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Assignment of mechanics. Submitted By :- Submitted To:- Rishav Sharma Mr. Hemant 109(minor) BSc. 1st.

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slide1

Assignment of mechanics

Submitted By :- Submitted To:-

Rishav SharmaMr. Hemant

109(minor)

BSc. 1st

slide2

Galilean Transformation Equations: The basic relation between the space and time co-ordinates in two inertial frames of reference were obtained by Galileo and are known as Galilean transformation equations.

z

Z’

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Let S and S’ be two inertial frame of reference , and let S’

moves with velocity v w.r.t. S, as show in the figure.Let the two frame of refrance be parallel to each other.Let us suppose that an event occurs at

A point P in space. The event can be described by four numbers x,y,z and t

In the frame S. The first three event are the cartestian co-ordinates of the point P and the fourth registers the instant of occurrence of the event.

In the frame S’ the event can be described by another set of similar four numbers x’, y’, z’ and t’. The time is measured from the instant

at two origins of of coordinates O and O’ momentarily coincide.

Then from the figure it is clear that the measurement in the

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X-direction made in frame S will exceed that made in frame S’ by the amount vt.That is x’ =x-vt ………………(1)

Since there is no re3lative motion in the Y and Z directions, therefore

Y’ =y ………….(2)

and z’ ………….(3)

We also assume that time of occurrence is same in both the framesi.e. t’ =t …………..(4)

The set of equations (1) to (4) is known as Galilean transformation equations. The inverce transformation equations may be written as

x = x’ + vt’

y = y’

z = z’

t = t’

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TRANSFORMATION OF VELOCITY AND ACCELERATION UNDER GALILEAN TRANFORMATION

1.Velocity transformations. The Galilean transformations from the frame S to S’ are given by

x’ =x-vt ………(1)

y’ = y ………(2)

z’ =z ……….(3)

t’ = t ………..(4)

Differentiating both sides of equation (1) w.r.t, we have

dx/dt =d/dt(x-vt)=dx/dt-v

Since t’=t,it follows that

dx’/dt=dx’/dt’

Therefore,the above equation becomes

dx’/dt’=dx/dt-v

Similarily, we can obtain

dy’/dt’=dy/dt

and dz’/dt’=dz/dt

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If u is velocity of the particle w.r.t. the observer of the frame S and ux,uyand uz denote the components of the particle along the three axes, then

ux=dx/dt; uy=dy/dt and uz=dz/dt

Similarly,if u’ is the velocity of the particle w.r.t. the observer of the frame S’ and ux’,uy’, and uz’ denote the components of the velocity of particle along the three axes of the frame S’, then ux’=dx’/dt’, uy’=dy’/dt’ and uz’=dz’/dt’

Using the above equations we have

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ux’=ux-v

uy’=uy

and uz’=uz ……….(5)

Further, the set of equations (5) can be written as

u’=u – v

The inverse Galilean transformations are

ux=ux’+v

uy=uy’ and uz=uz’ ……….(6)

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2.Acceleration transformations. Differentiating the set of equations (5) w.r.t.t, we have

dux’/dt=d/dt(ux-v)=dux/dt

Since t=t’, the above equ. may be written as

dux’/dt’=dux/dt

ax’=ax

Similarly, ay’=ay

and az’=az ………(7)

The set of equations (7) may be written collectively in the

form as

a’=a

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Since the components of the acceleration of the particle measured in the two inertial frames are independent of the uniform relative velocity of the two frames, acceleration of a particle is invariant under Galilean transformations.

invariance of newton s laws of motion

INVARIANCE OF NEWTON’S LAWS OF MOTION

We know that the measurement of acceleration of a particle is independent of d uniform relative motion of two inertial frames i.e.

a’=a

Let m be d mass of the particle. Newtonian mechanics, mass is a constant quantity and is independent of the motion of the observer. Multiplying both sides of above equation with m,we have

ma=ma

Thus if F=ma and F’=ma’,

then F’=F

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i.e. d observers of d two inertial frames of reference moving with velocity w.r.t. each other measure the force on d particle as to be d same.

Since measurement of force is not affected by the uniform relative relative motion between the two inertial frame of reference, Newton’s third law of motion i.e.

