Mechanics Part-1

Newton’s LawsThere are three main laws Newton expressedabout motion.1. 1st Law or the Law of inertia2. 2nd Law3. 3rd Law

The next slides will explain what these laws meanand how they are applied in nature

Newton’s First Law(Law of Inertia)

An object at rest remains at rest, unless acted upon by a net force. An object inmotion remains in motion, unless acted upon by a net force.(An object will remain at rest or continue motion unless an externalunbalanced(net force) force acts on it)

E.g.: A carom ball will not move until someone hits on it.Net force- The net force is the sum of the forces acting upon an object.

- FR

Net force= F + (-F) = 0 Net force = F + (-F) + R = R

Newton’s Second LawThe change of momentum per second is proportional to the applied forceand the momentum change takes place in the direction of the force.F α m × Change in velocity per secondF α ma ( a- acceleration= rate of change of velocity )F = k .maWhen F = 1N, m = 1 kg , and a = 1 ms-2 , then k= 1,

Therefore, F = maAlternate proof:

Since, force is equal to the rate of change of momentum,F = dP / dt . (Where P is the momentum)Thus, F= d(mv) / dt. (P=mv , m-mass, V- velocity)F= m dv/ dt.F= ma.

F = ma

Newton’s Third Law

To every action, there’s an equal and oppositereaction.Consider a tennis ball hitting the racket. Here the force exertedby the ball on the racket is equal to the force exerted on theball by the racket. Also the sum of momentums after andbefore are equal and thus the system obeys the conservationof linear momentum principle.

Force on theracket exertedby the ball Force on

the ballexertedby theracketBefore Collision After Collision

Some key termsImpulse- Impulse is the change of momentum in a collisionor the product of the force exerted and the time taken forthe collisionThus, I= Ft= Δmv (Δmv- change of momentum)Inertia- It is the tendency of an object to remain at aconstant velocity, or its resistance/reluctance of beingaccelerated.Terminal Velocity- It is the maximum velocity of anobject falling from a height or moving from rest through africtional medium. Thus, at terminal velocity, the totalresistance acting upon an object should be equal to thegravitational attraction/force.

Types of forces1. Weight

2. The normal force

W= mg

R The normalforce on anobject alwaysactsperpendicularto the plane inwhich theobject is kept

The product of themass and thegravitationalacceleration, theweight, always actstowards the center ofthe earth regardlessof the object’sposition or inclination

3. Friction4. Tension

R

WW

R1

R2

FrictionFriction is the force resisting the relative motion of anobject.There are two types of friction(dry friction):1. Static friction2. Kinetic friction

Static friction- Static friction is the friction that occursbetween two solid surfaces that do not move relativeto each other.Thus, the static frictional force should always begreater than the applied force on an object.

Kinetic friction- Kinetic friction or the kinetic frictional force occurswhen the two surfaces start to move relative to each other.

Force exerted by anexternal source

Staticfrictionalforceexerted onthe objectby thesurface

Frictional force > Force by external source. Therefore theobject remains stationary.

Object moving in thedirection of applied forceKinetic

frictionalforce on theobject

Frictional force < Force by external source. Therefore the object moves.

Objectdoes notmove

The frictional force increases with the appliedforce and reaches a maximum level. The frictionalforce exerted at this position is called the limitingfrictional force. It actually is the position of theobject beginning its motion or transition betweenimmobility and mobility. Once a greater force thanthe limiting frictional force is exerted on theobject, the object will start to move. However, thefrictional force acting upon the object is now aconstant value and is always less than the limitingfrictional force.These information can be summarized as follows:

Since, Kinetic frictional force is a constant value, it can be shown that,F kinetic / Normal force on the object also takes a constant value. Thisvalue varies according to the surface the object is placed in. That’s thereason as to why we can push a box easily in a more slippery surface, asthe ratio is smaller, the frictional force is smaller and the force neededto push the box is smaller in comparison to that in a rough surface

Kinetic frictional force is alwaysconstant and is less than the limitingfrictional force. However, alwaysapplied force > kinetic frictional force

Limiting frictional force(Maximum frictional force)

Static frictional force gradually increases with the applied force.However always, applied force < static frictional force

Frictional force

Applied force / Time theforce was applied

This ratio is called the coefficient of frictional force.Thus, F kinetic / Normal force = μ k .Therefore, F kinetic = μ k Normal force or F k = μ k NHowever, the limiting frictional force can also beexpressed in a similar notation as the frictional forceis a constant at the applied force.Thus, F limiting = μ N.

Linear Momentum, ImpulseNewton defined the force acting on an object asthe rate of change of its momentum.Momentum (mass × velocity) is a vector quantityand acts in the direction of the velocity.Change of momentum = mv – muTherefore, F =Therefore, F × t = mv – mu = momentum changeHowever, F × t = I (I-Impulse)Therefore, I= mv – mu = Δmv

mv – mut

I = Δmv

Momentum can be due to-1. Change in velocityE.g.: A ball 10g in weight increases velocity from10m/s to 20m/s on colliding with a wall causing amomentum change of 0.3kgms-1

10m/s

20m/s

Ft= mv- mu = 10 × 0.01 – 20 × 0.01= 0.3 kgms-1

2. Change of mass (e.g.: A rocket moves upwardsinto the air and losing mass as its fuel is burnt)E.g.: 0.1 kg of sand is allowed to fall onto a beltmoving at 0.1m/s. The sand is subjected tohorizontal momentum change of 0.01Ns.

