Download Circular motion and Gravitational (chapter no 7)

Circular motion and Gravitational (chapter no 7)
Centripetal Force: “The force which compels a body to move along a circular path”
Whenever an object moves in a circular path we know the object is accelerating because the
velocity is constantly changing direction. All accelerations are caused by a net force acting on
an object. In the case of an object moving in a circular path, the net force is a special force
called the centripetal force. Centripetal is Latin for "center seeking". So a centripetal force is a
center seeking force which means that the force is always directed toward the center of the
circle. Without this force, an object will simply continue moving in straight line motion
Equation for centripetal force: centripetal force depends upon on following things.
1. Velocity of the body: centripetal force is directly proportional to the square of the
velocity of a body.
Fc α V2
2. Radius of the circle: centripetal force is inversely proportional to the radius of the circle.
Fc α 1/ r
3. Mass of the body: centripetal force is directly proportional to the mass of the body
Fc α m
Combining all above equations Fc α mv2/r
Or
Fc = mv2/r
Centripetal Acceleration: The acceleration produced by centripetal acceleration.
According to Newton 2nd law of motion, the direction of centripetal acceleration is along
the direction of centripetal force (Fc).
1. Centripetal acceleration is perpendicular to the velocity of the body.
2. Acceleration directed towards the centre of the circle.
Fc = mv2/r
Fc = mac
2
there fore mac= mv /r
ac = V2/ r
ac α V2
and ac α 1/r
Centrifugal Force: According to Newton 3rd law of motion as a reaction of centripetal
force there appears another out ward reaction at the centre of the circle. This force is equal in
magnitude to the centripetal force but opposite in direction. This is called centrifugal force.
Fr = mv2/r
Centrifugal force exits so long as the centripetal force exits.
Application of centripetal force.
1. Banking of road.
2. Centrifuge.
3. Washing machine
4. Cream separator.
5. Extraction of juice from sugar cane.
GRAVITATION
Every object in our universe attracts the other object with certain fore towards its center. This
force of attraction is known as GRAVITATIONAL FORCE and the phenomenon is called
GRAVITATION. This is gravitational force which is responsible for the uniformity or
regularity in our daily astronomical life. The whole system of the universe is in order only due
to this force. Due to gravitation, the system of our universe is working uniformly and smoothly.
The planets around the earth or around the sun moves in an orderly motion due to gravitation.
NEWTON’S LAW OF
GRAVITATION
In order to explain the gravitational force between two bodies, Newton formulated a
fundamental law known after his name i.e. "NEWTON'S LAW OF GRAVITATION"
Newton’s law of gravitation states that every object in the universe attracts the other object with
a force and :
(1) The gravitational force of attraction between two bodies is directly proportional to the
product of their masses.
1 x m2 ------- (1)
(2) The gravitational force of attraction between two bodies is inversely proportional to the
square of the distance between their centers.
2 --------- (2)
MATHEMATICAL
REPRESENTATION
Combining (1) and (2)
/d2
F = G m1m2/d2
Where G = universal gravitational constant
Value of G:
G = 6.67 x 10-11 Nm2/kg2
MASS OF THE
EARTH
Consider a body of mass ‘m’ placed on the surface of the earth. Let the mass of the earth is ‘Me’
and radius of earth is ‘Re’ .
1m2
Gravitational force of attraction between earth and body is
F = G m Me/ Re2
We know that the force of attraction of the earth on a body is equal to weight the weight of
body.
i.e
F=W
therefore
W = G m Me/ Re2
But W = mg
mg = G m Me/ Re2
or
g = G Me/Re2
or
Me = g x Re2/G
From astronomical data:
g= 9.8 m/s2
Re = 6.4 x 106 m
G = 6.67 x 10-11 N-m2/kg2
Putting these values in the above equation.
Me = 9.8 (6.4 x 106)2/6.67 x 10-11
or
DENSITY OF EARTH
DENSITY OF
EARTH
We know that the earth is composed of numerous substances such as metals, non-metals,
water and other materials. Therefore we cannot determine exact density of the earth .For our
convenience, here, we are assuming that the earth is made of homogeneous substance and it
has a perfect spherical shape, however which is not true.
We know that the density of a substance is given by:
density = mass / volume
For earth:
Density = Me / Ve
Let the earth is a sphere of homogeneous composition , therefore its volume is given by:
Ve = 4/3(πR3e)
Hence the average density of the earth is:
Variation of g with altitude The value of ‘g’ is inversely proportional to the distance from
the centre of the earth. If distance from the centre of earth increases, the value of g decreases.
If value of ‘g’ at height ‘h’ from the earth is gh
gh = GM/(R+h)2
gh = gM/(R+h)2
It shows that if distance from the centre of the earth increase from the average radius of
the earth , the value of ‘g’ will decreases. This is reason due to which the value of ‘g’ is less on
mountain.
Orbital velocity of artificial satellite: Due to gravitational force artificial satellite
revolve round the earth in different orbits with uniform speed if satellite revolve around the earth
at height ‘h’ from the surface of earth.
Then centripetal force for satellite is
Fc = mv2/R+h ----1
Here m is mass , V is velocity and “R” is radius of earth
Gravitation force of earth provides this centripetal force to the satellite
F = G m Me/ (R+h)2----------2
Comparing 1 and 2 equations
mV2/R+h = GmMe/( R+h)2
V2 = GM(R+h) / ( R+h)2
V= √GM/(R+h)
If h the height of satellite is not very large than
Vorb= √ GM / R
G= 6.67x 10-11 N.m2/KG2
M = 6.4 x 1024 Kg
R = 6.4 x 106 m
After substituting the value
Vorb = 7900ms-1 or 7.9Kms-1