Class 9th Science: Chapter 10 Gravitation


Kepler's Law of Gravitation

Kepler’s Laws of Planetary Motion
The universe is one of the most fascinating subjects that were explored by many people around the world. Johannes Kepler, a student of Tyco Brahe, suggested three laws about the motion of the planets in the solar system, which revolutionised the comprehension about our solar system. These laws are named after him as Kepler’s laws of planetary motion.

According to Kepler’s theory, all the planets revolve around the sun in elongated orbits, rather than perfect circles.

Kepler’s three laws of planetary motion are:
•  The law of orbits
•  The law of areas and
•  The law of periods

Kepler’s first law
Kepler’s first law, the law of orbits, states that the orbit of a planet is an ellipse with the sun at one of the foci. An ellipse is a closed, curved shape that is defined by two points called foci representing an elongated circle.The closest point on a planet’s orbit from the sun is called perihelion and the farthest point from the sun is called the aphelion.

Kepler’s second law
Kepler’s second law of planetary motion, also known as the law of areas, states that the line joining the planet to the sun sweeps out equal areas in equal intervals of time as the planet travels in its orbit.

Kepler’s third law
Kepler’s third law, the law of periods, defines the relationship between the orbital period of a planet and the average radius of its orbit. The orbital period of a planet, denoted by T, is the time taken by the planet to make a complete revolution around the sun along its orbit.

The average radius of the orbit of a planet is also the mean distance of the planet from the sun.

The law of periods states that the square of orbital period, T, of a planet is proportional to the cube of its mean distance, R, from the sun.

The law of periods can be expressed as T2 α R3.

Universal Law of Gravitation

Every object in the universe attracts the other with a force. This is by virtue of the mass of the objects. This force of attraction was supposed to be thought upon by Newton while contemplating on the free fall of an apple towards the ground. The force of attraction, which is the gravitational pull due to mass of objects, exists universally. The factors that affect gravitational force were studied and a law was put forth which is known as 'Newton's 
Universal Law of Gravitation'.
The gravitational force between two objects in the universe is directly proportional to the mass of the objects and is inversely proportional to the square of the distance between them. Hence, the mathematical form of the law is

               F ∝  m1m2r2
where 'm1', 'm2' are the masses of the objects and 'r' is the distance between them. Equating both sides of the expression we get,

              F =  Gm1m2r2
where, G is the constant of proportionality called the 'universal gravitational constant'.

The second part of the law is called the 'inverse square rule' or 'inverse square law'. The force with which earth attracts any object on its surface is the weight (W) of the object, which is the product of the mass (m) of the object and its acceleration due to gravity (g). 'W' changes from place to place on the earth on account of variation in 'g'. Thus the mass of an object remains the same throughout the universe where the weight of an object changes from place to place.
Fundamental quantities like “mass” of a body is not easy to define. One way of defining mass is on the basis of the fact that the mass of an object is the measure of its inertia. Such a mass is known as “inertial mass”. If a force “F” acting on a body produces an acceleration “a” in the body, then, its inertial mass is defined as the ratio of “F” and “a”.
The SI unit for mass is the “kilogram”. When a body is placed in the earth’s gravitational field, the body is attracted by the earth. The force with which the earth attracts a body is known as the “weight of the body”.
                         Inertial mass = Fa .
If the acceleration gained by a body due to the earth’s gravitational attraction is “g,” then its weight is equal to “mg”.
                      Weight of a body = mg
Since the “weight” of an object is a force, its SI unit is the “newton”.
The weight of a body is a vector quantity, and always acts towards the centre of the earth. If a body is taken from the “earth” to the “moon”, then there will be no change in its mass, but its weight will decrease. This is because the moon attracts the body with less force than that exerted by the earth. In fact, the weight of a body on the surface of the moon is only one-sixth of its weight on the surface of the earth.

