Category : UPSC
FORCE OF GRAVITY, SOLIDS & FLUIDS
FORCE PF GRAVITY
Earth attracts everybody towards itself with a force known as ‘gravity’. Due to the force of gravity the ball thrown upwards doesn’t go upwards but it falls downwards after covering some vertical distance. Actually, every object attracts every other object towards itself with a force. This force is called the gravitational force. Gravitational force is one among the four fundamental forces. It is always attractive in nature.
NEWTON'S UNIVERSAL LAW OF GRAVITATION
Newton came to the conclusion that any two objects in the Universe exert gravitational attraction on each other.
Any two particles of matter anywhere in the universe attract each oilier with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between them,
i.e., \[F\propto \frac{{{m}_{1}}{{m}_{2}}}{{{r}^{2}}}\] or \[F=\frac{G{{m}_{1}}{{m}_{2}}}{{{r}^{2}}}\]
Here, the constant of proportionality G is known as the universal gravitational constant. It is termed a “universal constant” because it is thought to be the same at all places and all times.
\[G=6.673\times {{10}^{-11}}N{{m}^{2}}/k{{g}^{2}}.\]
Handy Facts
The value of universal gravitational constant, G is very small hence gravitational force is very small, unless one (or both) of the masses is huge.
Important Characteristics of Gravitational Force
Importance of the Universal Law of Gravitation
The universal law of gravitation successfully explained several
MASS AND WEIGHT
The quantity of matter in a body is known as the mass of the body. Mass is quantitative measure of inertia. Mass is an intrinsic property of matter and does not change as an object is moved from one location to another.
Weight, in contrast, is the gravitational force that the earth exerts on the object and can vary, depending on how far the object is above the earth’s surface or whether it is located near another body such as the moon.
The relation between weight W and mass m
\[E=\frac{G{{M}_{E}}m}{{{r}^{2}}};\,\,\,\,\,\,\,\,\,\,\,\,\,W=mg\]
As \[{{g}_{moon}}=\frac{1}{6}{{g}_{earth}}\] therefore,
\[{{w}_{moon}}=\frac{1}{6}{{w}_{earth}}\]
Inertial and Gravitational Mass
The mass of a body is the quantity of matter possessed by a body.
Inertial Mass: Inertial mass of a body is related to its inertia of linear motion, and is defined by Newton's second law of motion.
\[F={{m}_{i}}a\] or \[{{m}_{i}}=\frac{F}{a}\]
The mass \[{{m}_{i}}\]of the body in this sense is the inertial mass of the body.
In fact, inertial mass of a body is the measure of the ability of the body to oppose the production of acceleration in its motion by an external force.
Properties of inertial mass
Gravitational Mass
Gravitational mass of a body is related to gravitational pull on the body and is defined by Newton’s law of gravitation.
If a body of mass \[{{m}_{G}}\]is placed on the surface of earth of radius R and mass M, then gravitational pull on the body is given by
\[F=\frac{GM{{m}_{G}}}{{{R}^{2}}}\Rightarrow {{m}_{G}}=\frac{F}{\left( GM/{{R}^{2}} \right)}\]
The mass \[{{m}_{G}}\]of the body in this sense is the gravitational mass of the body.
ACCELERATION DUE TO GRAVITY OF THE EARTH
When a body is dropped from a certain height above the ground, it begins to fall towards the earth under gravity. The acceleration produced in the body due to gravity is called the acceleration due to gravity. It is denoted by g. Its value close to the Earth’s surface is\[9.8\,m/{{s}^{2}}\].
\[g=\frac{F}{m}\] or \[g=\frac{GM}{{{R}^{2}}}\]
This is the relation between acceleration due to gravity (g) and universal gravitational constant (G).
Acceleration due to gravity,
\[g=\frac{GM}{{{R}^{2}}}\] or \[g=\frac{4}{3}\pi G\,R\rho \]
VARIATION IN ACCELERATION DUE TO GRAVITY
The value of ‘g’ acceleration due to gravity, varies from place to place on the surface of earth. It also varies as we go above or below the surface of the Earth.
Variation in g with height or altitude:
\[g'=g\left( 1-\frac{2h}{{{R}_{e}}} \right)\]
i.e. The decrease in the value of g on going up a height ‘h’ above the surface of earth by a factor \[\left( 1-\frac{2h}{{{R}_{e}}} \right)\]
Variation in g with depth:
\[g'=g\left( 1-\frac{d}{{{R}_{e}}} \right)\]
Thus the value of g decreases by a factor \[\left( 1-\frac{d}{{{R}_{e}}} \right)\] as we go down below the surface of the earth.
Science in Action
ESCAPE SPEED
Escape speed is the minimum speed that should be given to the body to enable it to escape away from the gravitational field of earth.
