Category : UPSC
LAWS OF MOTION, FORCE, WORK, ENERGY & POWER, CENTRE OF MASS
LAWS OF MOTION AND FORCE
Everybody in this universe stays in a state of rest i.e., no change in position of a body wrt time or of uniform motion i.e., change in position of a body wrt time. This chapter is concerned about the cause of rest or motion (i.e., force) and its effect (i.e., acceleration or deceleration) and their relationship.
NEWTON'S LAWS OF MOTION
Newton’s First Law of Motion
According to this law, an object continues in a state of rest or in ci state of motion at a constant speed along a straight line, unless compelled to change that state by a net force. In other words, if ci body is in a state of rest, it will remain in the state of rest and if it is in the state of motion, it will remain moving in the same direction with the same velocity unless an external unbalanced force is applied on it. This law is also called law of inertia. It gives qualitative definition of force.
Handy Facts
A common misconception about Newton’s first law of motion is that a force is required to keep an object in motion. This is not so. Experiments done on air tracks (where there is a negligible friction) show that no force is required to keep an object moving with constant velocity. We get this misconception because friction is always present in our everyday lives.
Inertia and Mass
A greater net force is required to change the velocity of some objects than of others. The net force that is just enough to cause a bicycle to pick up speed will cause an imperceptible change in the motion of a freight train. In comparison to the bicycle, the train has a much greater tendency to remain at rest. Accordingly, we say that the train has more inertia than the bicycle. Quantitatively, the inertia of an object is measured by its mass. Inertia is the natural tendency of an object to remain at rest or in motion at a constant speed along a straight line. The mass of an object is a quantitative measure of inertia. The greater the mass, the greater is the inertia of body.
Types of Inertia
Inertia of rest: The tendency of the body to continue in state of rest even when some external unbalanced force is applied on it is called inertia of rest.
Science in Action
It is because of the property of inertia of rest, the coin continues in the state of rest.
Inertia of motion: The tendency of the body to continue in its state of motion even when some unbalanced force is applied on it is called the inertia of motion.
Science in Action
Newton’s Second Law of Motion
It states that the rate of change of momentum of a body is directly proportional to the applied unbalanced force. i.e., Rate of change of momentum \[\propto \] force applied
or, \[F\propto \frac{\Delta p}{\Delta t}\]
If a body is moving with initial velocity u and after applying a force F on it. Its velocity becomes v in time t, then
\[F\propto \frac{m(v-u)}{t}\]
Here \[\frac{(v-u)}{t}=a\](acceleration)
So F \[\propto \]ma or F = k ma, where A; is proportionality constant.
Momentum
The momentum of a moving body is defined as the product of its mass and velocity. If we represent the mass and velocity of a body by m and v respectively, then momentum is given by
\[\vec{p}=m\,\vec{v}\]
The direction of momentum of a body is same as that of its velocity.
The SI unit of momentum is kilogram meter per second (kg m/s).
Impulse or Change in Momentum
From Newton’s second law, \[\vec{F}=\frac{\Delta \vec{p}}{\Delta t}\] one can derive the impulse momentum theorem. This theorem states that impulse is equal to the change in momentum, or, \[\vec{F}=\Delta t=\Delta \vec{p}=\vec{p}-{{\vec{p}}_{0}}\]
where \[\vec{F}\Delta t\] is called impulse, \[\vec{F}\] is the average force and \[\Delta t\] is the time interval the force is in action.
NEWTON'S THIRD LAW OF MOTION
It states that to every action there is always an equal and opposite reaction.
This law of motion states that ‘if a body A exerts a force +F on a body B, then body B exerts a force -F on A, that is a force of the same magnitude and along the same line of interaction but in the opposite direction’.
Science in Action
CONSERVATION OF MOMENTUM
The principle of conservation of momentum states that “if there is a direction in which there is zero unbalanced force acting on a system then the total momentum of that system in that direction is constant even if the bodies act on each other”.
Also, the total momentum of the system remains constant, if no external force acts on a system of constant mass.
\[{{m}_{1}}\overrightarrow{{{v}_{1}}}+{{m}_{2}}\overrightarrow{{{v}_{2}}}+{{m}_{3}}\overrightarrow{{{v}_{3}}}+.......=\]constant
Ex. The pull of the Earth, do act on the bodies, but the result can still be used if there is a direction in which the external forces are balanced
Science in Action
Theses gases pass out through the tail nozzle of the rocket in downward direction with tremendous velocity. There for the rocket moves up with such a velocity so as to make the momentum of the system (rocket + emitted gases) zero.