Difference Between g and G
          Acceleration due to gravity (g)
 Universal Gravitational Constant (G)
1. The Acceleration produced on a freely falling body due to gravitational force is known as acceleration due to gravity.
1. The Force of attraction between any two objects of unit masses separated by unit distance in the universe Gravitational Constant.
2. It is denoted by g.
2. It is denoted by G
3. It changes from place to place.
3. Its value is constant everywhere in the universe.
4. Its units are m/sec2
4. Its units are Nm2/Kg2

Relation Between g and G
G stands for Newton's universal gravitational constant, whereas g stands for the acceleration due to gravity at a certain point.
G = 6.67300 × 10-11 N.m2.kg-2, G is a constant throughout space and time and it is a scalar quantity.
g = 9.8 m.s-2,   g is acceleration due to gravity which is a variable quantity and a vector qualtity.

According to Newton's law of universal gravitation the force of attraction between two bodies is given by
                     F = GMmr2         ---------- (i)
From Newton's second law of motion the weight of a body of mass m is
                     F = mg -----------------(ii)
From (i) and (ii)
                     mg = GMmr2      
                               or
                        g =   GMr2 .
Note:
g is a constant at a given location, which depends upon M and r.

Differences Between Mass and Weight
                 Mass
                           Weight
1. Mass is defined as the matter contained in body

2. Mass of a body is constant throughout the universe.

3. Mass is a scalar quantity.

4. S.I. Unit:  kilo gram.
1. Weight is defined as the gravitational force(pull) acted by the earth on the body.

2. Weight of the body changes from place to place depending on acceleration due to gravity.

3. Weight is a vector quantity.

4. S.I. Unit:  kilo gram weight.

Thrust and Pressure

Thrust and Pressure
Thrust
The upward force exerted on a body by the fluid in which the body is submerged is called the upthrust or buoyant force. The property of liquid to exert an upward force on a body immersed in it is called buoyancy. Being a force upthrust is measured in newton or kgf in the system of international units.

Examples
  • Pushing an empty can into water experiences the buoyant force.
  • Pushing a cork into water into water experiences the buoyant force.

Effect of Upthrust
The effect of upthrust is that the weight of body immersed in a liquid appears to be less than its actual weight.

Example
Lifting a bucketful of water appears lighter than its actual weightwhen it iswhen it is immersed in water.

Characteristic Properties of Upthrust
  • For the same volume inside the fluid more the density of fluid, greater is the upthrust.
  • Larger the volume of the bodysubmerged in fluid, greater is the upthust.
  • The upthrust acts on the body in upward direction at the centre of buoyancy i.e., the centre of gravity of the displaced liquid.

Factors Affecting the Upthrust
  • The magnitude of upthrust on a body due to a fluid (liquid or gas) depends on the volume of the body submerged in the fluid (liquid or gas) and
  • The magnitude of upthrust on a body due to a fluid (liquid or gas) depends on thedensity of the fluid (liquid or gas) in which the body is submerged.

Magnitude of the Upthrust
Up thrust = Weight of liquid displaced by the body i.e.,
Up thrust = Volume of the body submerged in the fluid × Density of the fluid × Acceleration due to gravity

FB = vρg, Where
FB = Up thrust
v =  Volume of the body submerged in the fluid
ρ = Density of the fluid in which the body is submerged
g = Acceleration due to gravity

Pressure
The suitcases or the handles of heavy luggage items have broad straps. Also the nails that are to be fixed into walls have pointed tips. Did you ever wonder if there is a scientific fact behind these mediocre observations?

The force that acts on an object perpendicular to its surface is the thrust, measured in newton in the SI system or dyne in the CGS system of units. Pressure is the thrust per unit surface, that plays an important role in these situations. Even though the luggage is heavy and it weighs more, the force acting per unit area decreases with broad straps of the handles. Thus lesser pressure facilitates a person holding these items.

Similarly, lesser surface area of the nails (hence the pointed tips) transfer a larger force exerted on them and facilitate easy fixing of them in walls. As pressure is force per unit area, its units are newton per metre square. One metre per metre square is termed Pascal (Pa) after the name of the scientist Blaise Pascal. The pressure exerted by solids depends on their weight and surface area through which the weight acts.

