CHARACTERTICS OF GRAVITATIONAL FORCE

i)                 It is always attractive

ii)             It is independent of the medium between the bodies. In contrast electric and magnetic forces are medium dependent.

iii)          It holds good over a wide range of distances ranging from interatomic distance to intergalactic distance. Its range is infinitive.

iv)           Gravitational force is conservative.

v)              The presence of solar system and planets revolve round the sun due to the presence of gravitational force.

vi)           The presence of atmosphere on the earth is due to the gravitational force.

vii)       The gravitational force of the earth keeps us firmly on the surface of the earth.

viii)    Tides in seas and oceans are due to the gravitational force of the sun and moon.

PROPERTIES OF G

1.   The value of G is independent of the nature and chemical composition of the masses of the bodies, and the medium in which the bodies are occurred.

2.   The value of G is unaffected by temperature and pressure.

VALUE OF G

A scientist henry Cavendish determined the value of G experimentally by using extra sensitive apparatus, The value of G is 6.67×10ˉˡNm²kgˉ².

GRAVITATIONAL CONSTANT G

If m1=m2=1 unit and r=1 unit, then, equation(7.1) reduces to G=F

Universal gravitational constant G is numerically equal to the force of attraction between two unit point masses separated by unit distance. Its value is 6.673×10-ˡˡ Nm²kg² in SI and 6.673×10-8 dyne cm²/g² in CGS system. Its dimensional formulae is mˉˡl³tˉ²

The value of G is same everywhere between all bodies in the universe. All our observation confirm this. Newton’s law of universal gravitation is applicable to bodies as small as electronic and atoms and as big as galaxies and cluster of galaxies.

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GRAVITATION

INTRODUCTION

We are already familiar with the term ‘gravity’ and ‘gravitation’ from chapter 3 and 5, in which kinematics of freely falling objects under gravity and conservation force were discussed. Galileo studied the motion of falling bodies systematically. Newton was the first to realize that the weight of the object on the earth’s surface was due to the force of attraction between the objects and the earth. He also realized that this attraction is universal between any two objects in the universe. In 1660, at the age of 23 he discovered his famous law of universal gravitation and published in his book ‘Principal Mathematica’

In this chapter, we will study Newton’s law of gravitation and apply it to the study of the motion of planets and satellites.

NEWTON’S LAW OF UNIVERSAL GRAVITATION

Historically, in 1665, the college at Cambridge, where Sir Isaac Newton studied, was closed due to plague. So, he went to his home in Lincolnshire. One day, according to a popular story, while sitting under an apple tree he saw an apple falling to the earth. That incident set him to think about gravitation. It occurred to him that the force responsible for a body(or apple) to fall to earth is also responsible for providing centripetal force to keep the moon moving round the earth in a close orbit. Accelerations possessed by the moon and the falling body might have the same origin.

Newton also assumed that both centripetal acceleration and acceleration of free fall are directed towards the centre of the earth acting on the moon and the apple respectively.

The acceleration due to gravity g was known to be equal to 9.8 m/s². Centripetal acceleration was calculated by Newton using expression a=ŵ²r=4Π²r/T² and it was found to be 2.72×10-³m/s². In this calculation T- period of rotation of the moon – was taken as 27.3 days and r-earth-moon distance as 3.84×10^8m.

This expression shows that acceleration, and hence, force is inversely proportional to the square of its distances from the centre of the earth. Newton was able to justify this result only by assuming the mass of the earth to be concerned at centre of the earth.

These ideas generalized by Newton in his well known law of gravitation, are given in the statement below:

“Everybody in the universe attracts every other body which is directly proportional to the product of masses and inversely proportional to the square of  distance between their centre.

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VECTORS

1. Scaler and Vectors

All physical quantities can be divided into two categories: Scaler and vectors.

SCALERS: Physical quantities which have only magnitude but no direction are called scalers. Mass, length, time, volume, speed, density, work, energy, power, temperature, specific heat, entropy, charge, electric current, electric potential, frequency etc. belong to this category.

Scaler quantity can be specified by a numerical value and a unit. For example, if one says that the average speed of a bicycle is 25 km/hr, 25 is the number(numerical value) and km/hr is the unit in which the speed has been expressed. This of course, does not say anything about the direction in which the bicycle is going. Scalers, being numbers, can be added algebraically.

VECTORS: Physical quantities, which has both magnitude as well as direction are called vectors. Velocity, acceleration, displacement, force, current density, electric and magnetic field, intensities etc belong to this category. Vector quantity can be specified by a unit, a numerical value and a statement about its direction. For example, if one says that the average velocity of a bicycle is 25 km/hr towards east, then this physical quantity has a numerical value25. a unit km/hr and a direction as east. Therefore, velocity is a vector.

The rule of ordinary algebra donot apply to vectors. For example, addition of two equal forces each of magnitude 2N can produce any resultant force from zero to 4N depending on direction of forces being added. In this chapter, we will define addition and multiplication rules for vectors and derive expressions for a resultant.