F12=-F21

must be hold in two inertial frames moving with uniform velocity relative to each other.

Hence, Newton’s laws of motion are invariant unde r Galilean transformations.

invariance of law of conservation of linear momentum
INVARIANCE OF LAW OF CONSERVATION OF LINEAR MOMENTUM

Consider two particles of masses m1 and m2 moving with velocities u1 and u2 respectively in an inertial frame S at rest. Suppose that two particle collide and after collision, they move with velocity v1 and v2.

According to d law of conservation of linear momentum in frame S, we have

m1u1+m2u2 = m1v1+m2v2 …………..(1)

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According to inverse Galilean transformation for velocities,

u1=u1’+v; u2=u2’+v; v1=v1’+v

and v2=v2’+v

In equation (1), substituting the above equations we have

m1(u1’+v)+m2(u2’+v)=m1(v1’+v)+m2(v2’+v)

m1u1’+m2u2’=m1v1’+m2v2’

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It is the law of conservations of linear momentum in frame S’. Hence d law of conservation of linear momentum is invariant under Galilean transformations.

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INVARIANCE OF LAW OF CONSERVATION OF ENERGY

Consider two particles of masses m1 and m2 moving with velocities u1 and u2 in an inertial frame S at rest. Suppose that the two particles collide and after collision ,they move with velocities v1 and v2 respectively. If Q is the amount of energy that appears as heat or in some other form then according to d law f conservation of energy in frame S,

1/2m1u12+1/2m2u22=1/2m1v12+1/2m2v22+Q ………(1)

1/2m1u1.u1+1/2m2u2.u2=1/2m1v1.v1+1/2m2v2.v2+Q……..(2)

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Suppose that d observer of the frame S’,the velocities of two particles before and after collision appear as u1’, u2’ and v1’, v2’ respectively.

According to inverse Galilean transformations for velocities,

u1=u1’+v; u2=u2’+v; v1=v1’+v and v2=v2’+v

slide18

According to the law of conservation of linear momentum in frame S’,we have

m1u1’+m2u2’=m1v1’+m2v2’

(m1u1’+m2u2’).v=(m1v1’+m2v2’).v

Using above equation, the equation()reduces to

1/2m1u1’2+1/2m2u2’2=1/2m1v1’2+1/2m2v2’2+Q

It is the law of conservation of energy in frame S’.

objective type questions
OBJECTIVE TYPE QUESTIONS

Q.1-What are Galilean transformation ?

Ans.-These are the simple equations given by Galileo to establish relation between space and time co-ordinate in two inertial frames.

Q.2-In Galilean trasformations, acceleration is:

(a)Variant (b) Invariant (c) Nome of these

Ans-Invariant

Q.3-Is earth inetial frame in real sense.

Ans-In true sense, earth is not an inertial system because it possesses acceleration due to its revolution and rotation motion. But the value of acceleration is so small that for all practicalreason it is taken as an inertial frame.

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Q.4-A Focault’s pendulum art latitude 30◦ has time period in hours

(a)12 (b)24 (c)48 (d) 14

Ans-(b) 24

Q.5-Which force is fictitious?

Gravitational (b) Centrifugal (c)Centripetal

Ans-(b)Centrifugal

Q.5-Name the various fictitious forces that start acting on a body, when frame of reference is rotating with uniform velocity .

Ans – (1)Centrifugal force (2) Coriolis force.

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Q.6-Can a particle be in equilibrium in a non-inertial frame ?

Ans- Yes. A particle can be in equilibrium in a non inertial frame. The particle will be in equilibrium when the external force acting on the particle is equal and opposite to the fictitious force acting on it due to the accelerated motion of a frame reference.

Q.7-What is the number of reference frames which may be inertial?

Ans- Infinite.

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Q.8-Why no cyclone are observed on equators ?

Ans- It is due to the fact there is no horizontal component of coriolis force acting on the winds at the equator. As a result of it, no whirling action takes place and cyclones are not formed at the equator.

Q.9- A body weighs more at poles as compare to equator. Why?

Ans-It is due to the effect of centrifugal force due to rotation of earth.

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Q.10-Focault’s pendulum demonstrates

(a)Effect of gravity (b) Rotation of earth (c)Stability

Ans-(b) Rotation of earth