However, the pressure (P) exerted by liquids depends on their density (r), acceleration due to gravity (g) and the height (h) of the liquid column. Mathematically it is given by P = hrg. The pressure exerted at a point in a liquid is equal in magnitude in all directions, hence it is scalar.

Pressure =    Thrust (F)Area (A)     

•  Pressure is measured in N m-2 or Pascal (Pa) in the SI system.

Pascal's Law
The increase in pressure of a liquid at a point is transmitted to all other parts of the liquid without any change. This is Pascal’s law and it is widely used in various applications like hydraulic brakes of vehicles, vehicle lift platforms in garages etc.

•  The thrust exerted by a body remains constant placed in any position.
•  The pressure exerted by the body changes if the surface area of contact of the body with another surface changes.
•  Thrust and pressure are also applicable to fluids i.e., liquids and gases.
•  Buoyancy is the upward force that a fluid exerts on an object when the object is immersed in that fluid.
•  Floating and sinking of objects depends on how the density of the object compares with the density of water.
•  The magnitude of the buoyant force depends on the density of the fluid.
•  Objects with a density less than that of a given liquid float when placed in that liquid.
•  Objects with a density greater than a given liquid sink when placed in that liquid.

Archimedes' Principle and Buoyancy

One of the most useful discoveries to mankind in the field of physics is Archimedes’ principle. Based on this principle, a device called a hydrometer was developed which helps in measuring the density of liquids with ease.

Archimedes’ principle states that “When a body is immersed completely or partially in a fluid, it experiences an upward force that is equal to the weight of the fluid displaced by the body.”

According to the principle, a solid that floats or is immersed in a liquid appears to lose its weight which is equal to the weight of the liquid displaced by the solid. Whether a given solid drowns or not in a given liquid depends on the density of the solid in comparison with that of the liquid.

For an easy approach, we consider the relative density of the substance, which is the ratio of the density of the substance to the density of water, which obviously has no units but a mere number.

If the relative density of the given solid is greater than that of the given liquid, it drowns in the liquid as there is a net downward force on the solid after it gets completely immersed in the liquid. If the relative density of the liquid and that of the solid are equal, then the solid just floats or drowns.

This implies that the solid immerses in the given liquid and stays suspended at the kept position. If the relative density of the given solid is lesser than that of the given liquid, it floats in the liquid.

This happens due to the up thrust or buoyant force of the liquid acting on the solid. The relative density of a floating solid in a given liquid gives the measure of the percentage of the solid that lies below the surface of the liquid.

The “relative density” or “specific gravity” of a substance is defined as the ratio of its density to the density of water at 4 degrees Celsius.

Relative density = Density of substanceDensity of water at 4℃

This can be expressed in many ways. If the numerator and the denominator are both multiplied by “volume”, then we get the expression relative density is equal to the ratio of “mass of the substance” to “mass of water of the same volume.”

Relative density =  Mass of substanceMass of water of same volume

Again, when the numerator and the denominator are both multiplied by acceleration due to gravity “g”, the expression becomes:

Relative density = Weight of substanceWeight of water of same volume

When the substance is immersed in water, it displaces water of volume equal to its own volume. According to Archimedes’ principle, the apparent loss of weight of a body immersed in water is equal to the weight of the water displaced. Therefore:

Relative density = Weight of substanceApparent loss of weight when immersed in water

This expression can be used to find the relative density of a solid body. In order to find the relative density of a liquid, a solid body is taken and its weight is found in air. Then, the weight of the same body is found when it is completely immersed in the liquid whose relative density is to be found. Finally, the weight of the body is found by immersing it completely in water. The relative density of the liquid can be found using the expression:

Relative density =  Loss of weight of a solid body when immersed in liquidLoss of weight of the same body when immersed in water
  • The relative density of a substance is the ratio of density of the substance to the density of water.
  • Relative density should be calculated using the same system of units for the substance as well as water.
  • A hydrometer uses Archimedes’ principle to determine the density of any liquid.
  • Archimedes’ principle is also used in designing ships and submarines.

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