2. CLASSICAL MECHANICS - CEED Physics Clinic Calicut

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i) Co – ordinate Systems – Two dimensional,

rectangle co – ordinates (x, y) or polar co –

ordinates (𝑟, 𝜃) can represent the position of particle

and related as

𝑥 = 𝑟 cos 𝜃

𝑦 = 𝑟 sin 𝜃

𝑟 = √𝑥2 + 𝑦2

and 𝜃 = 𝑡𝑎𝑛−1 𝑦

𝑥

in three dimensions, the cylindrical co – ordinates

(𝜌, 𝜃, 𝑧) and rectangular co – ordinates (𝑟, 𝜃, ϕ) and

their relation and Cartesian, co – ordinates as

follows:

𝑥 = 𝜌 cos 𝜃; 𝑦 = 𝜌 sin 𝜃, 𝑧 = 𝑧

𝜌 = √𝑥2 + 𝑦2 and 𝜃 = 𝑡𝑎𝑛−1 𝑦

𝑥= 𝑠𝑖𝑛−1 𝑦

𝜌

For spherical co – ordinates

𝑥 = sin 𝜃 cos 𝜙; 𝑦 = 𝑟 sin 𝜃 sin 𝜙; 𝑧 = 𝑟 cos 𝜃

𝑟 = (𝑥2 + 𝑦2 + 𝑧2)1/2

𝜃 = 𝑡𝑎𝑛−1 √𝑥2+𝑦2

𝑧; 𝜙 = 𝑡𝑎𝑛−1 𝑦

𝑥

ii) Degree of freedom – The minimum number of

independent variables co – ordinates required to

specify the position of a dynamical system,

consisting of one or more particles, is called the

number of degree of freedom of the system

e.g.a particle moving freely in a plane, can be

described by a set of two co – ordinates (x, y). A

system of two particles, moving freely in plane

requires two sets of two co – ordinates [e.g. (𝑥1, 𝑦1)

and (𝑥2, 𝑦2)] i.e. 4 co – ordinates to specify its

position. Similarly, for particle moving in space

requires three co – ordinates (𝑥, 𝑦, 𝑧) and a system

of 2 particles, moving freely in space requires two

sets of three co – ordinates [e.g. (𝑥1, 𝑦1, 𝑧1) and

(𝑥2, 𝑦2, 𝑧2)]

If system consists of N, particles moving freely in

plane, then 2N and if particles moving freely in

space then 3N independent co – ordinates to

describe its position. So degrees of freedom of

system in plane is 2N and that is for space is 3N.

Three point masses connected by three right

massless rods –

Degree of freedom = 3 × 3 − 3 = 6

A rigid body – A rigid body is a system with a

large number of particles not all lying on one line,

and with all its particles at fixed distance from each

others

degree of freedom = 5

if one fixes these three point the body is immovable.

Hence degree of freedom having N ≥ 3 is 6, which

is independent of N.

A rigid body fixed at one point – since it is fixed

at one point we lose 3 freedom. Hence the number

of Degree of freedom of this system is 6 – 3 = 3.

The body can rotate freely about this fixed point

with these degrees of freedom.

Mechanics of a System of Particles

i) Conservation of linear momentum – If the total

external force vanishes, the total linear momentum

is conserved.

ii) Conservation of angular momentum – If total

torque 𝑁(𝑒) = 0. 𝐿 the angular momentum is

constant in time or conserved.

iii) Conservation of energy – If the external forces

are derivable from scalar potential function and if

the internal forces are central, then total energy

(Kinetic + Potential) of the system is conserved

• Constraints – Constraints limit the motion

of the system

Constraints are of two type : holonomic and non –

holonomic

Holonomic constraints : (𝑟𝑖 − 𝑟𝑗)2

− 𝑐𝑖𝑗2 = 0

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Non – holonomic constraints : 𝑟2 − 𝑎2 ≥ 0

Further classified as : Rhenomous and

scleronomous

Rhenomous – Constraints contain time explicitly

Scleronomous – Constraints are not explicitly

dependent on time.

Also classified as

Conservative – Total mechanical energy of the

system is conserved during constraint motion and

constraint force do not do any work

Dissipative – The constraint force do work and the

total mechanical energy is not conserved

• Generalized co – ordinates – A set of

generalized co – ordinates is any set of co –

ordinates which describe to configuration.

• Generalized notations – Generalised

displacements,

𝛿𝑟𝑖 =3𝑁∑

𝑗 = 1

𝛿𝑟𝑖

ð𝑞𝑗𝑑𝑞𝑗

• If generalized co – ordinate has the

dimensions of momentum, then

generalized velocity will have the

dimension of force and so on.

• Generalized force

𝑄𝑗 =𝑁∑

𝑖 = 1𝐹𝑖 ,

𝛿𝑟𝑖

ð𝑞𝑗

• For a function 𝑓(𝑦) to have a stationary

value (extremum)or shortest 𝛿𝐼 = 0

• Hamiliton’s principle for conservative

system –

The motion of the system from time 𝑡1 to time 𝑡2 is

such that that the line integral

𝐼 = ∫ 𝐿𝑑𝑡𝑡2

𝑡1

where L = T – V, is an extremum for the path of

motion

• Lagrange’s equation for conservative

system

𝑑

𝑑𝑡(

𝑑𝐿

𝑑�̇�𝑗) −

𝜕𝐿

𝜕𝑞𝑗= 0

For non – conservative system

𝑑

𝑑𝑡(

𝜕𝐿

𝑑�̇�𝑗) −

𝜕𝐿

𝜕𝑞𝑗= 𝑄𝑗

• Lagrangian for a charge particle is an

electromagnetic field

𝐿 =1

2𝑚𝜈2 +

𝑞

𝑐𝜈. 𝐴 − 𝑞ϕ

• Charge particle in an Electromagnetic

field

Generalized potential

𝑈 = 𝑒𝜙 − 𝑒(�⃗�. 𝐴) …. (i)

where ϕ is scalar potential and 𝐴 is vector potential

⇒ The definition of energy E in the rotating frame

is

𝐸 = 𝑇 + 𝑉𝑒𝑓𝑓

=1

2𝑚𝜈2 −

1

2𝑚|𝜔 × 𝑟|2 + 𝑉 ….(ii)

Analogy between expression (i) and expression (ii)

i) The scalar potential energy 𝑒𝜙 ↔ 𝑉𝑒𝑓𝑓

ii) The velocity dependent potential energy

𝑒(�⃗�. 𝐴) ↔ 𝑚𝜈(𝜔 × 𝑟)

iii) The vector potential (momentum) 𝑒𝐴 ↔𝑚(𝜔 × 𝑟)

iv) The magnetic induction 𝐵 = (𝑉 × 𝐴) ↔

(𝑚

𝑒) 𝑉 × (𝜔 × 𝑟) = (

2𝑚

𝑒) 𝜔 if 𝜔 does not vary from

point to point

v) Magnetic force 𝑒(𝜈 × 𝐵) ↔ 2𝑚(𝜈 × 𝜔) =

carioles force

vi) Canonical momentum 𝑚𝜈 + 𝑒𝐴 ↔ 𝑚(𝜈 +𝜔 × 𝑟) and



vii) Energy 1

2𝑚𝜈2 + 𝑒ϕ ↔

1

2𝑚𝜈2 + 𝑉𝑒𝑓𝑓

This is particularly the analogy between the

magnetic field�⃗⃗� and the angular velocity vector 𝜔

Application of Lagrange’s equations of motion

• Linear harmonic oscillator

𝐿 =1

2𝑚𝑥2 −

1

2𝑘𝑥2

Equation of motion

𝑚�̈� + 𝑘𝑥 = 0

⇒ �̈� +𝑘

𝑚𝑥 = 0

• Simple pendulum

𝐿 =1

2𝑚𝑙2𝜃2 − 𝑚𝑔𝑙(1 − 𝑐𝑜𝑠 𝜃)

Equation of motion

�̈� +𝑔

𝑙𝜃 = 0

• Spherical pendulum

𝐿 =1

2𝑚𝑟2(�̇�2 + 𝑠𝑖𝑛2𝜃�̇�2) − 𝑚𝑔𝑟 𝑐𝑜𝑠 𝜃

• Isotropic oscillator (Three dimensional)

𝐿 =1

2𝑚(�̇�2 + 𝑟2�̇�2 + 𝑟2 𝑠𝑖𝑛2 𝜃�̇�2) −

1

2𝑘𝑟2

Dumb bell

𝐿 =1

2(𝑚1 + 𝑚2)(�̇�1

2 + �̇�12) +

1

2𝑚2{𝑙2�̇�2 −

2𝑙�̇�1�̇� sin 𝜃 + 2𝑙�̇�1�̇� cos 𝜃} − (𝑚1 + 𝑚2)𝑔𝑦1 −

𝑚2𝑔𝑙 sin 𝜃)

• Particle moving under a central force

𝐿 =1

2𝑚(�̇�2 + 𝑟2�̇�2) +

𝑘

𝑟

• Electric circuit

𝐿𝐸 = 𝑇𝑀 − 𝑉𝐸

𝑇𝑀 → Magnetic energy of the electric circuit

𝑉𝐸 → Electrical energy of the electric circuit

• Compound pendulum

𝐿 =1

2𝐼�̇�2 + 𝑚𝑔𝑙 𝑐𝑜𝑠 𝜃

• Atwood’s machine

𝐿 =1

2(𝑚1 + 𝑚2)�̇�1

2 + (𝑚1 − 𝑚2)𝑔𝑥1 − 𝑉0

Conservation Theorems for a System in Motion

• Cyclic or ignorable co – ordinate -

If the Lagrangian of a system does not contain a

particular co – ordinate 𝑞1, then for such a system 𝜕𝐿

𝜕𝑞𝑘 = 0, such co – ordinates are known as ignorable

co – ordinates

• Generalized momenta –

It is also termed as conjugate or canonical

momentum

𝑃𝑗 =𝜕𝐿

𝜕�̇�𝑗

• Conservation of linear momentum –

If a co ordinate corresponding to a displacement is

cyclic description of system motion remains

invariant under such a translation and linear

momentum is conserved

• Conservation of angular momentum -

If a co – ordinate corresponding to a rotation co –

ordinate is cyclic, then the component of the applied

torque along the given axis vanishes then the

component of angular momentum along that axis is

constant

• Conservation of energy

∑�̇�𝑗𝑝𝑗 − 𝐿 = 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡 = 𝐻

𝑗

where H = T + V and known as Hamiltonian

Hamilton’s canonical equations of motion

�̇�1 =𝜕𝐻

𝜕𝑝𝑖, 𝑝𝑖 = −

𝜕𝐻

𝜕𝑞𝑖


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• Hamiltonian for a charged particle in an

electro magnetic field

𝐻 =1

2𝑚𝜈2 + 𝑞ϕ

and also in temrs of p (momentum)

𝐻 =1

2𝑚(𝒑 −

𝑞

𝑐𝑨)

2

+ 𝑞ϕ

Canonical Transformations

Four generating functions -

𝐹1(𝑞, 𝑄, 𝑡): 𝐹2: (𝑞, 𝑝, 𝑡); 𝐹3(𝑝, 𝑄, 𝑡) and 𝐹4(𝑝, 𝑞, 𝑡)

For

First Form Second Form

𝑝𝑖 =𝜕𝐹1

𝜕𝑞𝑖 𝑝𝑖 =

ð𝐹2

𝜕𝑞𝑖

𝑝𝑖 =ð𝐹1

ð𝑄𝑖 𝑄𝑖 =

ð𝐹2

𝜕𝐹1

Third Form Fourth Form

𝑞𝑖 = −𝜕𝐹3

𝜕𝑝𝑖 𝑞𝑖 = −

𝜕𝐹4

𝜕𝑝𝑖

𝑃𝑖 =𝜕𝐹3

ð𝑄𝑖 𝑄𝑖 =

𝜕𝐹4

ð𝑃𝑖

Condition for a Transformation to be Canonical

a) An exact differential condition

∑(𝑝𝑖𝑑𝑞𝑖 − 𝑃𝑖𝑑𝑄𝑖) = 𝑑𝐹𝑖

b) Bilinear invariant condition

∑(𝛿𝑝𝑖𝑑𝑞𝑖 − 𝛿𝑝𝑖𝑑𝑝𝑖) =𝑖

∑(𝛿𝑃𝑖𝑑𝑄𝑖 − 𝛿𝑄𝑖 − 𝛿𝑄𝑖𝑑𝑃𝑖)𝑖

c) Invariance of Poisson brackets

and |𝑄, 𝑄| = [𝑃, 𝑃] = 0

[𝑄, 𝑃] = 1} condition

d) Invariance of Lagrange’s brackets

• Poisson brackets

[𝑋, 𝑌]𝑞.𝑝 =∑𝑖

(𝜕𝑋𝜕𝑌

𝜕𝑞𝑖𝜕𝑝𝑖−

𝜕𝑋

𝜕𝑝𝑖

𝜕𝑌

𝜕𝑞𝑖)

Lagrange’s brackets

(𝑢, 𝑣)𝑞.𝑝 =∑𝑖

(𝜕𝑞𝑖𝜕𝑝𝑖

𝜕𝑢 𝜕𝜈−

𝜕𝑝𝑖𝜕𝑞𝑖

𝜕𝑢 𝜕𝜈)

Relation between Poisson and Lagrange’s

brackets

2𝑛∑

𝑙 = 1

{𝑢𝑙 , 𝑢𝑖}[𝑢𝑢, 𝑢𝑗] = 𝛿𝑖𝑗 equations of

Equations of motion in Poisson bracket

�̇�𝑖 = [𝑞𝑖, 𝐻]; �̇�𝑖 = [𝑝𝑖, 𝐻]

• All functions whose Poisson bracket with

Hamiltonian vanish will be constants of

motion and cornversely Poisson brackets of

all constants of motion with H must be zero

Inertial Tensor

𝐽 = 1�⃗⃗⃗�

I – inertia tensor

In terms of component

𝐽𝑥 = 𝐼𝑥𝑥𝜔𝑥 + 𝐼𝑥𝑦𝜔𝑦 + 𝐼𝑥𝑧𝜔𝑧

𝐽𝑦 = 𝐼𝑦𝑥𝜔𝑥 + 𝐼𝑦𝑦𝜔𝑦 + 𝐼𝑦𝑧𝜔𝑧

𝐽𝑧 = 𝐼𝑧𝑥𝜔𝑥 + 𝐼𝑧𝑦𝜔𝑦 + 𝐼𝑧𝑧𝜔𝑧

where

𝐼𝑥𝑥 = ∑𝑚(𝑟2 − 𝑥2) = ∑ 𝑚 (𝑦2 + 𝑧2)

𝐼𝑦𝑦 = ∑𝑚(𝑟2 − 𝑦2) = ∑𝑚(𝑥2 − 𝑧2)

𝐼𝑧𝑧 = ∑𝑚(𝑟2 − 𝑧2) = ∑𝑚(𝑥2 − 𝑦2)

𝐼𝑥𝑦 = −∑𝑚𝑥𝑦 = 𝐼𝑦𝑥

𝐼𝑥𝑧 = −∑𝑚𝑥𝑧 = 𝐼𝑧𝑥

𝐼𝑦𝑧 = −∑𝑚𝑦𝑧 = 𝐼𝑧𝑦

since



𝐼𝑥𝑦 = 𝐼𝑦𝑥

𝐼 is a symmetric tensor

The quantity 𝐼𝑥𝑥 is called moment of inertia of the

body about X – axis. Similarly 𝐼𝑦𝑦 and 𝐼𝑧𝑧 defined

that about Y and Z axis respectively.

