3. CONTENTS
•History Of Gravity
•Newton’s Law Of
Gravitation
•Gravitation constant
•Acceleration due To gravity
•Mass And Weight to gravity
•Difference between gravity
and gravitation
•Gravitational Field
•Gravitational Potential
Difference
•Escape Velocity
•Satellite
4. Introduction
A popular story says that Newton came up with the theory of gravity instantly,
when an apple fell from a tree and hit him on the head. Actually, Newton saw
an apple falling from a tree, and it got him to thinking about the mysterious
force that pulls objects to the ground. Gravity was discovered in 1681. Thanks
to this 17th century English physicist and mathematician, our understanding
of the Universe and the laws that govern it would forever be changed.
5. Gravitation & Gravity
What is Gravitation?
Gravitation is the attractive force existing between any two objects that have mass. The force
of gravitation pulls objects together.
What is Gravity?
Gravity is the gravitational force that occurs between the earth and other bodies. Gravity
is the force acting to pull objects toward the earth.
6. Newton’s Law Of Gravitation
Newton’s law of gravitation, statement that any particle of matter in the
universe attracts any other with a force varying directly as the product
of the masses and inversely as the square of the distance between
them. In symbols, the magnitude of the attractive force F is equal
to G (the gravitational constant, a number the size of which depends on
the system of units used and which is a universal constant) multiplied
by the product of the masses (m1 and m2) and divided by the square of
the distance R: F = G(m1m2)/R2. Isaac Newton put forward the law in
1687 and used it to explain the observed motions of the planets and
their moons, which had been reduced to mathematical form
by Johannes Kepler early in the 17th century.
7. Vector Form Of Newton’s Law
Vector Form of Newton’s Law of Gravitation
The vector form of Newton’s law of gravitation signifies that
the gravitational forces acting between the two particles form
action-reaction pair.
From the above figure, it can be seen that the two particles of
masses and are placed at a distance, therefore according to
Newton’s law of gravitation, the force on m1 due to m2 i.e.
F12 is given by,
F12 = [-G m1m2]/|r12|2 r^
12
F12 = [-Gm1m2/|r1 – r2|3] (r1 – r2) . . . . . . . . . (i)
Where r^
12 is a unit vector pointing from m2 to m1.
The negative sign in Equation (i) indicates that the direction
of force F12 is opposite to that of r^
12.
Similarly, the force on m2 due to m1 i.e. F21 is given by,
F21 = -Gm1m2/|r21|2 r^
12 . . . . . . . . . . (ii)
From equations (i) and (ii) we get,
F12 = – F21
As F12 and F21 are directed towards the centres of the two
particles, so the gravitational force is conservative in nature.
8. Gravitational Constant
Gravitational constant is a constant relating the force of the gravitational attraction between two bodies to their
masses and their distance from each other in Newton's law of gravitation. The gravitational constant equals
approximately 6.67259 X 10-11 newton square meters per square kilogram.
Dimensional Formula of Gravitational Constant
The dimensional formula of gravitational constant is given by,
M-1 L3 T-2
Where, M = Mass, L = Length, T = Time
Derivation
From Newton’s law of gravitation,
Force (F) = [GmM] × r-2
Gravitational Constant (G) = F × r2 × [Mm]-1 . . . . (1)
Since, Force (F) = Mass × Acceleration = M × [LT-2]
∴ The dimensional formula of force = M1 L1 T-2 . . . . (2)
On substituting equation (2) in equation (1) we get,
Gravitational Constant (G) = F × r2 × [Mm]-1
Or, G = [M1 L1 T-2] × [L]2 × [M]-2 = [M-1 L3 T-2].
Therefore, the gravitational constant is dimensionally represented as M-1 L3 T-2.
9. Acceleration Due To Gravity
Application of Newton’s Law of Gravitation.
• Newton’s law of gravitation is applicable to all bodies from atomic particles to huge bodies.
•It is valid over a wide range of distance from the small terrestrial distance to the astronomical distance of
space.
Acceleration Due to Gravity.
Acceleration due to gravity is the acceleration gained by an object due to gravitational force. Its SI unit is
m/s2. It has both magnitude and direction, hence, it’s a vector quantity. Acceleration due to gravity is
represented by g. The standard value of g on the surface of the earth at sea level is 9.8 m/s2.
10. Gravitational mass is measured by comparing the force of gravity of an unknown mass to the force
of gravity of a known mass. The beauty of this method is that no matter where, or what planet, you
are, the masses will always balance out because the gravitational acceleration on each object will be
the same.
Let us suppose a body [test mass (m)] is dropped from a height ‘h’ above the surface of the earth
[source mass (M)], it begins to move downwards with an increase in velocity as it reaches close to
the earth surface.
Gravitational Mass
11. We know that velocity of an object changes only under the action of a force, in this case, the force is
provided by the gravity.Under the action of gravitational force, the body begins to accelerate toward
the earth’s centre which is at a distance ‘r’ from the test mass.