If the mass of the planet is M and its radius is R, then the escape speed from its surface will be
\[{{V}_{e}}=\sqrt{(2GM/R)}\] or \[{{V}_{e}}=\sqrt{(2gR)}\]
Escape speed from the surface of earth is 11.2 Km/sec.
Handy Facts
The escape velocity of a body from a depends upon the size (mass and radius) of the planet and hence the value of acceleration due to gravity on its surface. It does not depend upon mass of the body. To throw an ant or an elephant out of the gravitational field, the require velocity of projection is same
KEPLER’S LAWS OF PLANETARY MOTION
Kepler worked out three laws, which govern the motion of
planets and are known as Kepler’s laws of planetary motion.
Law of orbits (first law): All planets revolve in elliptical orbits around the sun and the sun is situated at one of the two foci of the elliptical path.
SATELLITES
Just as the planets revolve around the sun, in the same way few celestial bodies revolve around these planets. These bodies are called ‘Satellites’.
For example moon is the natural satellite of Earth. Artificial satellites are launched from the Earth. Such satellites are used for telecommunication, weather forecast etc. The path of these satellites are elliptical with the centre of Earth at a focus.
Characteristics of Motion of Satellites
Orbital velocity (\[{{v}_{0}}\]): Let a satellite of mass m revolves around the Earth in circular orbit of radius r with speed\[{{v}_{0}}\]
The gravitational pull between satellite and earth provides the necessary centripetal force.
Orbital velocity \[({{v}_{0}})=\sqrt{(Gm/R)}=\sqrt{(gR)}\]
Relation between escape speed (\[{{v}_{e}}\]) and orbital speed (\[{{v}_{0}}\]): \[{{v}_{e}}=\sqrt{2}{{v}_{0}}\]
Kinetic energy: The kinetic energy of the satellite is due to motion of the satellite.
\[K=\frac{GMm}{2r}\]
Potential energy: Potential energy of the satellite,
\[U=\frac{GMm}{2r}\]
The negative sign is because of zero potential energy at infinity.
Binding energy of satellite, \[E=\frac{GMm}{2r}\]
TYPES OF SATELLITES
Gee-stationary Satellite
A satellite which appears to be stationary for a person on the surface of the Earth is called geostationary satellite.
It is also known as Communication Satellite or Synchronous Satellite.
Features of Geo-stationary Satellite
i.e., 24 hours = 24 \[\times \] 60 \[\times \] 60 = 86400 sec.
Applications of Geo-stationary Satellite
Polar Satellite
Polar Satellites go around the poles of the earth in north-south direction and the earth rotates around its axis in east-west direction. The altitude of polar satellite is around 500 to 800 km and its time period is around 100 minutes.
FREE FALL
The motion of a body under the influence of gravity alone is called a free fall. When a body falls freely towards the earth, its velocity continuously increases. The acceleration developed in its motion is called acceleration due to gravity (g).
\[g=\frac{GM}{{{R}^{2}}}\]
This gives the acceleration due to gravity on the surface of the earth. \[g=9.8\,m/{{s}^{2}}\]
WEIGHTLESSNESS
The phenomenon of “weightlessness" occurs when there is no force of support on your body.
SOLIDS AMD FLUIDS
This chapter deals with an introduction to the mechanical properties of materials-solids: how they stretch and compress, fatigue, break and shear.
Liquids and gases flow, and we call both fluids. Their atoms and/ or molecules can move around fairly freely.
Viscosity is the internal resistance or friction, offered to an object moving through a fluid.
ELASTICITY
The property of the body by virtue of which it tends to regain its original shape and size after removing the deforming force is called elasticity. If the body regains its original shape and size completely, after the removal of deforming forces, then the body is said to be perfectly elastic.
The property of the body by virtue of which it tends to retain its deformed state after removing the deforming force is called plasticity. If the body does not have any tendency to recover its original shape and size, it is called perfectly plastic e.g., putty and mud.
Science in Action
STRESS AND STRAIN
Stress
The internal restoring force acting per unit area of a body is called stress.
i.e., Stress = Restoring force/ Area
Handy Facts
Materials behave differently under stress. When dropped, a glass tumbler shatters into pieces, a rubber ball deforms then bounces back and a metal suffers dents.
Strain
When a deforming force is applied on a body, there is a change in the configuration of the body. The body is said to be strained or deformed. The ratio of change in configuration to the original configuration is called strain.
i.e., Strain =\[\frac{Change\,\,in\,\,configuration}{Original\,\,configuration}\]
Strain being the ratio of two like quantities has no units and dimensions.
HOOKE’S LAW
Elastic limit is the upper limit of deforming force up to which, if deforming force is removed, the body regains its original form completely and beyond which if deforming force is increased, the body loses its property of elasticity and gets permanently deformed.