The quantities 𝐼𝑥𝑦, 𝐼𝑥𝑧 … etc are called products of

inertia

Value of moments of inertia of some important

cases

1) Thin uniform rod –

i) Through the centre and perpendicular to its length

= 𝑀𝑙2

12

ii) Through one end and perpendicular to its length

= 𝑀𝑙2

3

2) Thin rectangular lamina –

i) Through the centre and perpendicular to its plane

= 𝑀

12(𝑙2 + 𝑏2)

3) Hoop or circular ring –


= 𝑀𝑅2

ii) About a diameter = 𝑀𝑅2

2

4) Circular lamina or disc –


= 𝑀𝑅2/2

ii) About a diameter = 𝑀𝑅2/4

5) Annular ring or disc of outer and inner radii

𝑹𝟐 and 𝑹𝟏 -


= 𝑀

2(𝑅1

2 + 𝑅22)

6) Solid cylinder –

i) About its axis of cylindrical symmetry i.e. about

it own axis = 𝑀𝑅2

2

ii) Through its centre and perpendicular to its axis =

𝑀 (𝑙2

12+

𝑅2

4)

7) Hollow cylinder of external and internal radii

𝑹𝟐 and 𝑹𝟏 –

i) About its own axis = 𝑀

2(𝑅1

2 + 𝑅22)

ii) Through its centre and perpendicular to its axis =

M(𝑅1

2+𝑅22

4+

𝑙2

12)

8) Solid sphere –

i) About a diameter = 2

5𝑀𝑅2

ii) About a tangent = 7

5𝑀𝑅2

9) Spherical shell –

i) About a diameter = 2

5𝑀𝑅2

ii) About a tangent = 5

3𝑀𝑅2

iii) About a tangent = 5

3𝑀𝑅2

10) Solid right circular conc of base radius R –

i) About its axes = 3

10𝑀𝑅2

11) Thick shell or hollow sphere of external and

internal radii 𝑹𝟐 and 𝑹𝟏 -

i) About a diameter : 2

5 𝑀 (

𝑅25−𝑅1

5

𝑅23−𝑅1

3)

ii) About a tangent : 2

5[

𝑅25−𝑅1

5

𝑅23−𝑅1

3] + 𝑀𝑅22

12) Solid cone of altitude h and base of radius

R – (i) About its vertical axis : 3

10𝑀𝑅2

Rigid Body Dynamics

Rigid body – A rigid body is defined as a system of

points subject to the holonomic constraints where



the distances between all pairs of points remain

constant throughout the motion i.e. 𝑟𝑖𝑗 = 𝐶𝑖𝑗, where

𝑟𝑖𝑗 is the distance between ith and jth particle and

𝐶𝑖𝑗 is the constant.

Theorem – The number of degrees of freedom for

the general motion of a rigid body is six

Body copordinate system – A co – ordinate

system, fixed in rigid body, is called body set of

axes.

Space co – ordinate system – In this co – ordinate

system axes are fixed in space, called space set of

axes

• Eulerian angles –

They are defined as the three successive angles of

rotation of rigid body about a point fixed in the

body. We first rotate initial system of axes OXYZ

(fixed by space) by an angle ϕ counter clockwise

about the Z – axis and the resultant co – ordinate

system will be labelled as 𝜉, 𝜂, 𝜁 secondly the

intermediate axes 𝜉, 𝜂, 𝜁 are rotated about the 𝜉 axis

counter clockwise by an angle 𝜃 to produce set of

axes 𝜉′, 𝜂′, 𝜁′. The 𝜉′ - axis is at the intersection of

XY and 𝜉′, 𝜂′ plane is known as lines of nodes.

At last the axes 𝜉′, 𝜂′, 𝜁′ are rotated counter

clockwise by an angle Ψ about 𝜁′ axis to produce

𝑋′𝑌′𝑍′ system of axes (fixed in the body)

Here the angle (𝜙, 𝜃, Ψ) are known as Eulerian

angles.

• Components of angular velocity –

If 𝜙, 𝜃, Ψ represetns Euler’s angle, then �̇�, 𝜃, Ψ̇̇

represents angular velocity about the space Z – axis.

Lines of nodes and body 𝑍′ axis respectively.

𝜔𝜙, 𝜔𝜃, 𝜔Ψ represents �̇�, 𝜃, Ψ̇̇ respectively and

called components of angular velocity 𝜔.

Components of angular velocity along body set

of axes –

𝜔𝑥′ = �̇�𝑋′ + �̇�𝑋′ + Ψ̇𝑋′

= �̇� sin 𝜃 sin Ψ + �̇� cos Ψ̇

𝜔𝑌′ = �̇�𝑌′ + �̇�𝑌′ + Ψ̇𝑌′

= �̇� sin 𝜃 sin Ψ + �̇� cos Ψ̇

𝜔𝑍′ = 𝜙𝑍′ + �̇�𝑍′ + Ψ̇𝑍′

= �̇� cos 𝜃 + Ψ

Euler’s Theorem – The general displacement of a

rigid body with one point fixed, is a rotation about

some axis.

Euler’s equation of motion for a rigid body – The

x, y, z components of the torque is

𝑇𝑥 = 𝐼1�̇�1 + (𝐼3 − 𝐼2)𝜔2𝜔3

𝑇𝑦 = 𝐼2�̇�2 + (𝐼1 − 𝐼3)𝜔3𝜔1

𝑇𝑧 = 𝐼3�̇�3 + (𝐼2 − 𝐼1)𝜔1𝜔2

Motion Under Central Force

• The field in which its potential energy

depend only on the distance r from the same

fixed point, the field is called as central

field.

• As V is the function of 𝑟 only, then force is

always along 𝑟 and the problem has

spherical symmetry

• In this symmetry the solution is invariant

under the rotation of the system about any

fixed axis, so angle co – ordinate which

represents rotation is cyclic



• In the central force problem motion is

always in a plane, where polar axis is taken

along 𝐽 angular momentum

• The force between two interacting particles

is primarily la central force

• Two main features of central force

i) Conservation of energy

ii) Conservation of angular momentum

• For central force, the orbit always lies in a

plane which is perpendicular to the fixed

direction of angular momentum

• The total energy of the system

1

2𝑚�̇�2 +

𝑙2

2𝑚𝑙2 + 𝑉(𝑟) = Constant

• Equation of the path (or orbit)

𝑑2𝑢

𝑑𝜃2 = −𝑢 −𝑚

𝑙2𝑢2 𝑓 (𝑙

𝑢)

• The viral theorem

𝑇 = −1

2

∑𝐹𝑖𝑟𝑖

𝑖

and the inverse square law

2𝑇 + 𝑉 = 0

• Features of elliptic orbit

i) E = - 𝑘

2𝑎

All ellipses with the same major axis have the same

energy

ii) 𝜏2 ∝ 𝑎3

square the period of elliptic motion is proportional

to the cube of the semi major axis

• Condition for stable orbit

𝜕2𝑉′

𝜕𝑟2 |𝑟=𝑟0

> 0

• An orbit is said to be closed if the particle

eventually retraces its path

• The stable and closed orbits (circular and

non circular) for n = 1 and n = - 2 have the

force law as follows

For n = 1, 𝑓(𝑟) = −𝑘𝑟 Hooke’s law

𝑛 = −2, 𝑓(𝑟) = −𝑘/𝑟2 Inverse square law

• The Kepler problem : Inverse square law

–

The orbit is always a conic section, with

eccentricity

𝑒 = √1 +2𝐸𝑙2

𝑚𝑘2

𝑒 > 1. 𝐸 > 0 hyperbola

𝑒 = 1, 𝐸 = 0 parabola

𝑒 < 1, 𝐸 < 0 ellipse

𝑒 = 0, 𝐸 = −𝑚𝑘2

2𝑙2 circle

Scattering in a Central Field

• Relation between cross – section and impact

parameter is

𝜎(𝜙) =𝑝

𝑠𝑖𝑛 𝜙[

𝑑𝑝

𝑑𝜙] ….(i)

p = impact parameter

negative sign introduced because an increase of p

will decrease 𝜙

• For path of charged particles eccentricity

𝑒 = √1 +2𝐸𝐽2

𝑚𝑘2

Here force F = 𝑍𝑧𝑒2

4𝜋 0

1

𝑟2

= - 𝑘

𝑟2

So 𝑘 =𝑍𝑧𝑒2

4𝜋 0



⇒ 𝑒 = √1 +2𝐸𝐽2(4𝜋 0)2

𝑍2𝑧2𝑒4𝑚

⇒ Initial velocity is 𝜈0, ∴ its total energy

𝐸 =1

2𝑚𝜈0

2 or 𝜈0 = √2𝑚𝐸

Angular momentum

𝑚𝜈0𝑝 = 𝑚𝑟2𝜃 = 𝐽

𝐽 = 𝑃√2𝑚𝐸

∴ 𝑒 = √1 + (2𝐸𝑝4𝜋 0

𝑍𝑧𝑒2 )2

Since (2𝐸𝑝4𝜋 0

𝑍𝑧𝑒2 )2

is +ve quantity, so path of charged

particle is hyperbola

• Relation for scattering angle

ϕ = 𝜋 − 2𝛼

or 𝛼 =𝜋

2−

ϕ

2

where 𝛼 is angle between the direction of the

incoming asymptote and peripasis direction

• The functional relationship between the

impact parameter and the scattering angle is

𝑝 =𝑍𝑧𝑒2

2𝐸cot

ϕ

2

By differentiating p w.r.t ϕ and substituting in eqn

i) we have differential scattering cross section

𝜎(ϕ) =1

4[

𝑍𝑧𝑒2

(4𝜋 0)2𝐸]

2

𝑐𝑜𝑠𝑒𝑐4 𝜙

2

and known as Rutherford Scattering

Thus

Scattering cross – section or the number of particles

scattered per second along the direction ϕ are

proportional to

1. cosec2 𝜙

2

2. The square of the charge on nucleus (Ze)

3. The square of the charge on particle (ze)

4. Inversely proportional to the square of the initial

kinetic energy

Special Theory of Relativity

The theory of relativity consists of two parts

General theory deals with the problem involving

two reference frames having accelerated motion

with respect to each other.