Then, ma = GMm/r2 (Applying principle of equivalence)
⇒ a = GM/r2
The above acceleration is due to the gravitational pull of earth so we call it acceleration due to gravity,
it does not depend upon the test mass. Its value near the surface of the earth is 9.8 ms-2.
Therefore, the acceleration due to gravity (g) is given by = GM/r2.
Gravitational Mass
12. Formula of Acceleration due to Gravity
Force acting on a body due to gravity is given by, f = mg
Where f is the force acting on the body, g is the acceleration due to gravity, m is mass of the
body.
According to the universal law of gravitation, f = GmM/(r+h)2
Where,
f = force between two bodies,
G = universal gravitational constant (6.67×10-11 Nm2/kg2)
m = mass of the object,
M = mass of the earth,
r = radius of the earth.
h = height at which the body is from the surface of the earth.
As the height (h) is negligibly small compared to the radius of the earth we re-frame the
equation as follows,
f = GmM/r2
Formula Of Acceleration Due To
Gravity
13. Now equating both the expressions,
mg = GmM/r2
⇒ g = GM/r2
Therefore, the formula of acceleration due to gravity is
given by, g = GM/r2
Note: It depends on the mass and radius of the earth.
This helps us understand the following:
•All bodies experience the same acceleration due to
gravity, irrespective of its mass.
•Its value on earth depends upon the mass of the earth
and not the mass of the object.
Formula Of Acceleration Due To
Gravity
14. Acceleration due to Gravity on the Surface of Earth
Earth as assumed to be a uniform solid sphere with a mean density. We know
that,
Density = mass/volume
Then, ρ = M/[4/3 πR3]
⇒ M = ρ × [4/3 πR3]
We know that, g = GM/R2.
On substituting the values of M we get,
g = 4/3 [πρRG]
At any distance ‘r’ from the centre of the earth
g = 4/3 [πρRG]
The value of acceleration due to gravity ‘g’ is affected by
•Altitude above the earth’s surface.
•Depth below the earth’s surface.
•The shape of the earth.
•Rotational motion of the earth.
Formula Of Acceleration Due To Gravity
On Surface
15. Key Points about the value of g,
Above the surface of the earth value of g decreases.
Inside the earth the value of g dcreases.
Value of g decreases as one moves from equator to the poles.
Due to the rotation of the earth, value of g decreases more in
the equator regions compared to that in the poles.
At the poles g=9.832ms^-2, whereas at the equator g=9.790ms^-
2
The value of g above the sea-level and 45 degree latitude is
taken as the standard value which is 9.81ms^-2
Knowing the value of g, we can find the average density of the
earth.
Key Points Of g
16. Mass
We can define mass as the measure
of the amount of matter in a body. The SI unit of
mass is Kilogram (kg). The mass of a body does
not change at any time. Only for certain extreme
cases when a huge amount of energy is given or
taken from a body. For example: in a nuclear
reaction, tiny amount of matter is converted into a
huge amount of energy, this reduces the mass of
the substance.
Mass & Weight
17. Mass & Weight
Weight
It is the measure of the force of gravity acting on a body.
The formula for weight is given by:
w = mg
As weight is a force its SI unit is also the same as that of force, SI unit of weight is Newton
(N). Weight depends on mass and the acceleration due to gravity, the mass may not change
but the acceleration due to gravity does change from place to place.
Mass & Density of Earth
Mass: 5.972 × 10^24 kg
Density: 5.51 g/cm³
20. Centre Of Gravity & Centre of mass.
The center of gravity (CG) of an object is the point at which weight is evenly dispersed and
all sides are in balance. In a uniform gravitational field the centre of gravity is identical to
the centre of mass, a term preferred by physicists. The two do not always coincide, however.
For example, the Moon’s centre of mass is very close to its geometric centre (it is not exact
because the Moon is not a perfect uniform sphere), but its centre of gravity is slightly
displaced toward Earth because of the stronger gravitational force on the Moon’s near side.
Centre Of Gravity & Mass
21. When a body is in an uniform gravitational field, its centre of gravity is also its centre of
mass.
The location of a body’s centre of gravity may coincide with the geometrical centre of the
body, especially in a symmetrically shaped body composed of homogenous material.
For hollow bodies or irregularly shaped objects, the centre of gravity (or centre of mass)
may occur in space at a point external to the physical material (the centre of gravity can lie
OUTSIDE the object!) – for example, in the centre of a tennis ball or between the legs of a
chair.
Centre Of Gravity & Mass
22. What is Gravitational Potential Energy?
When a body of mass (m) is moved from infinity to a point inside the gravitational influence of a
source mass (M) without accelerating it, the amount of work done in displacing it into the source
field is stored in the form of potential energy this is known as gravitational potential energy. It is
represented with the symbol Ug.
Explanation: We know that the potential energy of a body at a given position is defined as the
energy stored in the body at that position. If the position of the body changes due to the
application of external forces the change in potential energy is equal to the amount of work done
on the body by the forces.