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TYPES OF UNCERTAINTIES

Uncertainties can be broadly classified into two categories.

1)   Uncertainties due to known causee systematic uncertainty

2)       Uncertainty due to known causes- Random uncertainty

 

Systematic uncertainty

They may arise due to various reasons. For example,(i) Incorrect design or calibration of the instrument gives certain uncertainty in the measurement. Limitations of the methods used for measurements causes this type of uncertainty. Lack of accuracy of the formula being used in the measurement, can also lead to this uncertainty. Usually the systematic uncertainty leads the measurement to consistently higher or consistently lower than the true value. In other words, systematic uncertainty is of the same sign( either +ve or –ve) and usually of the same size. Repeated observation will have no effect on this uncertainty. This uncertainty can be removed,

i)                 by careful design of the instrument and calibration with greater accuracy;

ii)                  by using improved methods of measurement.

RANDOM UNCERTAINTY

These uncertainties have no specific cause. They may arise due various factors. For example, carelessness of the observer, flexibility in the setting of apparatus, interference on the instrument from external physical factors like temperature, vibration, wind, moisture etc. These uncertainties have no set pattern and take place in a random manner, hence the name ‘random uncertainty’ or ‘random error’. A better experimenter can minimize these uncertainties but still they exist in the measurement to some extent. They can be reduced by taking a large number of observation and finding the average (arithmetic mean).This mean value can be supposed to be close to the ‘true’ value.

SIGNIFICANT FIGURE

Very often the uncertainty in the value of a quantity is not stated explicitly, specially when we are calculating (and not measuring). In those cases the uncertainty is indicated by the number of significant figures (meaningful digits). For example, the mass of a small marble is stated as 8.90 g. By writing it this way we mean that the first two digits are correct while the third digit is uncertain(but not incorrect). The uncertainty is about 0.01g. The rightmost or the doubtful digit is called the least significant digit and the leftmost digit is called the most significant digit. If we write the same value as 8.9g, the number of significant figures is two and the uncertainty is 0.1 g. The uncertainty in distance d=137 km is 1 km while the same uncertainty in d=137.0 km is about 100m

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TYPES OF PHYSICAL QUANTITIES

Physical quantities may be variable and constants. They may have dimension or could be dimensionless.

i)                 DIMENSIONAL VARIABLE: These are the quantities which are variable and have dimensions as well. example Velocity, acceleration, force, etc.

ii)             Dimensional constant: These are the quantities which have constant values and yet have dimensional example. Gravitational constant, coefficient of viscosity, Planck’s constant, velocity of light etc. Their numerical value does not change.

iii)          Non- dimensional variables: These are variable quantities and have no dimensions example specific gravity, strain, angle etc. They are typically ratio of two similar physical quantities.

iv)           Dimensionless constants: These are mere numbers like 2, 3, 4, ∏ . The numeric value of dimensionless physical quantities remains unchanged in any system of units.

USE OF DIMENSIONAL EQUATION

Dimensional equation have following uses:

i)                 To check the correctness of a physical equation

It is based upon the principal of homogeneity of dimensions. According to this principle, the dimensions of all terms on two sides of a dimensional equation must be the same. The problem are solved by writing the dimensions of all terms on both sides of an equation.

4.Uncertainty in measurement and uncertainty in figures

Whenever you measure the value of some physical quantity using a measuring device, the precision of the result always depends on the instrument used. For example, if you measure the length of a tile with a measuring tape with

Marks at 1 cm interval, the result could be 10 cm. If you measure the same tile with a regular scale(marks at 1 mm interval) the result might be 10.1 cm. In the first case it is wrong to write 10.0 because it will misrepresent the limitation or precision of the measuring instrument. If you use a vernier caliper(of precision 0.1 mm) the result might be 10.16 cm. The differences between these three measurements is in their uncertainty, which reflects the precision of the measuring device. In the first case the uncertainty is 1 cm, in second it is 0.1 cm and in third case it is 0.01 cm. It tells that measurements with a vernier caliper has low uncertainty in comparison to measurement with a tape. Any measurements with it will have better

