Chapter 9 Gravitation and Flotation
Gravitation and Flotation
Gravitation and Flotation, Universal Law of Gravitation, Force of Gravitation of the earth (gravity), Acceleration due to Gravity; Mass and Weight; Free fall. Thrust and Pressure. Archimedes’ Principle; Buoyancy.
Chapter 9 Gravitation and Flotation
Chapter 10 GRAVITATION INTRODUCTION
Have you ever wondered whether we would still be studying about with Gravitation if a stone had fallen on Newton’s head instead of an apple? Anyways, the real question is why does an apple fall down rather than go upward?
Isaac Newton was bornin Woolsthorpe near Grantham, England. He is generally regarded as the most original and influential theorist in the history of science. He was born in a Poor farming family. But he was not good at farming. He was sent to study at Cambridge University in 1661. In1665 a plague broke out in Cambridge and so Newton took a year off. It was during this year that the incident of the apple falling on him is said to have occurred. This incident prompted Newton to explore the possibility of connecting gravity with the force that kept the moon in its orbit. This led him to the universal law of gravitation. It is remarkable that many great scientists before him knew of gravity but failed to realize it. Newton formulated the well- known laws of motion. He worked on theories of light and color. He designed an astronomical telescope to carry out astronomical observations. Newton was also a great mathematician. He invented a new branch of mathematics, called calculus. Newton transformed the structure of physical science with his three laws of motion and the universal law of gravitation. As the key stone of the scientific revolution of the seventeenth century, Newton’s work combined the contributions of Copernicus, Kepler, Galileo, and others into a new powerful synthesis. It is remarkable that though the gravitational theory could not be verified at that time, there was hardly any doubt about its correctness. This is because Newton based his theory on sound scientific reasoning and backed it with mathematics. This made the theory simple and elegant. These qualities are now recognized as essential requirements of a good scientific theory
Newton’s Law of Gravitation
1 Gravitation:
The force of attraction between any two bodies in the universe is called gravitation or gravitational force. e.g.:- The attraction between sun and earth or the attraction between a book and table is due to gravitational force.
2 Newton’s Law of Gravitation: imp
According to this law, the force of attraction between any two bodies is directly proportional to product of masses and inversely proportional to square of distance between them.
Mathematically 
And 
Combining eq. (i) and (ii) we get
or 
Where 
is constant of proportionality called universal gravitational constant.
Its value is 
and its dimensional formula is 
Definition of G:
If 
and 
then 
Hence universal gravitational constant may be defined as the force of attraction between two unit masses separated by one meter distance apart.
Q. Why G is called universal gravitational constant?
Ans. Because the value of G does not depends upon nature, size, medium of the body and applicable at every place so it called universal constant.
The Universal gravitational constant G is an experimental value calculated by Cavendish 71 years after the law was formulated.
QUESTION 1: Two particles of masses 1.0 kg and 2.0 kg are placed at a separation of 50 cm. assuming that the only forces acting on the particles are their mutual gravitation; find the initial accelerations of the two particles.
Sol: The force of gravitation exerted by one particle on the other is
The acceleration of 1 kg particle is
toward 2 kg particle.
Similarly The acceleration of 2 kg particle is
toward 1kg particle.
- Evidences In support of Laws of Gravitation:
The rotation of earth around the sun and the rotation of moon around the earth can be explained by this law.
The formation of tides in oceans is due to gravitational force between moon and earth.
The limes of solar and lunar eclipses can be calculated by this law.
The orbits and periods of revolutions of artificial satellite can be predicted accurately by this law.
The value of 
varies from place on the surface of earth in accordance with this law.
4 Some Important Features of Gravitational Force:
It is independent of inter mediums.
It obeys Newton’s third law of motion.
It is strictly holed by point masses.
It is a central force acts at centers.
It is a conservative force.
This force is independent of the presence of the other bodies. Hence we can sys it follow principle of superposition of forces.
5 Gravity:
It is defined as the force between earth and a object lying on or near the surface of earth. A body falls on surface of earth under the effect of gravity.
Acceleration due to Gravity:
The acceleration produced in a falling body due to gravity of earth is called acceleration due to gravity. It is a vector quality and denoted by .
