Joint Entrance Examination

Graduate Aptitude Test in Engineering

Geotechnical Engineering

Transportation Engineering

Irrigation

Engineering Mathematics

Construction Material and Management

Fluid Mechanics and Hydraulic Machines

Hydrology

Environmental Engineering

Engineering Mechanics

Structural Analysis

Reinforced Cement Concrete

Steel Structures

Geomatics Engineering Or Surveying

General Aptitude

1

A cubical block of side 30 cm is moving with velocity 2 ms^{−1} on a smooth horizontal surface. The surface has a bump at a point O as shown in figure. The angular
velocity (in rad/s) of the block immediately after it hits the bump, is :

A

5.0

B

6.7

C

9.4

D

13.3

Before hitting point 0,

angular moment = mv $$ \times $$ $${a \over 2}$$

After hitting point 0,

Angular momentum = $${\rm I}\omega $$

$$ \therefore $$ $${\rm I}\omega $$ = $${{mva} \over 2}$$

$$ \Rightarrow $$ $$\omega $$ = $${{mva} \over {2{\rm I}}}$$

$${\rm I}$$ = moment of inertia about edge,

= $${{m{a^2}} \over 6} + m{\left( {{a \over {\sqrt 2 }}} \right)^2}$$

= $${{m{a^2}} \over 6} + {{m{a^2}} \over 2}$$

= $${{2m{a^2}} \over 3}$$

$$ \therefore $$ $$\omega $$ = $${{mva} \over {2 \times {{2m{a^2}} \over 3}}}$$ = $${{3v} \over {4a}}$$ = $${{3 \times 2} \over {4 \times 0.3}}$$ = 5 rad/s

angular moment = mv $$ \times $$ $${a \over 2}$$

After hitting point 0,

Angular momentum = $${\rm I}\omega $$

$$ \therefore $$ $${\rm I}\omega $$ = $${{mva} \over 2}$$

$$ \Rightarrow $$ $$\omega $$ = $${{mva} \over {2{\rm I}}}$$

$${\rm I}$$ = moment of inertia about edge,

= $${{m{a^2}} \over 6} + m{\left( {{a \over {\sqrt 2 }}} \right)^2}$$

= $${{m{a^2}} \over 6} + {{m{a^2}} \over 2}$$

= $${{2m{a^2}} \over 3}$$

$$ \therefore $$ $$\omega $$ = $${{mva} \over {2 \times {{2m{a^2}} \over 3}}}$$ = $${{3v} \over {4a}}$$ = $${{3 \times 2} \over {4 \times 0.3}}$$ = 5 rad/s

2

Concrete mixture is made by mixing cement, stone and sand in a rotating
cylindrical drum. If the drum rotates too fast, the ingredients remain stuck to the wall of the drum and proper mixing of ingredients does not take place. The maximum rotational speed of the drum in revolutions per minute(rpm) to ensure proper mixing is close to :

(Take the radius of the drum to be 1.25 m and its axle to be horizontal) :

(Take the radius of the drum to be 1.25 m and its axle to be horizontal) :

A

0.4

B

1.3

C

8.0

D

27.0

3

In the figure shown ABC is a uniform wire. If centre of mass of wire lies vertically below point A, then $${{BC} \over {AB}}$$ is close to :

A

1.85

B

1.37

C

1.5

D

3

Here AB = x

and BC = y

and $$\lambda $$ = linear mass density.

As centre of mass is below point A, so horizontal distance of the centre of mass from B is = xcos60

$$ \therefore $$ X

$$ \Rightarrow $$ $${x \over 2}$$ = $${{\left( {\lambda x} \right)\left( {{x \over 2}} \right)\cos {{60}^o} + \left( {\lambda y} \right)\left( {{y \over 2}} \right)} \over {\lambda \left( {x + y} \right)}}$$

$$ \Rightarrow $$ $${x \over 2}$$ = $${{{{{x^2}} \over 4} + {{{y^2}} \over 2}} \over {x + y}}$$

$$ \Rightarrow $$ x

$$ \Rightarrow $$ x

$$ \therefore $$ x = $${{ - 2y \pm \sqrt {{{\left( {2y} \right)}^2} - 4.1\left( { - 2{y^2}} \right)} } \over {2.1}}$$

= $${{ - 2y \pm \sqrt {12{y^2}} } \over 2}$$

= $$-$$ y $$ \pm $$ $$\sqrt 3 $$y

$${x \over y} \ne - \sqrt 3 - 1$$ as $${x \over y}$$ = positive.

$$ \therefore $$ $${x \over y}$$ = $$\sqrt 3 - 1$$

$$ \Rightarrow $$ $${y \over x}$$ = $${1 \over {\sqrt 3 - 1}} \times {{\sqrt 3 + 1} \over {\sqrt 3 + 1}}$$

= $${{\sqrt 3 + 1} \over 2}$$

= $${{2.732} \over 2}$$

= 1.366 $$ \simeq $$ 1.37

4

The moment of inertia of a uniform cylinder of length $$l$$ and radius R about its perpendicular bisector is $$I$$.
What is the ratio $${l \over R}$$ such that the moment of inertia is minimum?