FORCE
A force is that physical quantity which tries to change or changes the state of rest or of uniform motion of a body.
Units of force: The S.I. unit of force is newton.
In C.G.S. system, the unit of force is dyne.
1 newton = \[{{10}^{5}}\]dyne
In MKS system, the unit of force is the kilogramme force (kgf). 1 kgf= 9.8 newton (or 9.8 N)
Basic Forces in Nature
There are four basic forces in nature and they are
(i) Gravitational Force: Everybody in the universe attracts each other, this force is known as gravitational force. This is the weakest force among all other forces which is existing.
(ii) Weak Nuclear Force: These forces are \[{{10}^{25}}\] times stronger than gravitational force.
(iii) Electromagnetic Force: The electromagnetic forces are the forces between the charged particles. When charges are at rest, then the force is called as electrostatic force. This force is much stronger than gravitational force and it
(iv) Strong nuclear forces: This is the strongest force found in nature. These forces acts between the proton and the neutron in order to bind them in the nucleus.
This force is \[{{10}^{38}}\]times stronger than gravitational forces, \[{{10}^{2}}\]times stronger than electrostatic forces and \[{{10}^{13}}\]times stronger than weak nuclear forces.
FRICTION
Friction is a resistance to the relative motion between two objects in contact (in case of solid objects) or the body and its surroundings (in case object is moving in a fluid). Actually, when two objects are kept in contact, a reaction force R acts between the two objects as shown in the figure.
This reaction force R has two components -F, along the surface and M perpendicular to the surface. The force F which acts along the surface is called the force of friction.
The results of experimental investigation into the behaviour of frictional forces confirm that:
The constant \[\mu \] is called the coefficient of friction and each pair of surfaces has its own value for this constant.
Types of Friction
Static frictional force: When there is no relative motion between the contact surfaces, frictional force is called static frictional force. It is a self-adjusting force, it adjusts its value according to requirement.
The maximum value of static friction is called limiting friction.
Kinetic frictional force: Once relative motion starts between the surfaces in contact, the frictional force is called as kinetic frictional force. The magnitude of kinetic frictional force is also proportional to normal force
i.e., \[{{f}_{k}}={{\mu }_{k}}N\]
The coefficient of rolling friction (\[{{\mu }_{R}}\]) is the least and coefficient of static friction is maximum, i.e., \[{{\mu }_{R}}\prec {{\mu }_{K}}\prec {{\mu }_{S}}\].
Friction: A necessary Evil
Friction is necessary for doing various activities in our daily life.
Friction is an evil
Science in Action
Fast moving object such as cars, bullet trains, aeroplanes are all streamlined-designed with curved and sloping surfaces to cut through the air and reduce drag. Boats can also be streamlined to reduce water resistance.
Motion in a Lift
The weight of a body is simply the force exerted by earth on the body. If body is on an accelerated platform, the body experiences fictitious force, so the weight of the body appears changed and this new weight is called apparent weight. Let a man of weight W = Mg be standing in a lift.
Case (a): If the lift is moving with constant velocity v upwards or downwards.
Apparent weight, W = actual weight W
Case (b): If the lift is accelerated i.e., a = constant and in upward direction.
Apparent weight,
\[W'=W+{{F}_{0}}=Mg+Ma=M(g+a)\]
Case (c): If the lift is accelerated downward with acceleration a < g:
Apparent weight,
\[W'=W+{{F}_{0}}=Mg-Ma=M(g+a)\]
Case (d): If the lift is accelerated downward with acceleration a > g:
Apparent weight, W’ = M (g - a) is negative.
CENTRIPETAL FORCE
If m be the mass of object then it experiences a force which directs towards the centre of the circular path and has a magnitude given by
\[{{F}_{c}}=m{{a}_{c}}=\frac{m{{v}^{2}}}{r}\]or\[F=mr\,{{\omega }^{2}}\] [\[\therefore \,\,\,\,\,\,\,\,\,v=r\omega \]]
This force is known as centripetal force.