Gravitational Potential Energy
23. What is Gravitational Potential?
The amount of work done in moving a unit test mass from infinity into the gravitational influence
of source mass is known as gravitational potential.
Simply, it is the gravitational potential energy possessed by a unit test mass
⇒ V = U/m
⇒ V = -GM/r
⇒ Important Points:
The gravitational potential at a point is always negative, V is maximum at infinity.
The SI unit of gravitational potential is J/Kg.
The dimensional formula is M0L2T-2.
Gravitational Potential
24. What is gravitational potential difference ?
The amount of work done in moving a unit mass from one point to another
point in a gravitational field is called gravitational potential difference.
The gravitational potential at a point in a gravitational field is the work done
per unit mass that would have to be done by some externally applied force
to bring a massive object to that point from some defined position of
zero potential, usually infinity . Gravitational Potential of a Point Mass
Consider a point mass M, the gravitational potential at a distance ‘r’ from it
is given by;
V = – GM/r.
Gravitational Potential Difference
25. Gravitational Potential of a Uniform Solid Sphere
Gravitational Potential of a Uniform Solid Sphere
Consider a thin uniform solid sphere of the radius (R) and mass (M) situated in space. Now,
If point ‘P’ lies Inside the uniform solid sphere (r < R):
Inside the uniform solid sphere, E = -GMr/R3.
Using the relation V= over a limit of (0 to r).
The value of gravitational potential is given by,
V = -GM [(3R2 – r2)/2R2]
26. Escape Velocity
What is Escape Velocity?
Escape velocity is the minimum velocity required by a body to be projected to overcome the
gravitational pull of the earth. It is the minimum velocity required by an object to escape the
gravitational field that is, escape the land without ever falling back. An object that has this velocity
at the earth’s surface will totally escape the earth’s gravitational field ignoring the losses due to
the atmosphere.
For example, a spacecraft leaving the surface of Earth needs to go at 7 miles per second, or
around 25,000 miles per hour to leave without falling back to the surface.
27. Escape Velocity Formula
Formula of Escape Velocity
Escape velocity formula is given
Where,
•V is the escape velocity
•G is the gravitational constant is 6.67408 × 10-11 m3 kg-
1 s-2
•M is the mass of the planet
•R is the radius from the center of gravity
28. Escape Velocity Formula
Escape Velocity Formula:
Derivation:
Assume a perfect sphere-shaped planet of radius R and mass M. Now, if a body of mass m is
projected from a point A on the surface of the planet. An image is given below for better
representation:
In the diagram, a line from the center of the planet i.e. O is drawn till A (OA) and extended
further away from the surface. In that extended line, two more points are taken as P and Q at
a distance of x and dx respectively from the center O.
Now, let the minimum velocity required from the body to escape the planet b
ve
Thus, Kinetic Energy will be
At point P, the body will be at a distance x from the planet’s center and the force of gravity
between the object and the planet will be:
29. Escape Velocity Formula
To take the body from P to Q i.e. against the gravitational attraction, the work done will be-
Now, the work done against the gravitational attraction to take the body from the planet’s surface to
infinity can be easily calculated by integrating the equation for work done within the limits x=R to x=∞.
Thus,
By integrating it further, the following is obtained:
Thus, the work done will be:
Now, to escape from the surface of the planet, the kinetic energy of the body has to be equal to the work
done against the gravity going from the surface to infinity. So,
K.E. = W
30. Satellite
A satellite is a body that orbits around another body in space. There are two different types
of satellites – natural and man-made.
Natural satellite - A small or secondary planet which revolves around a larger one. Example : Moon is
the natural satellite of Earth.
Artificial satellite - A man-made object placed (or designed to be placed) in orbit around an
astronomical body (usu. the earth).
31. Orbital velocity
Orbital velocity is the velocity at which a body revolves around the other body. Objects that travel in the
uniform circular motion around the Earth are called to be in orbit. The velocity of this orbit depends on the
distance between the object and the centre of the earth.
orbital velocity vo, when the test mass is orbiting around the source mass in a circular path of radius ‘r’
having a centre of the source mass as the centre of the circular path, the centripetal force is provided by the
gravitational force as it is always an attracting force having its direction pointed towards the centre of a
source mass.
If the test mass is at small distances from the source mass r ≈
R(radius of the source mass)
Then,
32. A geostationary satellite is an earth-orbiting satellite, placed
at an altitude of approximately 35,800 kilometers (22,300
miles) directly over the equator, that revolves in the same
direction the earth rotates (west to east). The circular orbit
and angular velocity identical to that of the Earth,
the satellite is known as a geostationary satellite.
These satellites appear to be stationary above a particular
point which is due to the synchronization.
Geostationary Satellite
33. The equation for the acceleration of gravity was given as
g = (G • Mcentral)/R2
Thus, the acceleration of a satellite in circular motion about some central body is given by
the following equation
where G is 6.673 x 10-11 N•m2/kg2, Mcentral is the mass of the central body about which the
satellite orbits, and R is the average radius of orbit for the satellite.
Acceleration In Circular Motion