A

$${3 \over {\sqrt 2 }}$$

B

$$\sqrt {{3 \over 2}} $$

C

$${{\sqrt 3 } \over 2}$$

D

1

The volume of the cylinder V = $$\pi {R^2}l$$

$$\therefore$$ $${R^2} = {V \over {\pi l}}$$

We know, moment of inertia of a uniform cylinder of length $$l$$ and radius R about its perpendicular bisector is,

$$I = {{M{l^2}} \over {12}} + {{M{R^2}} \over 4}$$

[ Putting $${R^2} = {V \over {\pi l}}$$ in this equation]

$$ \Rightarrow $$ $$I = {{M{l^2}} \over {12}} + {{MV} \over {4\pi l}}$$

Here $$I$$ is a function of $$l$$ as M and V are constant.

$$I$$ will be maximum or minimum when $${{{dI} \over {dl}}}$$ = 0.

$$ \Rightarrow {{Ml} \over 6} - {{MV} \over {4\pi {l^2}}} = 0$$

$$ \Rightarrow {{Ml} \over 6} = {{MV} \over {4\pi {l^2}}}$$

$$ \Rightarrow {l \over 6} = {{\pi {R^2}l} \over {4\pi {l^2}}}$$ [ as $${V = \pi {R^2}l}$$ ]

$$ \Rightarrow {{{R^2}} \over {{l^2}}} = {4 \over 6}$$

$$ \Rightarrow {l \over R} = \sqrt {{3 \over 2}} $$

$$\therefore$$ $${R^2} = {V \over {\pi l}}$$

We know, moment of inertia of a uniform cylinder of length $$l$$ and radius R about its perpendicular bisector is,

$$I = {{M{l^2}} \over {12}} + {{M{R^2}} \over 4}$$

[ Putting $${R^2} = {V \over {\pi l}}$$ in this equation]

$$ \Rightarrow $$ $$I = {{M{l^2}} \over {12}} + {{MV} \over {4\pi l}}$$

Here $$I$$ is a function of $$l$$ as M and V are constant.

$$I$$ will be maximum or minimum when $${{{dI} \over {dl}}}$$ = 0.

$$ \Rightarrow {{Ml} \over 6} - {{MV} \over {4\pi {l^2}}} = 0$$

$$ \Rightarrow {{Ml} \over 6} = {{MV} \over {4\pi {l^2}}}$$

$$ \Rightarrow {l \over 6} = {{\pi {R^2}l} \over {4\pi {l^2}}}$$ [ as $${V = \pi {R^2}l}$$ ]

$$ \Rightarrow {{{R^2}} \over {{l^2}}} = {4 \over 6}$$

$$ \Rightarrow {l \over R} = \sqrt {{3 \over 2}} $$

Number in Brackets after Paper Name Indicates No of Questions

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Units & Measurements *keyboard_arrow_right*

Motion *keyboard_arrow_right*

Laws of Motion *keyboard_arrow_right*

Work Power & Energy *keyboard_arrow_right*

Simple Harmonic Motion *keyboard_arrow_right*

Impulse & Momentum *keyboard_arrow_right*

Rotational Motion *keyboard_arrow_right*

Gravitation *keyboard_arrow_right*

Properties of Matter *keyboard_arrow_right*

Heat and Thermodynamics *keyboard_arrow_right*

Waves *keyboard_arrow_right*

Vector Algebra *keyboard_arrow_right*

Dual Nature of Radiation *keyboard_arrow_right*

Electronic Devices *keyboard_arrow_right*

Practical Physics *keyboard_arrow_right*

Atoms and Nuclei *keyboard_arrow_right*

Communication Systems *keyboard_arrow_right*

Electrostatics *keyboard_arrow_right*

Current Electricity *keyboard_arrow_right*

Magnetics *keyboard_arrow_right*

Alternating Current and Electromagnetic Induction *keyboard_arrow_right*

Ray & Wave Optics *keyboard_arrow_right*