Within the elastic limit, stress is proportional to strain.
i.e.. Stress \[\propto \]strain or, stress = E \[\times \]strain
This constant E is known as modulus of elasticity or coefficient of elasticity. It depends upon the nature of the materials.
Types of Modulus of Elasticity
Corresponding to three types of strain, there are three types of modulus of elasticity:
Young’s Modulus of Elasticity (Y)
\[Y=\frac{normal\,\,stress}{longitudinal\,\,stress}\]
Bulk or Volume Modulus of Elasticity (K)
\[K=\frac{Normal\,\,stress}{Volumetric\,\,stress}\]
If p is the increase in pressure applied on me spherical body then, F/A = P
The reciprocal of bulk modulus of elasticity of a material is called its Compressibility.
\[Compressibility=\frac{1}{K}\]
Brittle, Ductile and Malleable solids
There are some materials which break as soon as the stress is increased beyond the elastic limit. They are called brittle, e.g. glass, ceramics etc.
Materials which have large plastic range of extension are called ductile. Using this property, materials can be drawn into thin wires, e.g. copper, aluminium etc.
Materials which can be hammered into thin sheets are called malleable e.g. gold, silver, lead, etc.
Elastomers: Rubber has a large elastic region. It can be stretched several times its original length. On the removal of stress it returns to its original state. But the stress-strain graph is not a straight line. This means, it does not obey Hooke’s law e.g. rubber, elastic tissue of aorta etc.
Handy Facts
Metals are polycrystalline materials. They are elastic for small strains and for large strains, metals become plastic.
FLUIDS
Fluids include liquids and gases. They begins to flow when a shearing stress is applied. Fluids have no definite shape. They assume the shape of containing vessel.
Density (\[\rho \])
Mass per unit volume is defined as density.
Density, \[t=\frac{mass\,\,(m)}{Volume\,\,(V)}\]
SI unit: \[kg/{{m}^{3}}\]
Specific Weight or Weight Density (W)
Specific weight, \[W=\frac{Weight}{Volume}\]
\[=\frac{mg}{V}=\left[ \frac{m}{V} \right]g=\rho g\]
SI Unit: \[N/{{m}^{3}}\]
Specific weight of pure water at \[4{}^\circ \]C is 9.81 \[kN/{{m}^{3}}\]
Relative Density
Relative density (R.D). =\[\frac{Density\,\,of\,\,given\,\,liquid}{Density\,\,of\,\,pure\,\,water\,\,at\,\,{{4}^{o}}C}\]
The density of water is maximum at \[4{}^\circ C\]and is equal to \[1.0\times {{10}^{3}}kg{{m}^{-3}}\].
Specific Gravity
It is defined as the ratio of the specific weight of the given fluid to the specific weight of pure water at \[4{}^\circ \]C.
Specific gravity =\[\frac{Specific\,\,weight\,\,of\,\,given\,\,liquid}{Specific\,\,weight\,\,of\,\,pure\,\,water\,\,at\,\,{{4}^{o}}C\,\,(9.81\,\,kN/{{m}^{3}})}\]
\[=\frac{\rho \ell \times g}{{{\rho }_{w}}\times g}=\frac{\rho \ell }{{{\rho }_{w}}}=\]R.D. of the liquid
PRESSURE IN A FLUID
The pressure exerted by a fluid is defined as the force per unit area at a point within the fluid.
\[{{P}_{av}}=\frac{\Delta F}{A\Delta }\]
When the force is constant over the surface, the above equation reduces to P=F/A
The SI unit of pressure is \[N{{m}^{-2}}\] and is also called Pascal (Pa). The other common pressure units are atmosphere and bar.
1 atm = \[1.01325\text{ }\times \text{ }{{10}^{5}}\]Pa, 1 bar = \[1.00000\text{ }\times \text{ }{{10}^{5}}\]Pa,
1 atm= 1.01325 bar
Expression for liquid pressure and total pressure
The liquid pressure at a depth h is given by
\[P\text{ }=\text{ }gh\rho \]where \[\rho \]is the density of the liquid.
and the total pressure at the same depth h
\[{{P}_{total}}={{P}_{atm}}+\rho gh,\,\,where\,\,{{P}_{atm}}\]is atmospheric pressure.
ATMOSPHERIC PRESSURE, ABSOLUTE PRESSURE AND GAUGE PRESSURE
Atmospheric Pressure
Force exerted by air column on unit cross-section area of sea level is called atmospheric pressure (\[{{P}_{0}}\])
\[{{P}_{0}}=\frac{F}{A}=101.3kN/{{m}^{2}}\]
Barometer is used to measure atmospheric pressure which was discovered by Torricelli.
Atmospheric pressure varies from place to place and at a particular place from time to time.