Special theory of relativity and several of its

predictions is based upon the following two

postulates

1. The laws of physics are the same in all intertial

frames of reference

2. The velocity of light in free space is constant and

is independent of the relative motion of the source

and the observer

• Lorentz transformation

𝑥′ = 𝑥

𝑦′ = 𝑦

𝑧′ =𝑧−𝜈𝑡

√1−𝛽2, 𝛽 =

𝜈

𝑐

𝑡′ =𝑡−𝜈𝑧/𝑐2

√1−𝛽2

In matrix form

𝐿 = (

1000

0100

00𝛾

−𝑖𝛽𝛾

00

𝑖𝛽𝛾𝛾

)

• Lorentz transformation are equivalent to

rotation of axes in four dimensional space

though an imaginary angle of tan−1(𝑖β)

• Two successive Lorentz transformation

corresponds to a single Lorentz

transformation with relative speed

𝛽" =𝛽+𝛽′

1+𝛽𝛽′

• 𝑋𝜇 → be the difference vector defined as



𝑋𝜇 = 𝑋1𝜇 − 𝑋2𝑝 [1 and 2 shows events]

• Note: For values of 𝜈 << 𝑐

and 𝜈

𝑐→ 0 ∴

1

√1−𝜈2

𝑐2

𝑙 = 𝑙0√1 −𝜈2

𝑐2

• 𝑙 < 𝑙0, ∴ length of rod appeared to be

contracted by the factor √1 −𝜈2

𝑐2

• Frame of reference at rest

Reference from in motion

Note

• There is no contraction in a direction

perpendicular to the direction of motion

World point and world line – A physical event in

Minkowski space is described by a point with four

co – ordinates (𝑥1, 𝑥2, 𝑥3, 𝑥4) (𝑥4 = 𝑖𝑐𝑡). This point

in four space is called world points

In this space, the motion of a particle corresponds to

a line, known as world line

Simultaneous events – A frame in which two

events occurs to a point

• Light like Interval

If [𝑟1 − 𝑟2] = 𝑐2(𝑡2 − 𝑡1)2

• World region and light conc

𝑋𝜇 is space like if the tow world points are

separated by

[𝑟1 − 𝑟2]2 < 𝑐(𝑡1 − 𝑡2)2

Small Oscillations

𝐹(𝑥) =𝑑𝑉

𝑑𝑥= 0

and a particle placed at such point with zero

velocity will continue to remain at rest

when the force acting on a particle vanishes, the

particle is said to be in equilibrium

Equilibrium is of Two Type

Stable Unstable

• 𝑑𝜈

𝑑𝑥 is positive at

a neighbouring

point

𝑑𝜈

𝑑𝑥 is negative at a

neighbouring point

• Potential energy

is minimum

Potential energy is equal

to total energy and is

maximum

• Normal co – ordinates – The generalized

co – ordinates, each one of them executing

oscillations of single frequency, are called

normal co – ordinates

Systems with Few Degree’s of Freedom:

• The parallel pendula – The normal

frequency associated with this mode

𝜔 = √𝑔

𝑙+

2𝑘

𝑚

• Double pendulum – Two normal

frequencies are

𝜔+2 =

𝑔

𝑙(2 + √2)

𝜔−2 =

𝑔

𝑙(2 − √2)

• Triple pendulum: a degenerate system –

Three normal frequencies

𝜔1 = √𝑔

𝑙+

2𝑘

𝑚



𝜔2 = √𝑔

𝑙+

2𝑘

𝑚

𝜔3 = √𝑔

𝑙−

𝑘

𝑚

𝜔1 = 𝜔2, the system possess two identical

frequencies and is, therefore, degenerate

• Linear triatomic molecule

Three value of 𝜔

𝜔1 = 0;

𝜔2 = √𝑘

𝑚

𝜔3 = √𝑘

𝑚(1 +

2𝑚

𝑀)

𝜔1 = 0, corresponds to → Translatory motion and

rest two oscillatory motion

• For the linear molecule there will be three

degree of freedom for rigid translation and

right rotation can account for only two

degrees of freedom

• Four true modes of vibration : Two of these

are longitudinal modes and two modes are

vibration are perpendicular to the axis

PART A

1. A solid sphere of mass m and radius a is rolling with a

linear speed 𝜐 on a flat surface without slipping. The

magnitude of the angular momentum of the sphere with

respect to a point along the path of the sphere on the

surface is

(a) 2

5 ma𝜐 (b)

7

5 ma𝜐 (c) ma𝜐 (d)

3

2 ma𝜐

2. An observer is sitting on a horizontal platform which

is rotating with a constant angular velocity. He puts an

object on the smooth frictionless floor of the platform,

away from the axis of rotation, with zero initial velocity

with respect to him. Let the time at this instant be t = 0.

In the frame of the platform, the object would

(a)remain at rest for all t > 0

(b)accelerate purely in a radial direction outwards for all

t > 0

(c)accelerate purely in a tangential direction for all t > 0

(d)accelerate radially in the outwards direction at t = 0,

however the direction of acceleration changes fot t > 0

3. The ratio of the inner radii of two glass tubes of same

length is √2 . A fluid of viscosity 8.0 cP flows through

the first tube, and another fluid of viscosity 0.8cP flows

through the second on when equal pressure difference is

applied across both of them. The ratio of the flow rate in

the first tube to that in the second tube is

(a) 1.6 (b)10√2 (c)√2 (d)0.4

4. A projectile is fired from the orgin O at an angle of

45° from the horizontal. At the highest point P of its

trajectory the radial and transverse components of its

acceleration in terms of the gravitational acceleration g

are

(a) ar = 2𝑔

√5 , 𝑎𝜃 =

𝑔

√5 (b) ar =

−2𝑔

√5 , 𝑎𝜃 =

−𝑔

√5

(c) ar = 𝑔

√5 , 𝑎𝜃=

2𝑔

√5 (d) ar =

−𝑔

√5 , 𝑎𝜃 =

−2𝑔

√5

5.A satellite moves around a planet in a circular orbit at

a distance R from its centre. The time period of

revolution of the satellite is T. If the same satellite is

taken to an orbit of radius 4R around the same planet,

the time period would be

(a)8T (b)4T (c)T/4 (d)T/8



6.In an inertial frame S, a stationary rod makes an angle

𝜃 with the x-axis. Another inertial frame S’ moves with

a velocity 𝜐 with respect to S along the common x –x’

axis. As observed from S’, the angle made by the rod

with the x’axis is 𝜃’. Which of the following statements

is correct?

(a) 𝜃’ < 𝜃 (b) 𝜃’ > 𝜃

(c) 𝜃’ < 𝜃 if 𝜈 is negative and 𝜃’ > 𝜃 if 𝜈 is positive

(d) 𝜃’ > 𝜃 if 𝜈 is negative and 𝜃’ < 𝜃 if 𝜈 is positive

7 .A thin massless rod of length 2ℓ has equal point

masses m attached at its ends (see figure). The rod is

rotating about an axis passing through its centre and

making angle 𝜃 with it. The magnitude of the rate of

change of its angular momentum |𝑑�⃗⃗�

𝑑𝑡| is

(a) 2mℓ2 𝜔2sin𝜃cos𝜃 (b) 2mℓ2 𝜔2sin𝜃

(c) 2mℓ2 𝜔2sin2𝜃 (d) 2mℓ2 𝜔2cos2𝜃

8. Moment of inertia of solid cylinder of mass m, height

h and radius r about an axis(shown in figure by dashed

line) passing through its centre of mass and

perpendicular to its symmetry axis is

(a)1

4mr2 +

1

12mh2 (b)

1

2mr2 +

1

8mh2

(c) 1

2mr2 +

1

12mh2 (d)

1

2mr2 +

1

4mh2

9. A circular platform is rotating with a uniform angular

speed w counter clockwise about an axis passing through

its centre and perpendicular to its plane as shown in the

figure. A person of mass m walks radially inwards with

a uniform speed 𝜈 on the platform. The magnitude and

the direction of the Coriolis force (with respect to the

direction along which the person walks) is

(a)2m 𝜔𝜈 towards his left

(b) 2m 𝜔𝜈 towards his front

(c)towards his right

(d)towards his back

10. A particle of mass m moving with a velocity �⃗� =

𝑣0(𝑖 ̂+ 𝑗̂), collides elastically with another particle of

mass 2m which is at rest initially. Here, 𝑣0 is a constant.

Which of the following statements is correct?

(a) The direction along which the centre of mass moves

before collision is – (�̂�+�̂̂�

√2)

(b) The speed of the particle of mass m before collision

in the center of mass frame is √2 𝑣0

(c)After collision, the speed of the particle with mass 2m

in the centre of mass frame is √2

3𝑣0

(d)The speed of the particle of mass 2m before collision

in the center of mass frame is √2𝑣0

11. A trapped air bubble of volume is released from a

depth h measured from the water surface in a large water

tank. The volume of the bubble grows to as it reaches

just below the surface. The temperature of the water and

the pressure above the surface of water remain constant

throughout the process. If the density of water is 1000

kg/m3 and acceleration due to gravity is 10 m/s2, then the

depth h is

(a) 1 m (b) 10 m (c)50 m (d)100 m

12. A rain drop falling vertically under gravity gathers

moisture from the atmosphere at a rate given by 𝑑𝑚

𝑑𝑡 =



kt2, where m is the instantaneous mass, t is time and k is

a constant. The equation of motion of the rain drop is

m𝑑𝑣

𝑑𝑡 + 𝜈

𝑑𝑚

𝑑𝑡 = mg. If the drop starts falling at t = 0, with

zero initial velocity and initial mass m0(given: m0 = 2

gm, k = 12 gm/s2), the velocity (𝜈) of the drop after one

second is

(a) 250 cm/s (b) 500 cm/s

(c) 750 cm/s (d) 1000 cm/s

13. Three masses m, 2m and 3m are moving in x – y

plane with speeds 3u, 2u and u, respectively, as shown in

the figure. The three masses collide at the same time at

P and stick together. The velocity of the resulting mass

would be

(a)𝑢

12(𝑥 + √3�̂�) (b)

𝑢

12(𝑥 - √3�̂�)

(c) 𝑢

12(−�̂� + √3�̂�) (d)

𝑢

12(−𝑥 + √3�̂�)

14. A particle is released at x = 1 in a force field �⃗�(x) =

(2

𝑥2 - 𝑥2

2)�̂�x , x ≥ 0. Which one of the following

statements is FALSE?

(a) �⃗�(x) is conservative

(b) The angular momentum of the particle about the

origin is constant

(c)The particle moves towards x = √2

(d)The particle moves towards the origin

15.The moment of inertia of a disc about one of its

diameters is IM. The mass per unit area of the disc is

proportional to the distance from its centre. If the radius

of the disc is R and its mass is M, the value of IM is

(a)1

2MR2 (b)

2

5MR2 (c)

3

10MR2 (d)

3

5MR2

15. A satellite is moving around earth in a circular orbit

of radius R. The time period T of the satellite is

(a)proportional to R (b)proportional to R2

(c) proportional to R3/2 (d)independent of R

16. In case of an inelastic collision which one of the

following is true

(a) Total energy is not conserved

(b)Momentum is not conserved

(c)Kinetic energy is conserved

(d)Kinetic energy is not conserved

17. An ideal fluid is flowing through a tube of

cylindrical cross section with smoothly varying radius.

The velocity of fluid particles at the point where tube’s

cross sectional area is 1 x 10-4m2 is given by 0.01 m/s.

The velocity at a point where cross sectional area is 2 x

10-4m2 is given by

(a) 0.0025 m/s (b) 0.005 m/s

(c) 0.02 m/s (d) 0.04 m/s

18. The coefficient of viscosity for a gas

(a) is independent of the pressure of the gas

(b) is proportional to T, the absolute temperature of the

gas

(c) is proportional to T2

(d)depends on the size of the vessel containing the gas

19. Moment of inertia of a uniform circular disk of

radius R and mass M about the tangential axis parallel

to its diameter is

(a) 𝑀𝑅2

4 (b)

𝑀𝑅2

2 (c)

5𝑀𝑅2

4 (d)

3𝑀𝑅2

2

20. The escape velocity from the earth is V0. For a planet

with radius three times and density twice that of the

earth, the escape velocity will be

(a)V0√2 (b)3V0√2



(c)2 V0√2 (d)V0√6

21.In the figure, the tension in the inelastic string is T

when all surfaces are frictionless. If 2 kg block is glued

on to the surface, the tension in the string will be

(a) zero (b)greater than T

(c)less than T (d)equal to T

22.Two metal wires A and B having lengths ℓ and 2ℓ

and radii R and 2R respectively are joined end to end

along their axis. When one end of the system is fixed

and other end is pulled with a constant force F, the

elongation in both the wires is equal. The ratio of their

young’s modulus YA : YB is

(a) 2 : 1 (b)4 : 1 (c)1 : 2 (d)1 : 4

23. A mass of 0.5kg moving with a speed of 2m/s hits

another mass 1kg moving in the same direction with a

speed of 1m/s.Kinetic energy of the centre of mass is

(a) 4/3 J (b) ¾ J (c) 0J (d) 8/3 J

24. After being hit, a golf ball reaches a maximum

height of 60m with a speed of 20m/s. Right after being

hit, the speed of the ball is (take g as 10 m/s2)

(a) 100m/s (b)80m/s (c)60m/s (d)40m/s

25. Under the influence if a force of 2kN, a wire of

diameter 2mm gets elongated by 4mm. What will be the

elongation in a wire of same material and same length

but of diameter 4mm?

(a) 0.5mm (b)1.0mm (c)1.5mm (d)2.0mm

26. A particle is acted upon by a force �⃗� = yz𝑖 ̂+ xz𝑗̂ +

xy�̂�. Which of the following statements is true?

(a)�⃗� is not conservative

(b) �⃗� is conservative and there exists a potential V such

that �⃗� = -∇⃗⃗⃗V, V = x2y + y2z + z2x

(c) �⃗� is conservative and there exists a potential V such

that �⃗� = -∇⃗⃗⃗V, V = -xyz

(d) �⃗� is not conservative and there exists a potebtial V

such that �⃗� = -∇⃗⃗⃗V, V = xyz

27. For a completely inelastic collision of two particles

in one dimension, which of the following statement is

NOT correct?

(a) The kinetic energy is conserved

(b)The linear momentum is conserved

(c) The particles move with a commom velocity after the

collision

(d) The total energy is conserved

28. Kepler’s second law of motion states that the rate

(𝑑𝑆

𝑑𝑡), at which the area (S) is swept out by the line from

the sun to the planet of mass m, is constant and its value

in terms of the angular momentum L of the planet is

given by

(a) 𝐿

𝑚 (b)

𝐿2

2𝑚 (c)

𝐿

2 (d)

𝐿

2𝑚

29. A monometer tube(U-shaped) is partially filled with

water. A non-mixing oil having density 20% less than

water density is poured in one of the arms until the oil-

water interface comes to the middle of the tube. If the

height of water column is 20 cm, the height of the oil

column will be

(a)16 cm (b)20 cm (c)22cm (d)25 cm

30. A satellite moves in an elliptical orbit around the

earth. The minimum and the maximum distances of the

satellite from the surface of the earth are 6.3 x 105m and

3.63 x 106m respectively. The radius of the earth is 37 x

106 m The ratio of speed of the satellite at apogee to its

speed at perigee is

(a) 0.3 (b) 0.35 (c)0.6 (d)0.7



31. An airplane, climbing with velocity 𝜐 and at an angle

𝜃 with respect to the horizontal, releases a projectile at

an altitude h

The projectile hits the ground after a time T after its

release. The horizontal distance travelled by the

projectile is

(a) (𝜐sin𝜃 + √𝜐2𝑠𝑖𝑛2𝜃 + 2𝑔ℎ) 𝜐

𝑔 cos𝜃

(b) (𝜐sin𝜃 - √𝜐2𝑠𝑖𝑛2𝜃 + 2𝑔ℎ) 𝜐

𝑔 cos𝜃

(c) (𝜐sin𝜃 + √𝜐2𝑠𝑖𝑛2𝜃 − 2𝑔ℎ) 𝜐

𝑔 cos𝜃

(d) (𝜐sin𝜃 - √𝜐2𝑠𝑖𝑛2𝜃 − 2𝑔ℎ) 𝜐

𝑔 cos𝜃

32.A crate of mass m is pulled at a constant speed down

a rough ramp inclined at angle 𝜃° by a horizontal force F

The coefficient of kinetic friction 𝜇 is

(a) 𝑚𝑔 𝑐𝑜𝑠𝜃

𝐹 (b)

𝐹

𝑚𝑔 tan𝜃

(c)𝑔 𝑠𝑜𝑛𝜃+(

𝐹

𝑚)𝑐𝑜𝑠𝜃

𝑔 (d)

𝑚𝑔 sin 𝜃+𝐹 cos 𝜃

𝑚𝑔 cos 𝜃−𝐹 sin 𝜃

33.Two blocks of masses m and 2m are hung from the

ceiling by three cables as shown

The cable in the middle is exactly horizontal and the

cable on the left makes an angle of 45° with respect to

the vertical. The angle 𝜃 made by the cable on the right

with respect to the vertical is

(a) tan-1 1 (b)2tan-1 1 (c)tan-1(1/2) (d)tan(1/√2)

34. A Rocket is fired vertically upwards from the earth’s

surface with an initial speed of 𝜐. If M is the mass of the

earth and R is the radius of the earth’s surface will it

rise?