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TABLE1.6: SOME DIMENSIONAL FORMULAE AND SI UNITS

S.N	Physical Quantities	Symbol	Relation with other physical quantities	Dimensional Formula	SI Unit	CGS Unit
1	AREA	A, s	Length ×breadth	L×L=[l]²	M²	Cm²
2	VOLUME	V	LENGTH× BREADTH× HEIGHT	L×L×L=[L³]	m³	cm³,cc
3	FREQUENCY	F, v, n	1/time period	1/[t]=[T-1]	Hertz=s-1 Hz cycle/s	Hertz=s-1 Hz cycles/s
4	DENSITY	ρ	Mass/volume	M/l³=[ml-3]	Kg/m³	G cm-³
5	VELOCITY	V,  u	Displacement/ time	[LT-1]	m/s, ms-1	Cm s-1
6	ACCELERATION	â	Displacement/time²	[LT-2]	M/S², MS-2	Cm s-2
7	FORCE	F	MASS×ACELERATION	[MLT-2]	Kgm/s²,N	G cm s-2
8	WORK	W	force× acceleration	[ML² T-2. ]	Nm, J	Dyne. Cm, erg
9	POWER	P	WORK/TIME	[ML²T-3]	J/S, w	ERG S-1
10	MOMENTUM	p	mass× velocity	[mlt-1]	K gm/s, ns	Gcms-1 dyne. S
11	ANGULAR DISPLACEMENT	Ɵ	ARCLENGTH/RADIUS	L/L=[LºMºTº]: dimensionless	rad	Rad
12	Angular  velocity	Ɯ	Angle/time	[T-1]	Rad/ s	Rad s-1
13	ANGULAR ACCELERATION	â	Angle/time²	[t-²]	Rad/s²	Rad/s-2
14	Moment of inertia	I	Mass ×(radius of gyration)²	[ML²]	kgm²	g.cm²
15	TORQUE	π	Force ×displacement	[ML²Tˉ²]	Nm	Dyne. Cm
16	Angular momentum	L	Moment of inertia× angular velocity	[ML²Tˉˡ]	Kgm²/s, Js	g.cm² sˉˡ
17	Gravitational constant	G	force× (displacement)² /mass× mass	[MˉˡL³Tˉ²]	Nm²/kg²	Dyne cm²gˉ²
18	Pressure	P	Force/area	[MLˉˡTˉ²]	N/m² or Pa	Dyne cmˉ²

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DEFINATION OF BASE UNITS

1.METER: In a1960 a meter was defined to be the equal to 1650763.73 wavelength of orange red light( of wavelength 6.0578021*10^-7m) emitted in vacuum by krypton-86 gas due to transition from 5d^5 to 2p^10 levels. In 1963 it was redefined in terms of speed of light in vacuum as the length of path traveled by light during a time interval of  1/2,99,792,458 of a second.

KILOGRAM: A kilogram is equal to the mass of a cylinder of platinum – iridium kept at all international bureau of weights and measures at serves, near Paris, France.

3.SECOND: A second is defined as the duration of 9,192,631,770 periods of radiation corresponding to transition between two hyperfine levels of the ground state of cesium-133.

Table 1.4: Some derived SI unit with special names

S.NO	PHYSICAL QUANTITIES	UNIT	SYMBOL OF UNIT
1	FORCE	NEWTON	N
2	ENERGY	JOULE	J
3	POWER	WATT	W
4	PRESSURE	NEWTON	N/M^2
/(METER)^2
5	FREQUENCY	HERTZ	HZ
6	SPECIFIC HEAT	JOULE/KG/	J/KG/K,JKG^-1 K^-1
KELVIN
7	LUMINOUS FLUX	LUMEN	LM
8	ILLIUMINANCE	LUX	LX
9	ELECTRIC CHARGE	COULOMB	C
10	ELECTRIC CURRENT	AMPERE	A
11	ELECTRIC POTENTIAL	VOLT	V
12	ELECTRIC INTENCITY	VOLT/METER	V/M OR NC^-1
OR NEWTON/
13	ELECTRIC CAPACITY,	FARAD	F
CAPACITANCE
14	ELECTRIC RESISTANCE	VOLT/AMPERE	V/M OR OHM
,OHM
15	PERMITTIVITY	FARAD/METER	F/M
16	RESISTIVITY	OHM-METER	OHM-M
17	INDUCTANCE	HENRY	H
18	MAGNETIC FLUX	WEBER	WB
19	MAGNETIC FLUX	WEBER/M^2	WB/M^2 OR T
DENSITY
20	MAGNETIC FIELD	AMPERE/	A/M
INTENCITY

AMPERE: An ampere is defined as that constant current which, when flowing in two infinitely long straight parallel conductors of negligible cross section and placed one meter apart in vacuum, would produce between them a force equal to 2*10^-7 Newton per meter of length of each conductor. We will learn more about it in electricity and magnetism.

KELVIN:A Kelvin is defined as the fraction of 1/273.16 of the thermodynamic temperature of the triple point of water. We will learn more about it in heat unit.

CANDELA: The luminous Intensity in a given direction of a source that emits monochromatic radiation of frequency 5.40*10^14 HZ having a radiation intensity in that direction of 1/683w per steradian.

7.MOLE:  A mole is the amount of substance which contains as many elementary entities(molecules) as there are atoms in 12*10^-3 kg of carbon-12. This units is very often used in chemistry.

ADVANTAGE OF SI UNITS

SI has following advantage over other system of units:

1)   It is a rational system of units. SI makes use of only one unit for one physical quantity. For example, for all types of energies example mechanical, heat, electrical etc.The units is joule.

2)   SI is a coherent system of units. In SI, all the derived units can be obtained by dividing and multiplying the base and supplementary units without introducing numerical factors.

• 3)   SI is a metric system. Like CGS and MKS system, the multiples and submultiples can be expressed as power of 10.

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