Particlesof sand Belt (Provides extra force

needed for the momentum asrequired by the sand particles )

Mass = 0.1 kg

Velocity gained = Velocity of Belt= 0.1 m/s

Therefore, momentum change per secondhorizontally = 0.01 N

F= mass ×

= × Velocity Change

F = Mass per second × Velocity Change

Velocity ChangeTime

Mass

Principle of Conservation of LinearMomentum(PCLM)

Principle-If no external forces act on a system of colliding objects, the totalmomentum of the objects in a given direction is a constant(Thatmeans the total momentum of the objects before collision= totalmomentum after collision)Explanation-From PCLM,

m1u1 + m2u2 = m1v1 + m2v2

E.g.: An object A of mass 5 kg is moving with a velocity of 2m/s.This object collides head-on with an object B of mass 1 kg movingin the opposite direction with a velocity of 4m/s. After collision,the objects stick. Calculate the final velocity of the composite.

From PCLM, 5 × 2 + 1 × 4 = 5 × VV = 1.2 ms-1

2ms-1 4ms-1V ms-1

A B A + B

E.g.: A snooker ball X of mass 0.03 kg, moving with a velocity of1m/s hits a stationary ball Y of mass 0.01kg. Y moves off with avelocity of 2.5 m/s at 60° to the initial direction of X. Find the finalvelocity of X and its direction.

PCLM : 0.03 × 1 = 0.01 × 2.5 + 0.02 × V Cos θ

0.02V Cosθ = .0.005 V Cosθ = 0.25 ①PCLM : 0 = 0.01 × 2.5 Sin60° + 0.03V Sinθ

V Sinθ = 0.72 ②

X 1ms-1Y

60°

θ

2.5ms-1

V ms-1

② / ①Tanθ = 0.72/ 0.25=2.88

θ = 70.85°

By substituting to ①, V= 0.25/ 0.32= 0.76ms-1

Final direction, 70.85° to the horizontal in the initialdirection.

V Cosθ = 0.25

V Sinθ = 0.72 V

Types of CollisionsCollisions can be divided into two groups based on whether thetotal kinetic energy is conserved in the system. In both of thesetwo types, the system in which the collisions takes place obeys theconservation of linear momentum.

1. Elastic CollisionsIt’s the type of collision of two or more objects where the total kineticenergy is conserved or total initial kinetic energy = total final kineticenergy.Also, in this type of collision, the kinetic energy does not get convertedto any other form of energy.

2. Inelastic CollisionsIt’s the type of collision where the total kinetic energy of the system isnot conserved.

Coefficient of Restitution (COR)When two objects collide directly, their relativevelocity after collision is in a constant ratio to theirrelative velocity before collision and is in the oppositedirection.That is , final velocity/initial velocity = -e ,Where e is the coefficient of restitution.For most of the collisions that take place in the realworld, 0 ≤ e ≤ 1. However, there could be incidentswhere e < o or e > 1. That means there could beincidents where there’s a total kinetic energy gainafter collision and a collision which has a negative CORmeans that the separation velocities of the objects arein the same direction as their approaching velocities.

However, collisions where,1. e = 1 are said to be elastic collisions, meaning

both the kinetic energy and the momentumare conserved.

2. 1> e > 0 are said to be inelastic collisions,meaning only the momentum is conserved,while there’s an kinetic energy loss.

3. e = 0 are said to be ‘stop’ at collision.

WorkWork is said to be done when a force displaces abody such that the component of the force actingalong the displacement is not zero.

Considering the dot or scalar product of the twovectors,W = F Cosθ. S W= FS Cosθ

θ

F

S

However, if the force is,Parallel or along the displacement, θ = 0. Thus,the above expression could be simplified as

W = FS

F

S

Special Points1. Since displacement is a vector quantity, a work

done in an opposite direction to an initial workdone will cause for the net work done to bezero.

E.g.: If you lift a weight S m and then lower down itto its initial position, the work done is zero.

S

2. When a force is perpendicular to the directionof an object’s motion, this force does no work onthe object (θ = 90°, W= FS Cos 90° W= 0 )

V

T

T is perpendicular to V.Therefore, rope does nowork on the ball.

EnergyIt’s the capacity of a body to do work. Thus, wecan say that the energy is an aid to do work orwork cannot stand alone.Forms of energy-There are many forms of energy and the mostcommon being1. Potential Energy(It is the energy stored in a

body due to its configuration or position)2. Kinetic Energy(It is the energy stored in a body

due to its movement in a plane)

Mechanical EnergyIt’s the sum of the potential energy and thekinetic energy in a system.Δ KE = ½ mv2 , ΔPE = mghE = Δ KE + ΔPE = ½ mv2 + mgh

E = ½ mv2 + mgh

PowerThe rate at which work is done or energy issuspended is called power.P= W/t and instantaneous power can bedescribed as P= dw/dt

Since, W= FS and P= W/t,It can be concluded that,

P= W/t

P = FVP = FV

Principle of conservation of EnergyPrinciple-The total energy in a closed system is always aconstant.Derivation-Any energy present in a closed system getsconverted to another form of energy. This type ofenergy includes mechanical energy, sound energy,light energy, thermal energy etc.Thus there’s no total energy gain or loss in a closedsystem and the sum of all energies at any moment isequal to that at another moment.

Principle of conservation ofmechanical energy

Principle-If mechanical energy(sum of kinetic energy andpotential energy) does not get converted to anyother form of energy the mechanical energy in aclosed system is conserved.Derivation-KE (initial) + PE(initial) = KE (final) + PE (final)

Thus any system that obeys the principle ofconservation of mechanical energy should alsoobey the principle of conservation of energy