(a)𝜈2

2𝑔−𝜈2

𝑅

(b) 𝜈2

𝐺𝑀−𝜈2

𝑅

(c) √2𝑔𝑅 (d) 𝑣2

𝑚𝑔

35. A uniform solid sphere of radius R produces an

acceleration of ag on its surface. At which distance inside

the sphere is the gravitational acceleration ag/2?

(a) R/4 (b) R2/2

(c) R/2 (d) R/√2

36. A satellite is placed in a circular orbit about the earth

whose radius is 1/100th the distance between the earth

and the moon. If the lunar period around the earth is 28

days, the period of revolution of the satellite is

approximately

(a) 10 min (b)40 min (c)100 min (d)2.8 days

37.An oscillating block-spring system has total

mechanical energy 2 Joules, amplitude of 20 cm and a

maximum speed of 2.4 m/s



The spring constant k is

(a)10 N/m (b)100N/m

(c)1 N/m (d)40 N/m

38. A physical pendulum consists of a solid cylinder

pivoted about one of its ends The moment of inertia of

the cylinder is 1

4MR2 +

1

12ML2 +

1

4ML2. The time period

of the pendulum is

(a) 2√2𝑔𝐿

𝑅2 (b) 2√𝑅2

2𝑔𝐿+2𝑔𝑅

(c) 2√𝑅2+4𝐿2/3

2𝑔𝐿 (d) 2√

𝑅

𝑔

39.A hollow sphere of moment of inertia 2MR2/3 and a

thin loop of moment of inertia MR2 roll without slipping

down an inclined plane. The ratio of their times of

arrival Tsphere/Tloop at the bottom of the incline is given by

(a) √1/2 (b)1/2 (c)3/√5 (d)√5/6

40. A chain of mass M and length L is hanging vertically

over a table, with its lowest point touching the surface of

the table. It is released and it falls on the table

completely inelastically. How much times does it take

for the chain to fall completely on the table

(a) L/g (b) 2L/g (c) √𝐿/𝑔 (d) √2𝐿/𝑔

41 .A planet of mass Mp is in a circular orbit around a

star of mass Ms at a radius R. If the star loses a fraction f

of its mass in a sudden explosion then what is the

minimum fraction of mass that it must lose for the planet

to escape to infinity?

(a) f = ½ (b) f = ¼

(c) f = Mp/Ms (d) f = 𝑀𝑝2/𝑀𝑠

2

42. A cylinder tied to a string of length L(see figure

below) takes time t to fully unwind. The size of the

cylinder is the doubled, thus making its mass 8 times and

moment of inertia 32 times their former values. If the

length of the string remains unchanged then the time

taken to fully unwind the string becomes

(a) 8t (b)4t (c)𝑡

4 (d)t

43. An object is made from a thin wire and is shaped like

a square with a side length L and a total mass M. What is

the moment of inertia of this object around an axis that

passes through the centre of the square and is

perpendicular to it?

(a)7

6 ML2 (b)

1

3 ML2 (c)

4

3 ML2 (d)

16

3 ML2

44.A soap bubble is attached to a very thin pipe through

which it slowly leaks. If the air loss(volume per unit

time) given by Q 𝛼 ∆P, where ∆P is the excess pressure

across the tube, then which of the following expressions

describe the evolution of radial speed , 𝜇r = dr/dt, of the

bubble with radius r ?

(a)r2𝑢r = constant (b) r𝑢r = constant

(c) 𝑢r/r = constant (d) r3𝑢r = constant

45. A thin rod is inclined to the vertical at an angle 𝜃 =

45° degree as shown below



The length of the rod is L and the angular velocity 𝜔 is

in the vertical direction. What is the magnitude of the

angular momentum?

(a) ML2𝜔/12 (b) ML2𝜔/12 √2

(c) ML2𝜔/6 (d) ML2𝜔/24

46. A point particle at rest is released from the top of a

sphere of radius R and slides down frictionlessly under

gravity. At what angle from the vertical does it leave the

sphere?

(a) cos-1(1/3) (b) cos-1(2/3)

(c)sin-1(1/√2) (d)cos-1(1/√2)

47.A heavy uniform rope of length L and mass per unit

length 𝜇 goes over a frictionless pulley of diameter R,

and has two masses M and m attached to its two ends as

shown. In terms of the distance x, the equilibrium

position is given by

(a)1

2(L-

𝜇𝐿

𝑚+𝑀 +

𝜋𝑅

2) (b)

1

2(L+

𝑚+𝑀

𝜇 )

(c) L - 𝜋𝑅

2 (d)

1

2(L-

𝑚−𝑀

𝜇 -

𝜋𝑅

2)

48. A triangle of uniform mass density of base L and

height h is shown below. The centre of mass of triangle

lies at this distance above the base:

(a)ℎ

2 (b)

2ℎ

3 (c)

ℎ

3 (d)

ℎ

6

49. A particle of mass m is located at a distance x along

the axis of a uniform disk of mass M and radius R. The

gravitational force felt by the mass m is given by

(a) 2𝐺𝑀𝑚

𝑅2 (𝑧

(𝑧2+𝑅2)1/2 - 1) (b) 2𝐺𝑀𝑚

𝑅2 (𝑧

(𝑧2−𝑅2)1/2 - 1)

(c) 2𝐺𝑀𝑚

𝑅2 (𝑧

(𝑧2+𝑅2)1/2 +1) (d) 2𝐺𝑀𝑚

𝑅2 (𝑧

(𝑧2+𝑅2)1/2 )

50. A large cylindrical container filled with water up to a

height h rests on a table. Neglecting the effect of

viscosity, at what height from the bottom of the

container should a hole be made such that the resulting

jet of water hits the surface of the table at the maximum

distance?

(a) h/√2 (b)h/2 (c)h/√3 (d)h/3

51.A wheel of radius R = 1 m is rolling on the ground

with slipping. Its angular velocity is 200 rad/s. If its

linear speed is 100 ms-1 in the positive x direction then

the bottom most part of the wheel is travelling with

respect to the ground at



(a)-300 m/s (b)-100 m/s

(c)100 m/s (d)300 m/s

52. A man climbs down a hemispherical hill, of radius

100m from the topmost point. If the coefficient of

friction between the shoes and the hill is 𝜇 = 0.1 then

approximately how much distance does he have to walk

before he slips?

(a) 5m (b)10m (c)20m (d)will never slip

53. A U shaped tube of uniform cross-section A

contains a liquid of density 𝜌. The total length of the

column is L. If the fluid is displaced, then the frequency

of oscillation is

(a) 1

2𝜋√𝑔/𝐿 (b)

1

2𝜋√𝑔𝐿/𝐴

(c) 1

2𝜋√𝑔𝜌𝐴 (d)

1

2𝜋√𝑔/𝐴

54.A horizontally placed hollow tube Has a cross-

sectional area A at the beginning of the tube that

gradually tapers of to A/2 at the end. An incompressible,

ideal fluid of density enters the tube with a velocity v at

the beginning of the tube. The difference in the pressure

across thetube is

(a)2𝜌𝑣2 (b)𝜌𝑣2

(c) 𝜌𝑣2/2 (d)3𝜌𝑣2/2

55. Seven uniform disks, each of mass m and radius r,

are inscribed inside a regular hexagon as shown. The

moment of inertia of this system of seven disks about an

axis passing through the central disk and perpendicular

to the plane of the disks, is

(𝑎)7

2𝑚𝑟2 (b)7𝑚𝑟2

(c) 13

2𝑚𝑟2 (d)

55

2𝑚𝑟2

56.There are three planets in a circular orbits around a

star at distances a ,4a and 9a respectively. At time t=𝑡0

the star and the three planets are in a straight line. The

period of revolution ofthe closest planet is T. how long

after 𝑡0will they again be in the same straight line?

(a)8T (b)27T

(c) 216T (d)512T

57.A raindrop falls under gravity and captures water

molecules from atmosphere. Its mass changes at the rate

𝜆m(t), where 𝜆is a positive constant and m(t) is the

instantaneous mass. Assume that acceleration due to

gravity is constant and water molecules are at rest with

respect to earth before capture. Which of the following

statements is correct?

(a)The speed of the raindrop increases linearly with time

(b)The speed of the raindrop increases exponentially

with time

(c)The speed of the raindrop approaches a constant value

when 𝜆t>>1

(d)The speed of the raindrop approaches a constant value

when 𝜆t<<1

PART B



1. If a generalised co-ordinate has the dimensions of

momentum , the generalised velocity will have the

dimension of

(A) velocity (B) Acceleration

(C) Force (D) Torque

2.The product of any generalised momentum and

the associated (or conjugate) co-ordinate must have

the dimensions of

(A) Energy (B) Angular momentum

(C) Linear momentum (D) Force

3.Whatever dimension a generalised co-ordinate has

the product of the generalised force and generalised

displacement (co-ordinate) must have the dimension

of

(A) Force (B) Torque

(C) Work (D) None of these

4.Equation of motion for bead sliding on a

uniformly rotating wire in a force free space is

(A) �̈� = 𝑟𝜔2 (B) �̈� +𝑚𝑔𝑙𝜃

𝐼= 0

(C) 2𝑚𝑟𝑟 𝜃 + 𝑚𝑟2�̈� = 0 (D) �̈� +𝑔

𝑙 𝜃 = 0

5.A hoop rolling down on an inclined plane without

slipping, its velocity at the bottom of the inclined

plane

(A) (4𝑔𝑙 sin 𝜙

3)

1

2 (B)

2𝑔𝑙 sin 𝜙

3

(C) (2𝑔𝑙 sin 𝜙

3)

1

2 (D)

4𝑔𝑙 sin 𝜙

3

6. For an electrical circuit comprising an inductance

L and capacitance C ,charged to q coulombs and the

current flowing in the circuit is i amperes,

Lagrangian can be represented as

(A) 𝐿𝑞2 −𝑞2

𝐶 (B) 𝐿𝑞2 −

1

2𝑞2𝐶

(C) 1

2𝐿𝑞2 −

1

2

𝑞2

𝐶 (D)

1

2𝐿𝑞2 +

1

2

𝑞2

𝐶

7.Lagrangian for charged particle in an

electromagnetic field is given as

(A) 1

2𝑚𝑣2 + 𝑞𝜙 +

𝑞

𝑐�̅�. �̅�

(B) 1

2𝑚𝑣2 − 𝑞𝜙 −

𝑞

𝑐�̅�. �̅�

(C) 1

2𝑚𝑣2 − 𝑞𝜙 +

𝑞

𝑐�̅�. �̅�

(D) 1

2𝑚𝑣2 + 𝑞𝜙 −

𝑞

𝑐�̅�. �̅�

8.Langrangian for compound pendulum is

(A) 1

2𝐼𝜃2 − 𝑚𝑔𝑙 cos 𝜃

(B) 1

2𝐼𝜃2 + 𝑚𝑔𝑙 cos 𝜃

(C) 1

2𝑚(𝑟2 + 𝑟2𝜃2) +

1

2 𝐼𝜃2 − 𝑚𝑔𝑙 cos 𝜃

(D) 1

2𝑚(𝑟2 + 𝑟2𝜃2) +

1

2 𝐼𝜃2 + 𝑚𝑔𝑙 cos 𝜃

9.The path followed by particle in sliding from one

point to another in the absence of friction in the

shortest time is

(A) Sphere (B) Sigmoid

(C) Cycloid (D) Catenary revolution

10.If co -ordinate corresponding to a rotation is

cyclic rotation of the system about given axis

remain invariant then the following quantity is

conserved

(A) Linear momentum

(B) Angular momentum

(C) Kinetic energy

(D) Potential energy

11.A physical system is invariant under rotation

about a fixed axis. Then the following quantity is

conserved



(A) Total linear momentum

(B) Linear momentum along axis of rotation

(C) Total angular momentum

(D) Angular momentum along the axis of rotation

12.The period of oscillation for compound

pendulum is

(A) 2𝜋√(𝑘2+𝑙2)

𝑔𝑙 (B) 2𝜋√

𝑔𝑙

𝑘2+𝑙2

(C) 2𝜋√(𝑘2+𝑙2)