Absolute Pressure
Sum of atmospheric and gauge pressure is called absolute pressure.
\[{{P}_{abs}}={{P}_{atm}}+{{P}_{gauge}}\]
\[\Rightarrow \] \[{{P}_{abs}}={{P}_{0}}+h\rho g\]
Pascal’s Law
According to Pascal’s law- A pressure applied to a confined fluid at rest is transmitted equally undiminished to every part of the fluid and the walls of the container. This principle is used in a hydraulic presses, brakes, jack or lift, etc.
Science in Action
ARCHIMEDES’ PRINCIPLE
A body immersed in a fluid partly or wholly experiences an upward buoyant force equivalent to the weight of the fluid displaced by it. The buoyant force acts through the centre of gravity of the displaced fluid. The phenomenon of force exerted by fluid on the body is called buoyancy and the force is called buoyant force.
A body experiences buoyant force whether it floats or sinks, under its own weight or due to other forces applied on it.
Body float, if weight of it is less than buoyant force and sink, if weight of the body is greater than buoyant force.
SURFACE TENSION
Surface tension can be defined in the form of an imaginary line on the liquid surface or by relating it to the work done. The force acting per unit length of an imaginary line drawn on the free liquid surface at right angles to the line and in the plane of liquid surface, is known as surface tension.
Surface tension, \[T=\frac{F}{L}\]
Its SI unit: \[N/m\,\]or\[J/{{m}^{2}}\,\].
Examples of surface tension
Factors Affecting Surface Tension
Surface Energy
According to molecular theory of surface tension the molecules in the surface have some additional energy due to their position. This additional energy per unit area of the surface is called surface energy.
i.e.. Surface energy =\[\frac{Work\,\,done}{Increase\,\,in\,\,surface\,\,area}\]
Angle of Contact
The angle enclosed between the tangent plane at the liquid surface and the tangent plane at the solid surface at the point of contact inside the liquid is termed as the angle of contact.
The angle of contact depends on the nature of the solid and liquid in contact.
Shape of liquid surface: When a liquid is brought in contact with a solid surface, the surface of the liquid becomes curved (concave or convex) near the place of contact.
The free surface of a liquid which is near the walls of a vessel and which is curved because of surface tension is known as meniscus. Meniscus concave for glass water and meniscus convex. For glass-mercury.
Angle of contact depends upon the surfaces in contact.
Water proofing agent: Angle of contact increases due to water proofing agent. It gets converted from acute to obtuse angle.
Capillary Rise
A glass tube with fine bore and open at both ends is known as capillary tube. The property by virtue of which a liquid rise or fall in a capillary tube is known as capillary rise or fall or capillarity. Rise or fall of liquid in tubes of narrow bore (capillary tube) is called capillary action.
Handy Facts
When the capillary tube is of insufficient length, the liquid will not overflow. It rises up to the top end of the tube and then adjusts the radius of curvature of its meniscus.
FLOW OF LIQUID
Streamline Flow
When a liquid (fluid) flows, such that each particle of the liquid passing a point moves along the same path and has the same velocity as its predecessor then the flow is called stream line flow. It is also called laminar flow.
Turbulent Flow
When the velocity at a point in the liquid changes with time the flow is called unsteady flow.
BERNOULLI’S THEOREM
When incompressible, non-viscous, irrational liquid i.e., ideal liquid flow from one position to other in streamline path then in its path at every point, the sum of pressure energy, kinetic energy and potential energy per unit volume remains constant.
Science m Action
Dynamic lift: The wings of the aeroplane are having tapering. Due to this specific shape of wings when the aeroplane runs, air passes at higher speed over it as compared to its lower surface. This difference of air speeds above and below the wings, in accordance with Bernoulli’s principle, creates a pressure difference, due to which an upward force called ‘dynamic lift’ acts on the plane, if this force becomes greater than the weight of the plane, the plane will rise up. During storms, the wind blows with very high speed which leads to blowing off the tinned roofs. This is because the pressure above the roof is less than the pressure below the roof.
VISCOSITY
“The property of a fluid by virtue of which it opposes the relative motion between its different layers is known as viscosity and the force that is into play is called the viscous force”.
Effect of Temperature and Pressure on Viscosity
Effect of temperature: On increasing temperature viscosity of a liquid decreases.
Effect of pressure: On increasing pressure viscosity of a liquid increases except water whose viscosity decreases with pressure rise.
Science in Action
An oil, when used as lubricant in a machine, forms a thin layer of the oil over the metallic parts of the machinery. During working of the machinery, the metallic parts do not come in direct contact with each other. The friction between solid- solid surfaces. So that the oil layer is effective as lubricant for a long time, the oil should be of high viscosity.
Stoke’s Law
When a solid moves through a viscous medium, its motion is opposed by a viscous force depending on the velocity and shape and size of the body.
Importance of Stokers law
Terminal Velocity
It is the maximum constant velocity acquired by the body while falling freely in a viscous medium.
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