The value of does not depend upon mass, size and shape of the body but varies from place to place on the surface of earth due to shape of earth and position of the body.
Expression for acceleration due to gravity or Relation between 
and 
:
Suppose earth is a sphere of mass 
and radius 
and a body of mass 
is placed on the surface of earth, then according to Newton’s law of gravitation, the force of attraction between earth and body is
And the force experienced by the body due to its weight is
Comparing eq. (i) and (ii) we have
or 
Thus value of g does not depend upon the mass of body but depends upon the mass of the planet and radius of the planet.
7 Mass and density of the earth:
Mass of Earth:
Cavendish was the first which successfully determined 
and
.
As we know 
⟹ 
Where 
so 
or
Average density of Earth:
If earth is supposed to be sphere of 
s 
and 
then average density of earth may be given as
As 
or
also
⟹ 
Or 
Where 
& 
⟹ 
or 
or 
Density of earth on upper layer is smaller 
while density at inner layers is larger then 
Kepler’s Law of Planetary Motion:
Kepler proposed three laws to explain the motion of planets around sun as given below:
(i) Kepler’s First Law(Law of Orbit):-
According to Kepler’s first law, every planet revolves around the sun is an elliptical orbit and sun is at any one of its foci. As shown in fig.
(ii) Kepler’s Second Law (Law of Area):-
According to Kepler’s second law of planetary motion, every planet swept out equal area in equal interval of time.
Suppose a planet is at position 
and moves up to 
in time 
then area swept out by the planet is 
In the same time is planet move from 
to 
then area swept out by the planet 
As from diagram
⟹ 
Clearly 
⟹ 
Hence if a planet is near the sun then its velocity beams larger and is planet is away from sun then its velocity decreases.
(iii) Kepler’s Third Law(Law of Period):-
According to Kepler’s third law of planetary motion, the square of time period is equal to the cube of semi major axis.
i.e. 
or 
Where K is a constant of proportionality for two different planets,
we can write 
Thus the larger the distance of planet from Sun larger will be the period of rotation.
8 16. A Saturn year is 29.5 times the Earth year. How far is the Saturn from the Sun if the Earth is 1.50
108 km away from the Sun?
Answer: we know that 
also 
so 
33 Inertial and Gravitational Mass:
The mass of the body which measure inertia of the body called inertial mass
i.e. 
Properties:-
- It is directly proportional to matter possessed by the body.
- It is independent on size, shape and state of body.
- It is conserved in both physical and chemical processes.
- It is not affected by presence of other bodies.
- Inertial mass increase with speed as
Gravitational Mass:
The mass of a body which determine the gravitational pull of earth acting upon it is called gravitational mass.
As 
Larger the 
larger is the gravitation pull of the earth on it. Both 
and are equals.
Similarity:-
Both represent quantity of matter of the body.
Both do not depend on shape and state of matter.
Both are equal in magnitude.
Both are not affected by presence of other bodies.
Both are scalar quantity.
Difference:-
Inertial mass is determined by Newton’s second law of motion and gravitation mass is determined by Newton’s law of gravitation.
Inertial mass can be measured only when body is motion which is neither convenient nor practical while gravitational mass can be measured using common balance.
What is meant by the term ‘buoyancy’?
Ans: The tendency of a liquid to exert an upward force on an object placed in it, is called
buoyancy.
Define buoyant force. Name two factors on which buoyant force depends.
Ans: Buoyant force: The upward force acting on an object immersed in a liquid is called buoyant force.
Factors affecting buoyant force:
• Volume of object immersed in the liquid. If the volume of the object is more, the buoyant force acting on it is more and if the volume of the object is less, then the buoyant force acting on it is less.
• Density of the liquid.
Why does an object float or sink when placed on the surface of water?
Ans: The density of the objects and water decides the floating or sinking of the object in water. The density of water is 1gm/ cm3.
• If the density of an object is less than the density of water then the object will float.
• If the density of an object is more than the density of water then the object will sink.
State Archimedes’ principle. List its two applications.
• Archimedes’ principle states that the upward buoyant force that is exerted on a body immersed in a fluid, whether fully or partially submerged, is equal to the weight of the fluid that the body displaces and acts in the upward direction.