𝑚𝑔𝑙 (D) 2𝜋√

𝑚𝑔𝑙

𝑘2+𝑙2

13.A point mass,m,under no external forces is

attached to a weightless cord fixed to a cylinder of

radius R . Initially the cord is completely wound up

so that mass touches the cylinder. A radially

directed impulse is now given to the mass, which

starts unwinding, then the angular momentum of the

mass about the cylinder axis

(A) 𝑚𝑅𝑣02𝑡 (B) 𝑚 × (2𝑅𝑣0

3𝑡)1/2

(C) 𝑚 × (2𝑅𝑣03𝑡)1/3 (D) 𝑚 × (2𝑅𝑣0

3𝑡)

14.A solid homogeneous cylinder of radius, r, rolls

without slipping on the side of stationary large

cylinder of radius R the period of small oscillation

about the stable equilibrium position is

(A) 2𝜋√(3(𝑅−𝑟)

2𝑔) (B) 2𝜋√(

2(𝑅−𝑟)

3𝑔)

(C) 2𝜋√2𝑔

3(𝑅−𝑟) (D) 2𝜋√

3𝑔

2(𝑅−𝑟)

15.Hamilton’s canonical equations of motion are

(A) �̇�𝑖 =𝜕𝐻

𝜕𝑝𝑖 and �̇�𝑖 =

𝜕𝐻

𝜕𝑞𝑖

(B) �̇�𝑖 =𝜕𝐻

𝜕𝑝𝑖 and �̇�𝑖 = −

𝜕𝐻

𝜕𝑞𝑖

(C) 𝑞𝑖 =𝜕𝐻

𝜕�̇�𝑖 and 𝑝𝑖 =

𝜕𝐻

𝜕�̇�𝑖

(D) 𝑞𝑖 =𝜕𝐻

𝜕�̇�𝑖 and 𝑝𝑖 = −

𝜕𝐻

𝜕�̇�𝑖

16.If co-ordinate is cycle, Hamiltonian would

reduce the number of variable in formulation by

(A) one (B) Two

(C) Three (D) Four

17.For a charged particle in an electromagnetic

field the canonical momenta are

(A) 𝑚𝑣 +𝑞

𝑐𝐴 (B)

1

2𝑚𝑣2 +

𝑞

𝑐A

(C) 𝑚𝑣 −𝑞

𝑐𝐴 (D)

1

2𝑚𝑣2 −

𝑞

𝑐A

18.For a charged particle in a electromagnetic field,

the Hamiltonian �̅� is represented as

(A) 1

2𝑚 (

�̅�

𝑚+

𝑞

𝑚𝑐− �̅�)

2

+ 𝑞𝜙

(B) 1

2𝑚 (

�̅�

𝑚+

𝑞

𝑚𝑐�̅�)

2

− 𝑞𝜙

(C) 1

2𝑚 (

�̅�

𝑚−

𝑞

𝑚𝑐− �̅�)

2

+ 𝑞𝜙

(D) 1

2𝑚 (

�̅�

𝑚−

𝑞

𝑚𝑐− �̅�)

2

− 𝑞𝜙

19.The Jacobi’s form of the least action principle

(A) Δ ∫ √2[𝐻 − 𝑉(𝑞] 𝑑𝜌 = 0

(B) Δ ∫ √2[𝐻 + 𝑉(𝑞] 𝑑𝜌 = 0

(C) Δ ∫ √2[𝐿 − 𝑉(𝑞] 𝑑𝜌 = 0

(D) Δ ∫ √2[𝐿 + 𝑉(𝑞] 𝑑𝜌 = 0

20.An artificial satellite revolves about the earth at

height H above the surface, the orbital period so that

a man in the satellite will be in the state of

weightlessness is

(A) 2𝜋√𝑔

𝑅 (B) 2𝜋√

𝑅

𝑔

(C) 1

2𝜋√

𝑔

𝑅 (D)

1

2𝜋√

𝑅

𝑔



21.If a body is thrown vertically upwards, it strikes

the ground at

(A) 16

3 𝜔ℎ cos 𝜙 (

2ℎ

𝑔)

1/2

to the west

(B) 16


2ℎ

𝑔)

1/2

to the east

(C) 2


2ℎ

𝑔)

1/2

to the west

(D) 2


2ℎ

𝑔)

1/2

to the west

22. The 𝛼-particle scattering cross-section and

hence the number of 𝛼-particle scattered must be

proportional to

(A) E (B) 𝐸−1

(C) 𝐸2 (D) 𝐸−2

23.The mechanical equivalent of an LCR series

circuit with an voltage source is a

(A) Damped harmonic oscillator

(B) Forced harmonic oscillator

(C) Free linear harmonic oscillator

(D) Damped and forced harmonic oscillator

24. The value of m and n for which the

transformations are 𝑄 = 𝑞𝑚 cos 𝑛𝑝 ; 𝑃 =

𝑞𝑚 sin 𝑛𝑝 represents a canonical transformations

are

(A) m=1, n=2 (B) 𝑚 =1

2, 𝑛 = 2

(C) 𝑚 = 2, 𝑛 =1

2 (D) m=2, n=1

25. Jacobi identity for Poisson bracket

(A) [X,[Y,H]]+[Y,[H,X]]+[H,[X,Y]]=0

(B) [X,[Y,H]]-[Y,[H,X]]+[H,[X,Y]]=0

(C) [X,[Y,H]]+[Y,[H,X]]-[H,[X,Y]]=0

(D) [X,[Y,H]]-[Y,[H,X]]-[H,[X,Y]]=0

26. The operator which represents the two variables

should commute if the Poisson bracket of two

variables have value

(A) i (B) 0 (C) ih (D) -ih

27. An inverted pendulum consists of a particle of

mass m supported by rigid mass less rod length 𝑙.

The pivot o has a vertical motion given by

z=𝐴𝑠𝑖𝑛 𝜔𝑡 the Lagrangian of the system is

(A) 1

2𝑚𝑙2𝜃2 + 𝑚𝑔𝑙 cos 𝜃 − 𝑚𝑙 𝐴𝜔2 sin 𝜔𝑡 cos 𝜃

(B) 1

2𝑚𝑙2𝜃2 − 𝑚𝑔𝑙 cos 𝜃 − 𝑚𝑙 𝐴𝜔2 sin 𝜔𝑡 cos 𝜃

(C) 1

2𝑚𝑙2𝜃2 − 𝑚𝑔𝑙 cos 𝜃 + 𝑚𝑙 𝐴𝜔2 sin 𝜔𝑡 cos 𝜃

(D) 1

2𝑚𝑙2𝜃2 + 𝑚𝑔𝑙 cos 𝜃 + 𝑚𝑙 𝐴𝜔2 sin 𝜔𝑡 cos 𝜃

28. Lagrange’s equations of motion are second

order equations, the degree of freedom for this are

(A) 2n (B) 2n-1

(C) 2n+1 (D) 2n+2

29. For the transformation 𝑄 = log (1 +

𝑞1/2 cos 𝑝); 𝑃 = 2𝑞1/2(1 + 𝑞1/2 cos 𝑝). The

generating function is

(A) −(𝑒𝑄 − 1)2 tan 𝑝 (B) (𝑒𝑄 − 1)2 cot 𝑝

(C) (𝑒𝑄 − 1)2 tan 𝑝 (D) −(𝑒𝑄 − 1)2 cot 𝑝

30.for generating function 𝐹1 =1

2𝑚𝜔2𝑞2 cot 𝑄 an

expression for the displacement of linear harmonic

oscillator is given by

(A) √𝟏

𝒎𝝎𝟐sin(ωt + β) (B) √

𝟏

𝒎𝝎𝟐sin(ωt − β)

(C) √𝟐𝑬

𝒎𝝎𝟐 sin(𝜔𝑡 + 𝛽) (D) √𝟐𝑬

𝒎𝝎𝟐 sin(𝜔𝑡 − 𝛽)

31. The force which is always directed away or

towards a fixed centre and magnitude of which is



function only of the distance from the fixed centre,

known as

(A) Coriolis force (B) Centripetal force

(C) Centrifugal force (D) Central force

32.Differential equation for planetary motion is

given as

(A) 𝑑2𝑢

𝑑𝜃= 𝑢 −

𝑚

𝑖2𝑢2 𝑓(1

𝑢)

(B) 𝑑2𝑢

𝑑𝜃= 𝑢 +

𝑚

𝑖2𝑢2 𝑓(1

𝑢)

(C) 𝑑2𝑢

𝑑𝜃= −𝑢 −

𝑚

𝑖2𝑢2 𝑓(1

𝑢)

(D) 𝑑2𝑢

𝑑𝜃= −𝑢 +

𝑚

𝑖2𝑢2 𝑓(1

𝑢)

33. For obits under inverse square law of force, the

effective potential energy is given is given by

(A) 𝑘

𝑟+

𝑙2

2𝑚𝑟2 (B) 𝑘

𝑟−

𝑙2

2𝑚𝑟2

(C) 𝑘

𝑟2 +𝑙2

2𝑚𝑟2 (D) −𝑘

𝑟2 −𝑙2

2𝑚𝑟2

34.The momentum of an electron (mass m) which

has the same kinetic energy as it rest mass energy is

(A) √3 𝑚𝑐 (B) √2 𝑚𝑐

(C) mc (D) 𝑚𝑐

√2

35.Two events are seperated by a distance of

6 × 105km and the first event occurs 1s before the

second event. The Interval between the two events

(A) Is time like (B) Is light like(null)

(C) Is space like (D) Cannot be determined from

the information given

36.Which of the following equation is relatively

invariant (𝛼, 𝛽, 𝛾 and 𝛿 are constant of suitable

dimension ) ?

(A) 𝜕𝜙(𝑥,𝑡)

𝜕𝑡= 𝛼

𝜕2𝜙

𝜕𝑥2 (𝑥, 𝑡)

(B) 𝜕2𝜙

𝜕𝑡2(𝑥, 𝑡) = 𝛽2 𝜕2𝜙

𝜕𝑥2 (𝑥, 𝑡)

(C) ) 𝜕2𝜙

𝜕𝑡2(𝑥, 𝑡) = 𝛾

𝜕𝜙

𝜕𝑥2 (𝑥, 𝑡)

(D) 𝜕𝜙

𝜕𝑡= 𝛿

𝜕3𝜙

𝜕𝑥3(𝑥, 𝑡)

37.Although mass-energy equivalence of special

relativity allows conversion of a photon to an

electron-positron pair such a process cannot occur

in free space because

(A) The mass is not conserved

(B) The energy is not conserved

(C) The momentum is not conserved

(D) The charge is not conserved

39.The Lagrangian for a three particle system is

given by 𝐿 =1

2(𝜂1

2 + 𝜂22 + 𝜂3

2) − 𝑎2(𝜂12 + 𝜂2

2 +

𝜂32 − 𝜂1𝜂3) where a is real then of the normal co-

ordinates has the frequency 𝜔

(A) 𝜔2 = 𝑎2 (B) 𝜔2 =𝑎2

2

(C) 𝜔2 = 2𝑎2 (D) 𝜔2 = √2𝑎2

40. Identify the points of unstable equilibrium for

the potential shown in the figure

(A) p and s (B) q and t

(C) r and u (D) r and s

41.For repulsive inverse squre forces, the shape of

erbit

(A) Elliptic (B) Parabolic

(C) Hyperbolic (D) All of these



42.In the case of elliptic orbits energy is

Proportional to

(A) a (B) 𝑎−1

(C) 𝑎−3 (D) 𝑎3

where a is semi major axis of elliptic orbit

43.A particle of mass m,moves under the under the

action of a central force whose potential is 𝑉(𝑟) =

𝑘𝑚𝑟3(𝑘 > 0) then energy for which the orbit will

be a circle of radius a, about the orgin is

(A) 3

2𝑚𝑘𝑎3 (B)

3

2𝑚𝑘𝑎2

(C) 1

2𝑚𝑘𝑎 (D)

1

2𝑚𝑘𝑎2

44.A particle of mass m moves under the action of

central force is potentional is 𝑉(𝑟) = 𝑘𝑚𝑟3(𝑘 > 0)

then the period of circular motion is

(A) 2𝜋

√𝑘𝑎2 (B)

2𝜋

√3𝑘𝑎

(C) 2𝜋

√𝑚𝑘𝑎 (D)

2𝜋

√𝑘𝑎

45.A particle of mass m, under the action of central

force whose potential 𝑉(𝑟) = 𝑘𝑚𝑟3(𝑘 > 0) then

angular momentum for which the orbit will be a

circle of radius a ,about the origin is

(A) 𝑚𝑎√3𝑘𝑎 (B) 𝑚𝑎2√𝑘𝑎

(C) 𝑚𝑎2√3𝑘𝑎 (D) 𝑚𝑎√𝑘𝑎

46. A particle of mass m, under the action of central

force whose potential 𝑉(𝑟) = 𝑘𝑚𝑟3(𝑘 > 0) then

angular frequency is

(A) √3𝑘𝑎 (B) √𝑘𝑎

(C) √5𝑘𝑎 (D) √15𝑘𝑎

47.The mutual potential energy V, of two particles

depends on their mutual distance, r, as follows

𝑉 =𝑎

𝑟2 −𝑏

𝑟; 𝑎 > 0, 𝑏 > 0 if the particles are in

static equilibrium , then the seperation is

(A) 2𝑎

𝑏 (B)

2𝑏

𝑎

(C) 𝑎

𝑏 (D)

𝑏

𝑎

48.A particle of mass m moves in a central force

field defined by �̅� = −𝑘�̅�/𝑟4, if �̅� is the total

energy supplied to the particle, then its speed is

given by

(A) 𝑘

𝑚𝑟2+

2𝐸

𝑚 (B)

