## Laws of Motion, Force, Work, Energy & Power, Centre of Mass

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

• When a carpet is suddenly jerked the dust fly off, because due to the sudden jerk the carpet moves but the dust on account of inertia of rest is left behind.
• The passenger standing in a bus tends to fall backwards when the bus suddenly starts, this is because his feet are in direct contact with the floor of the bus and the friction at the contact is high this faction does not allow the feet to slip on the floor, the feet therefore move forward with the floor and the upper part of the body is still at rest for a while thus the passenger gets a backward jerk.
• Coin drops into the glass when sudden force is applied on the cardboard (see figure).

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

• It Is dangerous to jump out of a moving vehicle (bus/train), this is because inside the train/bus, complete body of the passenger is in a state of motion with the train/bus and

• on reaching the ground his feet come to rest but upper part of the body continues to move with the speed of vehicle and the person falls forward  on the ground. It is dangerous to jump out of a moving train and it is better to come out when it halts. However if in case of some emergency if  some person wants to jump safety from a moving  vehicle he should run for quite a while in the  direction of motion of the vehicle after the jump  so that his entire body remains in motion for some time.

• When a running car stops suddenly, the passenger is jerked forward. The reason is that in a running car, the whole body of passenger is in the state of motion. As the car stops suddenly, the lower part of his body being in contact with the car, comes to rest but his upper part remains in the state of motion due to the inertia of motion. Thus he gets jerked forward.

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

• Motor cars are able to move along a road because the reaction of the road pushes the car along in response to the action of the wheels pushing on the road.
• Swimming in a pond- a swimmer pushes (or applies force) the water with his hands and feet to move in the forward direction in water. It is the reaction to this force that pushes the swimmer forward.
• Propulsion of aeroplane. The propellers of an aeroplane pushes the air backwards and the forward reaction of the air makes the aeroplane move forward.

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

• Recoiling of a gun: guns recoil when fired, because of the law of conservation of momentum. The positive momentum gained by the bullet is equal to negative recoil momentum of the gun and so the total momentum before and after the firing of the gun is zero.
• Propulsion of Jet and Rockets: A rocket standing at the launching pad has zero momentum. When the propellants inside the rocket burn, a high velocity blast of hot gases is produced.

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:

• frictional force opposes the movement of an object across the surface of another with which it is in rough contact.
• the direction of the frictional force is opposite to the potential direction of motion.
• the magnitude of the frictional force is only just sufficient to prevent movement and increases as the tendency to move increases, up to a limiting value. When the limiting value is reached, the frictional force cannot increase any further and motion is about to begin (limiting equilibrium). When the frictional force F reaches its limit, its value then is related to the normal reaction N in the following way $F=\mu N$or $\mu =F/N$

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.,            ${{f}_{k}}={{\mu }_{k}}N$

• Sliding friction: When one body slides over the surface of another body, the resistance to its motion is called as sliding friction. It is always more than rolling friction.
• Rolling friction: When one body rolls over the surface of another body, the resistance to its motion is termed as rolling friction. Friction in this case is very small.

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.

• We could not hold articles such as glass tumbler and other things without friction. It becomes very difficult to hold a greasy glass.
• We could not write with pen or pencil if there is no friction.
• Friction helps objects to move, stop or to change the direction of motion. We cannot walk without friction.

Friction is an evil

• It causes wear and tear. For example, soles of shoes, ball bearings, steps of a stair, parts of machines etc.
• Friction produces heat. When a machine is operated, heat generated causes damage to the machinery.

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

• Cream Separator: It is device which works on the principal of centrifugal force. It contains a vessel which has milk, when it rotated the lighter particles i.e. the cream is collected in a cylindrical layer around the axis and the milk is drained through an outlet attached to the vessel.

• Washing Machine Drier: When wet clothes are packed tightly in a cylindrical vessel with perforated walls and rotated with very high speed, water particles move out through the walls of the vessel.

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

• motion of moon around the earth.
• motion of satellite round its planet.

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

• When a vehicle moves on a curved road it requires centripetal force. Outer edge of the curved road is raised above the inner edge in order to provide centripetal force.
• Electrons which moves around the nucleus requires centripetal force.
• Earth experiences centripetal force in order to move around the sun.

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:

• A coolie with a luggage on his head, moving on a horizontal platform, does no work, since the direction of force is vertically up and displacement horizontal i.e., angle $\theta$between them is ${{90}^{o}}$(even though he might feel physically tired).

Positive work:

• When a horse pulls a cart, the applied force and the displacement are in the same direction. So, work done by the horse is positive.

Negative work:

• When brakes are applied to a moving vehicle, the work done by the braking force is negative. This is because the braking force and the displacement act in opposite directions.

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

• When a boy runs, the internal energy in his body is converted into kinetic energy.
• When we throw an object, the muscular energy stored in our body is converted into kinetic energy.
• Electric energy into light energy-Electrical Bulb.
• Chemical energy into electrical energy-Cell.
• Electrical energy into heat energy- Heater.
• Electrical into mechanical – Electrical motor
• Mechanical into electrical – Dynamo
• Sound energy into electrical energy – Microphone

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

• Elastic collision
• Inelastic collision

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

• linear momentum is conserved
• kinetic energy is conserved
• total energy is conserved
• coefficient of restitution is unity (e = 1)

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:

• linear momentum is conserved
• kinetic energy is not conserved
• total energy is conserved
• coefficient of restitution is 0 < e < 1

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:

• The geometrical shape of the body
• The distribution of mass in the body

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.

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.

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

$=\frac{load}{effort}=\frac{effort\,arm}{load\,arm}$

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

• A handle or knob is provided at the free edge on the plank of the door as torque is maximum, when d is maximum. Thus we can close or open the door easily by applying less force at the edge of the door.
• When we open the lid of a bottle by turning it, our fingers are applying a couple to the lid.

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

• The angular velocity of a planet revolving in an elliptical orbit around the sun increases, when it comes near the sun and vice-versa. When the planet moving along its elliptical orbit is near the sun, its moment of inertia about the axis through the sun decreases and therefore its angular speed increases. On the other hand, when it is far away from the sun, its moment of inertia increases and hence angular speed decreases.
• An ice-skater or a ballet-dancer can increase her angular velocity by folding her arms and bringing the stretched leg close to the other leg.

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