that its radius decreases by 50%, then the value of acceleration due to gravity at its surface
Q53.
53. At what depth below earth surface, value of acceleration due to gravity becomes (If R = radius of the earth) [MOE]
Q54.
54. The ratio of values of g at height ? to that at depth ? is (R = radius of earth)
Q55.
55. How fast the earth would rotates so that particles at its equator fly off?
Q56.
56. 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:
Q57.
57. The ratio of escape velocity of a particle at height R to orbital velocity of a satellite in orbit close to earth is
Q58.
58. Two planets of equal densities have radii R and 2R. Then ratio of escape velocities of particles from their surface is
Q59.
59. In a missile launched with velocity less than escape velocity, the sum of its KE and PE is always [IOM]
📅IOM
Q60.
60. 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
Q61.
61. 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
Q62.
62. By what percentage is escape speed greater than the speed of satellite close to earth surface
Q63.
63. The maximum height reached by a rocket fired with 90% of escape velocity from surface of earth is
Q64.
64. A body is projected vertically from th earth's surface with the K.E. equal to ha the minimum value needed for it to escape. The height through which it rises above the earth is
Q65.
66. 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
Q66.
67. Time taken by a radiowave to go and come back from a communication satellite to earth is nearly [BPKIHS]
📅BPKIHS
Q67.
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
Q68.
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
Q69.
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?
Q70.
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
Q71.
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:
Q72.
Which of the given is constant?
Q73.
`The acceleration due to gravity in the planet A is 9 times the acceleration due to gravity on earth. A man jumps to a height of 2m on the surface of planet What is the height of jump by the same person on earth?
Q74.
Which one of the following statement is correct?
📅KU 2015
Q75.
Escape velocity of a body projected from earth:
📅KU 2016
Q76.
Geostationary satellite of earth is:
📅KU 2016
Q77.
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
Q78.
A 5kg mass with a string Im length moves in a vertical circle with the velocity of 4m/s. The net tension in the string is 130N. Then, the position of body is.
📅KU 2016
Q79.
At particular point, the acceleration due to gravity is always constant for same or different mass due to :
📅KU 2016
Q80.
Star that appear stationary form the earth
📅KU 2017