𝑘

𝑚𝑟2−

2𝐸

𝑚

(C) √𝑘

𝑚𝑟2+

2𝐸

𝑚 (D) √

𝑘

𝑚𝑟2−

2𝐸

𝑚

49.A particle moving in a central force located at

r=0 describes the spiral 𝑟 = 𝑒−𝜃 the magnitude of

force is inversely proportional to

(A) r (B) 𝑟2 (C) 𝑟3 (D) 𝑟4

50.A particle describes a circular orbit under the

influence of an attractive central force directed

towards a point on the circle. The force inversely

proportional to

(A) 𝑟2 (B) 𝑟3 (C) 𝑟4 (D) 𝑟5

51. For orbits the conic depends on the value of

eccentricity given

(A) √1 −2𝐸𝑙2

𝑚𝑘2 (B) √1 +

2𝐸𝑙2

𝑚𝑘2

(C) √1 −𝑚𝑘2

2𝐸𝑙2 (D) √1 +𝑚𝑘2

2𝐸𝑙2

52.The impact parameter,s, defind as the

perpendicular distance between the centre of force

and the incident velocity this parameter proportional

to



(A) E (B) 𝐸1/2

(C) 𝐸−1 (D) 𝐸−1/2

53.The orbit is symmetric about the direction of

peripherals. the scattering angle is given by

(A) 𝜃 = 𝜋 − 2𝜓 (B)𝜃 = 𝜋 + 2𝜓

(C)𝜃 = 2𝜋 − 𝜓 (D)𝜃 = 2𝜋 + 𝜓

54.The disired relationship between the impact

parameter and the scattering angle is

(A) 𝑍𝑍′𝑒2

2𝐸 𝑐𝑜𝑠𝑒𝑐

𝜃

2 (B)

𝑍𝑍′𝑒2

2𝐸sin

𝜃

2

(C) 𝑍𝑍′𝑒2

2𝐸cot

𝜃

2 (D)(

𝑍𝑍′𝑒2

2𝐸) 𝑐𝑜𝑠𝑒𝑐4 𝜃

2

55.I n the famous Rutherford scattering cross

section,differential scattering cross scattering

section proportional to

(A) e (B)𝑒2

(C) 𝑒3 (D)𝑒4

56. I n the famous Rutherford scattering cross

section,differential scattering cross scattering

section is inversely proportional to

(A) sin 𝜃 (B)sin2 𝜃

(C) sin3 𝜃 (D)sin4 𝜃

57.The angle of recoil of the target particle relative

to the incident direction of the scattered particle is

(A) 1

2(𝜋 − 𝜃) (B)

1

2(𝜋 + 𝜃)

(C) 𝜋 − 𝜃 (D)𝜋 + 𝜃

58.A body is freely fallinf on the earth’s surface, the

body deflects by

(A) 3 cm towards west (B) 3 cm towards east

(C) 3 cm towards south (D) 3 cm towards north

59.A ball is released from rest from a great height

above the ground in Delhi . It will fall on the ground

(A) Exactly below the point of release

(B) Slightly east of the vertical

(C) Slightly west of the vertical

(D) Slightly north of the vertical

60.The LorentZ transformation matrix where the

relative velocity is along the Z-axis given by

(A)(

0 0 0 00 0 0 010

01

𝛾−𝑖𝛽𝛾

𝑖𝛽𝛾𝛾

)

(B) (

1 0 0 00 1 0 000

00

−𝛾𝑖𝛽𝛾

𝑖𝛽𝛾𝛾

)

(C)(

1 0 0 00 1 0 000

00

𝛾−𝑖𝛽𝛾

𝑖𝛽𝛾𝛾

)

(D)(

0 0 0 00 0 0 010

01

𝑖𝛽𝛾−𝑟

𝛾𝑖𝛽𝛾

)

61.The length contraction

(A) Predicts that the length of an object approaches

zero as its speed approches the speed of light in

vaccum

(B) Predicts that there is no change in the length of

an object when it speed approaches the speed of

light in vaccum

(C) Predicts that the length of an object reduce to

half when its speed approaches the speed of light in

vaccum

(D) Predicts that the length of the object is directly

proportional to its velocity



62.An astronaut moves in a super spaceship

traveling at a speed of 0.8c.The astronaut observes a

photon approaching him from space.The speed of

photon with respect to the astronaut is

(A) 1.8c (B) c

(C) 0.2c (D) 0.9c

63.A spaceship is travelling with a velocity 0.4c

where c is the velocity of light . A person

performing an experiment in this spaceship

observes a particle moving with a velocity 0.4c in

the same direction as that of the motion of the

spaceship . A stationary observer on the earth would

observe the particle to have the velocity

(A) 0.69c (B) 0.50c

(C) 0.80c (D) 0.73c

64.LorentZ transformation assume

(A) Space and time are both relative

(B) Spase is relative,but time is absolute

(C) Spase is absolute , but time is relative

(D) Space and time are both absolute

65.The difference vector 𝑋𝜇, is space like if the two

world points are seperated such that

(A) |�̅�1 − �̅�2|2 ≥ 𝑐2(𝑡1 − 𝑡2)2

(B) |�̅�1 − �̅�2|2 ≤ 𝑐2(𝑡1 − 𝑡2)2

(C) |�̅�1 − �̅�2|2 > 𝑐2(𝑡1 − 𝑡2)2

(D) |�̅�1 − �̅�2|2 < 𝑐2(𝑡1 − 𝑡2)2

66. The difference vector 𝑋𝜇, is time like if the two

world points are seperated such that

(A) |�̅�1 − �̅�2|2 ≥ 𝑐2(𝑡1 − 𝑡2)2

(B) |�̅�1 − �̅�2|2 ≤ 𝑐2(𝑡1 − 𝑡2)2

(C) |�̅�1 − �̅�2|2 > 𝑐2(𝑡1 − 𝑡2)2

(D) |�̅�1 − �̅�2|2 < 𝑐2(𝑡1 − 𝑡2)2

67. The LotrentZ tranformations are equivalent to

rotation of axes in four dimensional space through

an imaginary angle

(A) tan(𝑖𝛽) (B) sin (𝑖𝛽

√1−𝛽2)

(C) tan−1(𝑖𝛽) (D) cos−1 (𝑖𝛽

√1−𝛽2)

68. The proper length of space vehicle 𝐼0.According

to an observer on earth , the length of the spaceship

is 25% of its proper length. The speed of the

spaceship according to the observer on earth is

(A) 𝑐√3

2 (B) 𝑐 √

3

2

(C) 0.968c (D) 0.87c

69.On the annihilation of a particle anf its anti-

particle, the energy released is E,mass of each

particle

(A) 𝐸

𝑐2 (B) 𝐸

2𝑐2

(C) 𝐸

𝑐 (D)

𝐸

2𝑐

70.A cube has side 𝐼0 when at rest .If the cube

moves with velocity v parallel to its one edge then

its volume becomes

(A) 𝑙03 (B) 𝑙0

3 (1 −𝑣2

𝑐2)

−1

2

(C) 𝑙0 3 (1 −

𝑣2

𝑐2) (D) ) 𝑙03 (1 −

𝑣2

𝑐2)

1

2

71.In the normal co-ordinate of the system each of

the new co-ordinates involving..,………resonant

frequencies

(A) One (B) Two

(C) Three (D) Four

72.Normal frequencies for free vibration of linear

triatomic molecules



(A) √𝑘

𝑚(1 +

2𝑀

𝑚) (B) √

𝑀

𝑚(1 +

2𝑘

𝑚)

(C) √𝑘

𝑚(1 +

2𝑚

𝑀) (D) √

𝑘

𝑚(1 +

2𝑀

𝑘)

73.Ppssible longnitudinal mormal modes of the

linear symmetric triatomic molecule is/are

(A) One (B) Two (C) Three (D) Four

74.Number of possible modes of vibration

perpendicular to the axis in linear symmetric

triatomic molecules

(A) Two (B) Three (C) Four (D) Four

75.Normal frequency for free vibrations of the

parallel pendula is given by

(A) √𝑔

𝑙−

2𝑘

𝑚 (B) √

𝑔

𝑙+

2𝑘

𝑚

(C) √𝒈

𝒍−

𝑚

𝟐𝒌 (D) √

𝒈

𝒍+

𝑚

𝟐𝒌

76.A masseless spring of force constant k has

masses 𝑚1 and 𝑚2 attached it two ends. The system

rests on a horizontal table . The angular vibrational

frequency 𝜔 of this system is

(A) [𝑘/(𝑚1 − 𝑚2)]1/2 (B) [𝑘/(𝑚1 + 𝑚2)]1/2

(C) [𝑘(𝑚1 + 𝑚2)/𝑚1𝑚2)1/2 (D) [𝑘 (1

𝑚1−

1

𝑚2)]

77. there are six particles lyimg on a plane . the

degrees of freedom associated with them are

(A) 6 (B) 18

(C) 12 (D) None of these

78.rolling down from the top of a fixed spheres is

an example of

(A) Scleromic, non holonomic and conservative

system

(B) Only conservative system

(C) Only Scleromic system

(D) Only non-holomic system

79.A cylinder rolling without slliping down a rough

inclined plane of 𝑎𝑛𝑔𝑙𝑒 𝜃 is an example of

(A) Scleromic,conservative system only

(B) Scleronomic,holonomic,conservative system

(C) Only conservative system

(D) Only Scleronomic system

80.A particle moving on a very long frictionless

wire which rotates with constant angular velocity

about a horizontal axis is an example of

(A) Rhenomic,holonomic ,conservative system

(B) Only conservative system

(C) Only honomic and conservative system

(D) Rhenomic,non-holomic and non-conservative

81.How many degree’s of freedom a rigid body

posses

(A) 3 (B) 6 (C) 9 (D) infinity

82.When a rigid body rotates about a given axis the

degree’s of freedom it will have, is

(A) 1 (B) 2 (C) 3 (D) 4

83.When a cylinder rolls without slliping on a

plane, how many degrees of freedom it has

(A) 1 (B) 2 (C) 3 (D) 4

84.Two particles moving in space curve and have

fixed distance between them, have degrees of

freedom numbering



(A) 1 (B) 2 (C) 3 (D) 4

85.Three particles moving in space so that the

distance between any two of them always remain

fixed have degrees of freedom equal to

(A) 1 (B) 3 (C) 3 (D) 9

86.The number of degrees of freedom for a system

for a system of a rigid rod moving freely in space

and a particle is constraint to move opn that rod is

equal to

(A) 1 (B) 2 (C) 3 (D) 4

87.Sclerenomous constraints are

(A) Independent of time (B) Dependent on time

(C) Both (A) and B (D) None of these

88.Constraint in the case of rigid body is

(A) Dynamic constraint

(B) Sclerenomous constraint

(C) Rhenimous constraint

(D) Static constraint

89.An example of a rheonomous constraint is

(A) Bread rotating on a wire loop

(B) Bread on a rotating wire loop

(C) A simple pendulum

(D) A torisional pendulum

90.name the type of constraint that may be

expressed in the form of as equation relating co -

ordinates of the system and time

(A) Holonomic (B) Non-holonomic

(C) Scleronomous (D) All of these

91.A non-holonomous constraint may be expressed

in the form of

(A) Equality (B) Inequality

(C) Vector (D) nonre of these

92.The constraint on the motion of a particle in a

plane reduces the number of degrees of freedom by

(A) One (B) Three

(C) Four (D) None of these

93.A particle is constrained to move along the inner

surface of a hemisphere number of degrees of

freedom of the particle is

(A) One (B) Two

(C) Three (D) Four

94.The generalised co-ordinate 𝜃 for the motion of

a simple pendulum oscillating in sa vertical plane is

(A) cos−1 𝑥

𝑙 (B) sin−1 𝑦

𝑙

(C) (A) and (B) both (D) None of these

95.The Lagrangian method of undetermined

multiplers can be used for the holonomic constraints

if

(A) The forces of constraints are required

(B) It is inconvient to reduce all the co-ordinates of

the system to independent ones

(C) Both (A) and (B)

(D) None of these

96.Let two unequal masses 𝑚1 and 𝑚2(𝑚1 < 𝑚2)

be connected by a string of length l, passes over a

frictionless pully such that the distance of 𝑚2from

the pully be x, then the Langrangian of the system is

(A) 1

2(𝑚1 − 𝑚2)𝑥2 + (𝑚1 + 𝑚2)𝑥



(B) 1

2(𝑚1 + 𝑚2)𝑥2 + (𝑚2 − 𝑚1)𝑥

(C) 1

2(𝑚1 + 𝑚2)𝑥2 − (𝑚2 − 𝑚1)𝑥

(D) 1

2(𝑚2 − 𝑚1)𝑥2 − (𝑚2 − 𝑚1)𝑥

97.A particle of mass m moves along a straight line

and is attracted towards a point on this line with a

force proportional to the distence x from that point.

The Langragian of the system is

(A) 1

2𝑚𝑣2 +

1

2𝑘𝑥2 (B)

1

2𝑚𝑣2 −

1

2𝑘𝑥2

(C) 1

2𝑚𝑣2 + 𝑘𝑥2 (D) 𝑚𝑣2 +

1

2𝑘𝑥2

98.A particle of mass m, moves in a plane,its

motion defined by (𝑟, 𝜃) under the influence of a

force 𝐹 = −𝑘𝑟 directed towards the origin. The

Lagrangian of the system is given by

(A) 1

2𝑚�̇�2 +

1

2𝑚𝑟2�̇�2 −

1

2𝑘𝑟2 (B)

1

2𝑚�̇�2 +

1

2𝑘𝑟2

(C) 1

2𝑚�̇�2 +

1

2𝑚𝑟2�̇�2 +

1

2𝑘𝑟2 (D) None of these

99.In Q.98 the r-component of the motion is given

by

(A) 𝑚�̈� + 𝑘𝑟 = 0 (B) 𝑚�̈� + 𝑚𝑟2�̇�2 + 𝑘𝑟 = 0

(C) 𝑚�̈� − 𝑚𝑟2�̇�2 + 𝑘𝑟 = 0 (D) None o0f these

100.The equation of motion for a small particle of

mass m at position x is 𝑚�̇� + 𝛾�̇� − 𝑚𝑔 = 0.