CENTRIFUGAL FORCE
The virtual force which balances the centripetal force in uniform circular motion is called as centrifugal force. It is not the real force as it is due to the acceleration of rotating frame. When a body is rotating in a circular path and the centripetal force vanishes, the body would leave the circular path.
Science in Action
CIRCULAR MOTION
Motion of a particle along a circle or circular path is called a circular motion. If the body covers equal distances along the circumference of the circle, in equal intervals of time, the motion is said to be a uniform circular motion. A uniform circular motion is a motion in which speed remains constant but direction changes so velocity.
Examples of uniform circular motion are
BANKING OF ROAD
The tilting of the vehicle is achieved by raising the outer edge of the circular track, slightly above the inner edge. This is known as banking of curved track.
CONDITION OF OVERTURNING
If speed is greater than limiting speed, then condition of overturning is occurred.
Science in Action
WORK, ENERGY AND POWER
The meaning of work in physics is different from its meaning in common language. Actually, in physics work has a meaning only when a displacement is caused in a body by the applied force on it. If there is no displacement in a body by an applied force in the direction of force, no work is said to be done.
WORK
Work is defined as the product of the force and displacement in the direction of applied force or product of displacement and force in the direction of displacement.
W= Force \[\times \] displacement in the direction of force
= F.S = FS cos \[\theta \]
where \[\theta \] is the angle between F and S.
The SI unit of work is newton-metre is also called joule (J)
1 joule = \[{{10}^{7}}\] erg
Work done by a force applied at an angle
W = component of force in the direction of displacement\[\times \]magnitude of displacement = F cos \[\theta \,s\]
Work done by a force can be positive, negative or zero as the value of cos \[\theta \,\]is positive, negative or zero.
(\[\therefore \] F and s, being magnitudes, are always positive)
Work is a scalar quantity but you can have positive and negative work.
Science in Action
Zero work:
Positive work:
Negative work:
ENERGY
Energy is defined as the capacity to do work.
The SI unit of energy is the joule (J) same as that of work. The commonly used unit for electrical-energy consumption is the kilowatt-hour (kWh).
Thus, 1 kWh = 1 kW\[\times \]1 hour
= (1000 W)\[\times \](3600 s)
= (1000 J/s)\[\times \] (3600 s)
= (3600000 joules) =\[3.6\times {{10}^{6}}j\].
For electrical-energy consumption in houses, factories, shops, etc., kilowatt-hour is simply called ‘unit’ (Board of trade unit B.O.T.U.).
Kinetic Energy (K.E.)
Energy possessed by a body by virtue of its state of motion is called kinetic energy. Kinetic energy is always positive and is a scalar.
\[K.E.=\frac{1}{2}m{{v}^{2}}=\frac{{{P}^{2}}}{2m}\]
Potential Energy (RE.)
Potential energy is energy due to position. If a body is in a position such that if it were released it would begin to move, it has potential energy.
P.E. = mgh
For example, energy of water in a water tank on the roof, energy of small spring in ball-pen, etc.
Gravitational potential energy
When an object is allowed to fall from higher level to a lower level it gains speed due to gravitational pull, i.e. it gains kinetic energy.
The magnitude of its gravitational potential energy is equivalent to the amount of work done by the weight of the body in causing the descent.
If a mass m is at a height h above a lower level, the PE. possessed by the mass is mgh.
Since h is the height of an object above a specified level, an object below the specified level has negative potential energy
Work-Energy Theorem
According to the work-energy theorem, total work done on a system by forces equals to the change in kinetic energy.
LAW OF CONSERVATION OF ENERGY
According to this law, energy can only be converted from one form to another, it can neither be created nor destroyed. The total energy before and after the transformation always remains the same.
Transformation of Energy
The conversion of one form of energy to the other form is termed as transformation of energy. The phenomenon in which energy is transformed from useful from to useless form is known as dissipation of energy.
Science in Action
Mass-Energy equivalence Relation
According to this relation mass (m) and energy (E) are inter convertible
\[E=m{{c}^{2}}\]
Where, c =\[3\times {{10}^{8}}m{{s}^{-1}}\] is the velocity of light in vacuum or air.