Assuming initial speed to be 𝑣0 the terminal speed

of particle will be

(A) 𝑚𝑔

𝛾 (B) √𝑣0 + 2𝑔𝑥

(C) 𝑣0 + 𝑔𝑡 (D) 𝑚𝑔

𝛾2𝑡

101.langrangian of the sun-earth system is (where r

is the sun and earth distence M and m are the mass

of sun and earth respectively �̇� is the angular speed

and G is gravitational constant.)

(A) ) 1

2𝑚�̇�2 +

1

2𝑚𝑟2�̇�2 −

𝐺𝑀𝑚

𝑟

(B) ) 1

2𝑚�̇�2 +

1

2𝑚𝑟2�̇�2 +

𝐺𝑀𝑚

𝑟

(C) ) 1

2𝑚�̇�2 −

𝐺𝑀𝑚

𝑟

(D) 1

2𝑚𝑟2�̇�2 +

𝐺𝑀𝑚

𝑟

102.The generalised v elocity co-ordinate 𝑞𝑘 of a

classifical system with Lagrangian ‘L’ is said to be

cycle if

(A) 𝜕𝐿

𝜕𝑞𝑘= �̇�𝑘 (B)

𝜕𝐿

𝜕𝑞𝑘=

𝑑

𝑑𝑡(

𝜕𝐿

𝜕�̇�𝑘)

(C|) 𝜕𝐿

𝜕𝑞𝑘= 0 (D)

𝜕𝐿

𝜕�̇�𝑘= 0

103.The Lagrangian of a particle moving in a plane

under the influence of a central potential is given b

y 𝐿 =1

2𝑚(�̇�2 + 𝑟2�̇�2) − 𝑉(𝑟). The generalised

moments corresponding to r and 𝜃 are given by

(A) 𝑚�̇� and 𝑚𝑟2�̇� (B) 𝑚�̇� and 𝑚𝑟�̇�

(C) 𝑚�̇�2 and 𝑚𝑟2�̇� (D) 𝑚�̇�2 and 𝑚𝑟2�̇�2

104.The Hamiltonian corresponding to the

Lagrangian 𝐿 = 𝑎𝑥2 + 𝑏𝑦2 − 𝑘𝑥𝑦 is

(A) 𝑃𝑥2

2𝑎+

𝑃𝑦2

2𝑏+ 𝑘𝑥𝑦 (B)

𝑃𝑥2

4𝑎+

𝑃𝑦2

4𝑏− 𝑘𝑥𝑦

(C) 𝑃𝑥2

4𝑎+

𝑃𝑦2

4𝑏+ 𝑘𝑥𝑦 (D)

𝑃𝑥2+𝑃𝑦2

4𝑎𝑏+ 𝑘𝑥𝑦

105.The Lagrangian of particle of mass m moving

in a plane is given by 𝐿 =1

2𝑚(𝑣𝑥

2 + 𝑣𝑦2)𝑎(𝑥𝑣𝑦 −

𝑦𝑣𝑥) where 𝑣𝑥 and 𝑣𝑦 are velocity components and

a is a constant. The canonial momenta of the

particle are given by

(A) 𝑝𝑥 = 𝑚𝑣𝑥 and 𝑝𝑦 = 𝑚𝑣𝑦

(B) 𝑝𝑥 = 𝑚𝑣𝑥 + 𝑎𝑦 and 𝑝𝑦 = 𝑚𝑣𝑦 + 𝑎𝑥

(C) 𝑝𝑥 = 𝑚𝑣𝑥 − 𝑎𝑦 and 𝑝𝑦 = 𝑚𝑣𝑦 + 𝑎𝑥



(D) 𝑝𝑥 = 𝑚𝑣𝑥 − 𝑎𝑦 and 𝑝𝑦 = 𝑚𝑣𝑦 − 𝑎𝑥

106.A system of two particles having masses 𝑚1

and 𝑚2 are connected by an in extensible, masseless

string of length 𝑙passing throygh a small hole

horizontal table, the Lagrangian of the system is

(A) (𝑚1 + 𝑚2)�̇�2

2+

𝑚2𝑟�̇�2

2− 𝑚1𝑔(𝑟 − 𝑙)

(B) (𝑚1 + 𝑚2)�̇�2

2+ (𝑚2 − 𝑚1) 𝑔(𝑟 − 𝑙)

(C) (𝑚1 + 𝑚2)�̇�2

2+ (𝑚2 − 𝑚1)

𝑟�̇�2

2

(D) (𝑚1 + 𝑚2)�̇�2

2+ 𝑚1

𝑟�̇�2

2− 𝑚1𝑔(𝑟 − 𝑙)

107.Which of the following is incorrect for

conservative system ?

(A) 𝛿𝐿 = 𝑝𝑖𝛿𝑞𝑖 (B) 𝛿𝐿 −𝑑

𝑑𝑡(𝑝𝑖𝛿𝑞𝑖)

(C) 𝜕𝑇

𝜕𝑝𝑖= 𝑞𝑖 (D) 𝜕𝑇 + 𝜕𝑉 = 0

108.Initially two co-cordinates system are

coincident .The primed rotates with angular velocity

𝜔 with respect to the other non-rotating fame. If ‘i’

is the one of unit vector in the rotating co-ordinate

system . Then 𝑑𝑖

𝑑𝑡 is non-rotating frame is given be

(A) i’ (B) �⃗⃗⃗� × 𝑖′

(C) �⃗⃗⃗�. 𝑖′ (D) �̅� × (�̅� × 𝑖′)

109.In Q.108 𝑑2𝑖′

𝑑𝑡2 is non-rotating frame

(A) zero (B) 𝜔2𝑖′

(C) �⃗⃗⃗� × 𝑖′⃗⃗⃗ (D) �̅� × (�̅� × 𝑖̅′)

110.In Q.108 𝑑𝑖′

𝑑𝑡 in rotating co-ordinate system is

given by

(A) Zero (B) 𝑖′⃗⃗⃗ (C) �⃗⃗⃗� × 𝑖′⃗⃗⃗ (D) �̅� × (�̅� × 𝑖̅′)

111.An xyz co-ordinate system ,initially coinciding

with an intial frame xyz. rotetes with an angular

velocity �̅� = 2𝑖̂ + 𝑡2𝑗̂ + (2𝑡 + 4)�̂� where t=time .

The position vector of a particle at time t in (xyz)

system is given by 𝑟 = (𝑡2 + 1)𝑖̂ + 6𝑡𝑗̂ + 4𝑡3�̂� its

apparent velocity at time t=1 sec

(A) 𝑣′ = 2𝑖̂ + 6𝑗̂ + 4�̂�

(B) 𝑣′ = 2𝑖̂ − 6𝑗̂ + 12�̂�

(C) 𝑣′ = 2𝑖̂ + 6𝑗̂ + 12�̂�

(D) 𝑣′ = 2𝑖̂ + 6𝑗̂ − 4�̂�

112.Which one of the following particles

experience a coriolis

(A) A particle at rest w.r.t earth at Bhopal

(B) A particle thrown vertically upward at Bhopal

(C) A particle thrown vertically upward at the north

pole

(D) A particle moving horizontally along the north -

south direction at the Bhopal

113.If the escape velocity from the surface of a

spherical planet of mass M is given by √𝐺𝑀

2𝑅 the

radius of the planet is

(A) R/2 (B) R (C) 2R (D) 4R

114. A linear transformation of a generalised co-

ordinates q and the corresponding momentum p to

Q and P given by 𝑄 = 𝑞 + 𝑝 ; 𝑃 = 𝑞 + 𝛼𝑝 is

canonical if the value of the constant 𝛼 is

(A) -1 (B) 0 (C) +1 (D) +2

115.A particle of mass m, is constrained to move on

the plane curve xy=C(C>0) under gravity (y-axis

vertical ) .The Lagrangian of the particle is given by

(A) 1

2𝑚�̇�2 (1 +

𝐶2

𝑥4) +𝑚𝑔𝐶

𝑥



(B) 1

2𝑚�̇�2 (1 +

𝐶2

𝑥4) −𝑚𝑔𝐶

𝑥

(C) 1

2𝑚�̇�2 (1 +

𝐶

𝑥4) +

𝑚𝑔𝐶

𝑥

(D) 1

2𝑚�̇�2 (1 +

𝐶

𝑥4) −𝑚𝑔𝐶

𝑥

116.A particle of mass m falls a given distance 𝑧0 in

time 𝑡0 = √2𝑧0

𝑔 and the distance travelled in time t is

given by 𝑧 = 𝑎𝑡2 + 𝑏𝑡2, where constant a and b are

such that the time 𝑡0 is always the same . The

integral ∫ 𝐿𝑑𝑡𝑡

0 is an extremum for real values of the

coefficient only when

(A) 𝑎 = 0 𝑎𝑛𝑑 𝑏 =𝑔

2 (B) 𝑎 =

𝑔

2𝑎𝑛𝑑 𝑏 = 0

(C) 𝑎 = 𝑔, 𝑏 = 0 (D) 𝑎 =𝑔

2𝑎𝑛𝑑 𝑏 = 𝑔

117.The Hamiltonian corresponding to Lagrangian

𝐿 −1

2�̇�2 −

1

2𝜔2𝑥2 − 𝛼𝑥3 + 𝛽𝑥�̇�2

(A) H=𝑝2

2(1+2𝛽𝑥)−

1

2𝜔2𝑥2 + 𝛼𝑥3

(B) H=𝑝2

2(1+2𝛽𝑥)+

1

2𝜔2𝑥2 + 𝛼𝑥3

(C) H=𝑝2

2(1+2𝛽𝑥)−

1

2𝜔2𝑥3 + 𝛼𝑥2

(D) H=𝑝2

2(1+2𝛽𝑥)−

1

2𝜔2𝑥 + 𝛼𝑥3

118. 𝑞1 𝑎𝑛𝑑 𝑞2 are fgeneralised co-ordinate and

𝑝1, 𝑝2 are the corresponding generalised

momentum. The Poisson bracket [X,Y] of 𝑋 =

𝑞12 + 𝑞2

2 and 𝑌 = 2𝑝1 + 𝑝2 is

(A) (𝑞12 + 𝑞2

2)𝑝1 (B) 3(𝑞12 + 𝑞2

2 )

(C) 4𝑞1 + 2𝑞2 (D) 0

119.The tranmformation 𝑞 = 𝑃𝑄2, 𝑝 =1

𝑄 is

canonical. Generating function is

(A) F=qp (B) 𝑃 =𝑃

𝑞

(𝐶) 𝐹 = 𝑃𝑞 m (D) 𝐹 = 𝑝2𝑞

120. A particle moving in a circular orbit about the

origin under the actio of central force 𝐹 = −𝑘�̂�

𝑟3 If

the potential energy iz zero at infinity, the totel

energy of the particle is

(A) −𝑘

𝑟2 (B) −𝑘

2𝑟2

(C) zero (D) +𝑘

𝑟2

121. A particle moving in 1

𝑟 potential . Which of the

following statements is incorrect in this case ?

(A) Angular momentum of the particle is always

conserved

(B) Kinetic energy of the particle is always

conserved

(C) The particle always follows a closed path

(D) Force on the particle is always radical

122.A planet is revolving around a star in an elliptic

orbit . The ratio of the farthest distance to the closet

distance of the planet from the star is 4. The ratio of

kinetic energies of the planet at the farthest to the

closest position is

(A) 1:16 (B) 16:1 (C) 1:4 (D) 4:1

123.A particle moves in a central force field 𝑓̅ =

−𝑘𝑟𝑛�̂�. where k is a constant,r the distance of the

particle from the origin and �̂� is the unit vector in

the direction of position vector 𝑟 .Closed stable

orbits are possible for

(A) n=1 and n=2 (B) n=1 and n=-1

(C) n=2 and n=-2 (D) n=1 and n=-2

124.The Lagraaqngian for the Kepler problem is

given by 𝐿 =1

2(�̇�2 + 𝑟2�̇�2 +

𝜇

𝑟(𝜇 > 0)) where 𝑟, 𝜃

denote the polar co-ordinates and the mass of the

particle is unity. Then



(A) 𝑝𝜃 = 2𝑟2�̇� (B) 𝑝𝑟 = 2�̇�

(C) The angular momentum of the particle about the

centre of attraction is a constant

(D) The total energy of the particle is time

dependent

125.the mean distance of Mars from the sun being

1.524 times that of the earth, the time of revolution

of Mars about sun

(A) 1 year (B) 10.24 years

(C) 1.8814 years (D) 18.814 years

126. A particle describes the curve is luminscate

under a force P toward the pole

(A) 𝑃 =1

𝑟5 (B) 𝑃 =1

𝑟7

(C) 𝑃 =1

𝑟3 (D) 𝑃 =1

𝑟9

127.The speed v of a particle moving in an elliptical

path in an inverse square field is given by

(A) 𝑣2 =𝑘

𝑚(

2

𝑟−

1

𝑎) (B) 𝑣2 =

𝑘

𝑚(

2

𝑟+

1

𝑎)

(C) 𝑣2 =𝑘

𝑚(

𝑟

2+

1

𝑎) (D) 𝑣2 =

𝑘

𝑚(

2

𝑟+

𝑎

2)

128.For orbits under inverse square law of force the

effective potential energy is given by

(A) 𝑘

𝑟2+

𝑙2

2𝑚𝑟2 (B) −

𝑘

𝑟2−

𝑙2

2𝑚𝑟2

(C) 𝑘

𝑟2−

𝑙2

2𝑚𝑟2 (D)

𝑘

𝑟+

𝑙2

2𝑚𝑟2

129.The force which always directed away or

towards a fixed centre and magnitude of which is a

function only of the distance from the fixed centre

is known as

(A) Central force (B) Coroils force

(C) Centrifugal force (D) Centripetal force

130.The motion in which the distence between two

bodies never exceeds a finite limit is

(A) unbounded motion (B) Bound motion

(C) Fixed motion (D) Rotational motion

131.Two solid spheres of radius R and mass M each

are connected by a thin rigid rod of negligible

mass. The distence between the centres are 4R . The

moment of inertia about an axis passing through the

centre of symmetry and perpendicular to the line

joining the spheres is

(A) 11

5𝑀𝑅2 (B)

22

5𝑀𝑅2

(C) 44

5𝑀𝑅2 (D)

88

5𝑀𝑅2

132.A particle of mass moves in a potential 𝑉(𝑥) =1

2𝑚𝜔2𝑥2 +

1

2𝑚𝜇𝑣2 where x is the position co-

ordinates v is the speed ,and 𝜔 𝑎𝑛𝑑 𝜇 are constants

The canonical(conjugate) momentum of the particle

is

(A) 𝑝 = 𝑚(1 + 𝜇)𝑣 (B) p=mv

(C) 𝑝 = 𝑚𝜇𝑣 (D) 𝑝 = 𝑚(1 − 𝜇)𝑣

133.A circular hoop of mass M and radius a rolls

without slliping with constant angular speed 𝜔

along horizontal x-axis in the x-y plane When the

centre of the loop is at a distance 𝑑 = √2 a from the

origin , the magnitude of the total angular

momentum of the hoop about the origin is

(A) 𝑀𝑎2𝜔 (B) √2 𝑀𝑎2𝜔

(C) 2𝑀𝑎2𝜔 (D) 3𝑀𝑎2𝜔

134.if a particle moves outward in a plane along a

curved trajectory described by 𝑟 = 𝑎𝜃, 𝜃 = 𝜔𝑡

where a and 𝜔 are constants then its

(A) Kinetic energy is conserved



(B) Angular momentum is conserved

(C) Total momentum is conserved

(D) Radical momentum is conserved

135.For a particle moving in a central field

(A) The kin etic energy is a constant of motion

(B) The potential energy is velocity depedent

(C) The motion is confined in a plane

(D) The total energy is not conversed

136.A bead of mass m slides along a straight

frictionless rigid wire rotating in a horizontal plane

with angular speed 𝜔. The axis of rotation is

perpendicular to the wire and passes through one

end of the wire. If r is the distance the mass from

the axis of rotation and v is its speed then the

magnitude of the coroils force is

(A) 𝑚𝑣2

𝑟 (B)

2𝑚𝑣2

𝑟

(C) 𝑚𝑣𝜔 (D) 2𝑚𝑣𝜔

137.A particle of charge q, mass m and linear

momentum �̅� enters an electromagnetic field of

vector potential �̅� and scalar potential 𝜙. The

Hamiltonian of the particleis

(A) 𝑝2

2𝑚+ 𝑞𝜙 +

𝐴2

2𝑚 (B)

1

2𝑚(�⃗� −

𝑞

𝐶�̅�)

2

+ 2𝜙

(C) 1

2𝑚(�⃗� −

𝑞

𝐶�̅�)

2

+ �⃗�. 𝐴 (D) 𝑝2

2𝑚𝑞𝜙 − �⃗�. 𝐴

138.A particle is moving in an inverse square force4

field. If the total energy of the particle is positive

the trajectory of the particle is

(A) Circular (B) Elliptical

(C) Parabolic (D) Hyperbolic

139.A particle of mass 2kg is moving such that at

time t,its position in metre is given by 𝑟(𝑡) = 5𝑖̂ −

2𝑡2𝑗 ̂. The angular momentum of the particle t=2s

about the origin in kg𝑚−2𝑠−1 is

(A) −40�̂� (B) −80�̂�

(C) 80�̂� (D) 40�̂�

140. A system four particles in x-y plane,Of

these,two particles each of mass m are located at (-

1,1) and (1,-1) . The remaining two particles each of

mass 2m are located at (1,1) and (-1,-1) . The xy-

component of the moment of inertia tensor of this

system of particles

(A) 10m (B) -10m

(C) 2m (D) -2m

141. For the given transformations (i) Q=p, P=-q

and (ii) Q=P,P=q, where p and q are canonically

conjugate variables , which of the following

statements are true?

(A) Both(i) and (ii) are canonical

(B) only (i) is canonical

(C) Only (ii) is canonical

(D) Neither (i) nor (ii) is canonical

142.The mass m of a moving particle is 2𝑚0

√3 where

𝑚0 is its rest mass. The linear momentum of the

particle

(A) 2𝑚0𝑐 (B) 2𝑚0𝑐

√3 (C) 𝑚0𝑐 (D)

𝑚0𝑐

√3

143.A particle of mass m is constrained to move in

a vertical plane along a trajectory given by 𝑥 =

𝐴 cos 𝜃, 𝑦 = 𝐴 sin 𝜃 where A is a constant . The

Lagrangian of the particle is

(A) 1

2𝑚𝐴2𝜃2 − 𝑚𝑔𝐴 cos 𝜃

(B) 1

2𝑚𝐴2𝜃2 − 𝑚𝑔𝐴 sin 𝜃

(C) 1

2𝑚𝐴2𝜃2



(D) ) 1

2𝑚𝐴2𝜃2 + 𝑚𝑔𝐴 cos 𝜃

144.A rigid frictionless rod rotates anti clockwise in

a vertical plane with angular velocity �⃗⃗⃗� .A bead of

mass move outward along the rod with constant

velocity �⃗⃗�0. The bead will experience a coriolis

force

(A) 2𝑚𝑢0𝜔𝜃 (B) −2𝑚𝑢0𝜔𝜃

(C) 4𝑚𝑢0𝜔𝜃 (D) −𝑚𝑢0𝜔𝜃

145.The Hamiltonian corresponding to the

Lagrangian 𝐿 =1

2(𝑞1

2 + 𝑞1𝑞2 + 𝑞22) − 𝑉(𝑞) is

(A) (𝑝12 − 𝑝1𝑝2 + 𝑝2

2) + 𝑉(𝑞)

(B) 2

3(𝑝1

2 − 𝑝1𝑝2 + 𝑝22) + 𝑉(𝑞)

(C) 2

3(𝑝1

2 − 𝑝1𝑝2 + 𝑝22) + 𝑉(𝑞)

(D) None of these

146.The value of the Poisson bracket [�⃗�, 𝑟, �⃗⃗�, �⃗�]

where �⃗� 𝑎𝑛𝑑 �⃗⃗� are constant vectors is

(A) �⃗��⃗⃗� (B) �⃗� − �⃗⃗� (C) �⃗� + �⃗⃗� (D) �⃗�. �⃗⃗�

147.A mass m is connected on either side with a

spring each of spring constants 𝑘1 𝑎𝑛𝑑 𝑘2 . The

Free ends of springs are tied to rigid supports. The

displacement of the mass is x from equilibrium

position which one of the following is TRUE?

(A) The force acting on the mass is −(𝑘1𝑘2)1/2𝑥

(B) The angular momentum of the mass is zero

about the equilibrium point its Lagrangian is 1

2𝑚�̇�2 −

1

2(𝑘1 + 𝑘2)𝑥2

(C) The total energy of the system is 1

2𝑚�̇�2

(D) The angular momentum of the mass is 𝑚𝑥�̇� and

the Lagrangian of the system is 𝑚

2�̇�2 +

1

2(𝑘1 + 𝑘2)𝑥2

148.The distance between the two bodies in infinite

initially and finally in

(A) Unbounded motion (B) Rotational motion

(C) Bounded motion (D) Traslational

motion

149.If the total energy of a particle in a conservative

force field is zero then the velocity obtained to such

case is

(A) zero (B) Escape velocity

(C) Recoil velocity (D) None of these

150.The force exerted by one particle on the other

varies inversely as the squre of the distance between

them by

(A) 𝐹(𝑟) = −𝑘

𝑟2 (B) 𝑓(𝑟) = −𝑘2

𝑟2

(C) 𝐹(𝑟) = −𝑘�̂�

𝑟2 (D) 𝐹(𝑟) =𝑘

𝑟2

151.What is the nature of the orbit if energy is less

than zero?

(A) Ellipse (B) Circle

(C) Hyperbola (D) Parabola

152.What is the nature of the orbit if the value of

eccentricity is equal to one?

(A) Hyperbola (B) Ellipse

(C) Parabola (D) Circle

153.The value of eccentricity for an elliptic orbit is

(A) 𝜖 > 1 (B) 𝜖 = 1

(C) 𝜖 = 0 (D) 0 < 𝜖 < 1



154.A particle moving in a central force located at

r=0 describes the spiral 𝑟 = 𝑒−𝜃 the magnitude of

force is inversely proportional to

(A) 𝑟3 (B) r (C) 𝑟2 (D) 𝑟4

155.In Rutherford scattering , an 𝛼 −particle of

energy E is scatterd through an angle 𝜃, the

differential scattering cross-section is proportional

to

(A) 𝐸 cot𝜃

2 (B) 𝐸2 sin4 𝜃

2

(C) 𝐸−2 (sin𝜃

2)

−4

(D) 𝐸2 (sin𝜃

2)

−4

156.According to special theory of relativity a

particle cannmot travel with the speed of light

because its

(A) Velocity will soon be infinite

(B) Mass will be infinite

(C) Mass will reduce to zero

(D) None of these

157. if the Galien transformations were correct then

aberration angle will be given by

(A) tan 𝜃 =𝑣

𝑐 (B) sin 𝜃 =

𝑣

𝑐

(C) cos 𝜃 =𝑣

𝑐 (D) None of these

158.Photographs of rapidly moving distant objects

will

(A) Not show Lorentz contraction

(B) show Lorentz contraction

(C) Not show any change

(D) none of these

159.A small sphere of radius R in its proper frame

is moving with half the velocity of light, when

viewed by the observer in a laboratory frame it

looks like

(A) A sphere (B) An ellipsoid

(C) A paraboloid (D) Hyperboloid

160. If the speed of light were 2

3 of its present value

the energy released in a given atomic explosion will

be decreased to a factor

(A) 2

3 (B)

4

9 (C)

5

9 (D) √

5

9

161.A body of mass 𝑚0 is placed in a rocket . The

rocket is moving with velocity v=0.6c. Then the

mass of the rocket as observed by a person sitting in

the rocket is

(A) 𝑚0 (B) 5

4𝑚0 (C)

4

5𝑚0 (D) 2𝑚0

162.A body with a charge q starts from rest and

acquire a velocity v=0.5c.Then the new charge on it

is

(A) q (B) 𝑞√1 − (0.5)2

(C) 𝑞√1 − (0.5) (D) 𝑞

√1−(0.5)2

163.A slowly moving electron collides with a

position at rest and amnihilates it producing two

photons . If the rest of the electron and position be

𝑚0. Then the frequency of each photon is

(A) 2𝑚0𝑐2 (B) 𝑚0𝑐2

(C) 𝑚0𝑐2

ℎ (D)

2𝑚0𝑐2

ℎ

164.A gamma ray of energy 2.2MeV produce an

electron positron pair. Then the energy inparted to

each of charge particles is nearly

(A) 1.1MeV (B) 0.51MeV

(C) 0.59MeV (D) 1.18MeV



165.A particle with a mean proper life time of

1𝜇𝑠𝑒𝑐. moves through the laboratory at a velocity

of 2.7 × 1010 cm/sec. What is the life time , as

measured by an observer in the laboratory ?

(A) More than one micro second

(B) Same as above

(C) Less than one micro second

(D) Data appears to insufficient

166. When an observer moves so fast that the

lengths that he measures are reduced to half , his

time interval measurements

(A) Be variant (B) Reduced to half

(C) Becomes twice (D) Reduced o 1

4

𝑡ℎ

167.The rest mass of electron is 𝑚0 when it moves

with a velocity v=0.6c then its rest mass is

(A) 𝑚0 (B) 5

4𝑚0

(C) 4

5𝑚0 (D) 2𝑚0

168.The mass of an electron is double its rest mass

then the velocity of the electron is

(A) 𝑐

2 (B) 2c (C)

√3

2𝑐 (D) √

3

2 c

169.Rest mass of energy of an electron is 0.51MeV,

A moving electron has a kinetic energy of 9.69MeV

. The ratio of mass of the moving electron to its rest

mass is

(A) 19:1 (B) 20:1

(C) 1:19 (D) 1:20

170. Rest mass of electron is 9.1 × 10−31kg. The

mass equivalent energy of its electron is

(A) 0.511 ergs (B) 0.511J

(C) 0.511eV (D) 0.511MeV

171.A Galilean transformation applies between two

frames of teference P and Q is

(A) Q is rotating with uniform angular velocity

relative to P

(B) Q is rotating with uniform acceleration relative

to P

(C) Q is rotating with uniform velocity relative to P

(D) Q goes round P at constant distance with a

constant speed

172.Which of the following equation is relatively

invariant (𝛼, 𝛽, 𝛾 and 𝛿 are constant of suitable

dimension ) ?

(A) 𝜕𝜙(𝑥,𝑡)

𝜕𝑡= 𝛼

𝜕2𝜙

𝜕𝑥2 (𝑥, 𝑡)

(B) 𝜕2𝜙

𝜕𝑡2(𝑥, 𝑡) = 𝛽2 𝜕2𝜙

𝜕𝑥2 (𝑥, 𝑡)

(C) ) 𝜕2𝜙

𝜕𝑡2(𝑥, 𝑡) = 𝛾

𝜕𝜙

𝜕𝑥2 (𝑥, 𝑡)

(D) 𝜕𝜙

𝜕𝑡= 𝛿

𝜕3𝜙

𝜕𝑥3(𝑥, 𝑡)

173. Although mass-energy equivalence of special

relativity allows conversion of a photon to an

electron-positron pair such a process cannot occur

in free space because

(A) The mass is not conserved

(B) The energy is not conserved

(C) The momentum is not conserved

(D) The charge is not conserved


2. CLASSICAL MECHANICS - CEED Physics Clinic Calicut

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Transcript of 2. CLASSICAL MECHANICS - CEED Physics Clinic Calicut