POWER
The time rate of doing work is defined as power (P). If equal works are done in different times, power will be different. More quickly work is done, power will be more.
\[power\,(P)=\frac{work\,(W)}{time\,(t)}\]
The S.I. unit of power is the joule per second and is called the watt (W). 1 H.P (Horse power) = 746 W
COLLISIONS
Collision is an event in which two or more than two bodies interact with one another for a short time and exchange momentum and kinetic energy. Collisions are of two types
Handy Facts
Linear momentum is always conserved, in collision whereas kinetic energy is conserved only in elastic collision.
Elastic Collision
A collision in which there is no loss of kinetic energy is called elastic or perfectly elastic collision. The basic characteristics of perfectly elastic collision are
Handy Facts
The collision in one dimension is also known as head-on collision.
Inelastic Collision
In an inelastic collision kinetic energy is lost during collision.
The basic characteristic of an inelastic collision are:
In case of perfectly inelastic collision the two bodies get stuck together and move with common velocity that is why for perfectly inelastic collision.
CENTRE OF MASS AND ROTATIONAL MOTION
The motion through space in which the position of the centre of mass of the object changes is considered as translational motion.
CENTRE OF MASS (COM)
For a system of particles, centre of mass, is that point at which its total mass is supposed to be concentrated.
Rigid Body
A body which does not deform on the application of whatsoever large force is called a rigid body. Ideally such type of body will not exist but practically, large, extended object can be treated as rigid body. For example, door is a rigid body.
Centre of Mass of Some Symmetrical Regular Shaped Objects
When bodies are symmetrical in shape and have uniform densities then their centre of mass would lie on their geometrical centres.
The position of centre of mass depends on following two factors:
ROTATIONAL MOTION
A rigid body performs a pure rotational motion, if each particle of the body moves in a circle, and the centre of all the circles lie on a straight line, called the axis of rotation.
Examples: Motion of a ceiling fan, motion of a potter’s wheel, etc.
Angular Displacement (\[\Delta \theta \])
The change in position of a particle moving in a circular path with respect to the centre is known as it’s angular displacement.
Its SI unit is radian.
Angular Velocity (\[\omega \])
The rate of change of angular displacement of a body.
Average angular velocity\[\omega =\frac{{{\theta }_{2}}-{{\theta }_{1}}}{{{t}_{2}}-{{t}_{1}}}=\frac{\Delta \theta }{\Delta t}\]
Also, angular velocity, \[\omega =2\pi n\] where, n = number of revolutions per second.
Its SI unit is radian/s
Relation between angular velocity (\[\omega \]), linear velocity (v) and radius of circular path (r)
\[v=r\,\omega \] or \[\omega =\frac{v}{r}\]
MOMENT OF INERTIA
The property of a body by virtue of which it opposes any change in its state of rest or of rotational motion is defined as its moment of inertia. The moment of inertia of a particle in rotational motion is equal to the product of its mass (m) and square of its distance (r) from the axis.
Moment of inertia, \[I=m{{r}^{2}}\]
It is neither a scalar nor a vector but it is considered as a tensor.
Its SI unit is \[kg\,{{m}^{2}}\]
TORQUE
Torque is the turning or twisting action on a body about the axis of rotation due to a force\[\vec{F}\].
\[\vec{\tau }=(\vec{r}\times \vec{F}).\hat{n},\]
Rigid body in Equilibrium
A rigid body is in equilibrium, if it has zero translational acceleration and zero angular acceleration.
Principle of Moments for a Lever
Load \[\times \] load arm = effort \[\times \] effort arm
Mechanical advantage (M.A.) of lever
\[=\frac{load}{effort}=\frac{effort\,arm}{load\,arm}\]
Couple
When two equal and parallel forces having different line of action acts on a body then it makes a couple. It has always the unidirectional rotational effect.
Couple = force \[\times \] force arm
Science in Action
ANGULAR MOMENTUM
In translational motion the measure of quantity of motion possessed by a body is linear momentum and the physical quantity analogous to it in rotational motion is angular momentum, it is represented by L and it is a vector quantity. Angular momentum\[L=I\omega \]. Its S.I. unit is joule-second.
Relation Between Torque (r) and Angular Momentum (L)
\[\frac{dl}{dt}=\tau \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(As\,\,\tau =I\,\alpha )\]
Conservation of Angular momentum
Suppose on a system of particles of a rigid body no external force is acting then its angular momentum remains conserved, this is known as conservation of angular momentum.
Science in Action
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