Joint Entrance Examination

Graduate Aptitude Test in Engineering

Geomatics Engineering Or Surveying

Engineering Mechanics

Hydrology

Transportation Engineering

Strength of Materials Or Solid Mechanics

Reinforced Cement Concrete

Steel Structures

Irrigation

Environmental Engineering

Engineering Mathematics

Structural Analysis

Geotechnical Engineering

Fluid Mechanics and Hydraulic Machines

General Aptitude

1

A standing wave is formed by the superposition of two waves travelling in
opposite directions. The transverse displacement is given by

y(x, t) = 0.5 sin $$\left( {{{5\pi } \over 4}x} \right)\,$$ cos(200 $$\pi $$t).

What is the speed of the travelling wave moving in the positive x direction ?

(x and t are in meter and second, respectively.)

y(x, t) = 0.5 sin $$\left( {{{5\pi } \over 4}x} \right)\,$$ cos(200 $$\pi $$t).

What is the speed of the travelling wave moving in the positive x direction ?

(x and t are in meter and second, respectively.)

A

160 m/s

B

90 m/s

C

180 m/s

D

120 m/s

Standard equation of standing wave,

y(x, t) = 2a sin kx cos $$\omega $$t

Given,

y(x, t) = 0.5 sin $$\left( {{{5\pi } \over 4}x} \right)$$ cos (200$$\pi $$t).

So, k = $${{5\pi } \over 4}$$ and $$\omega $$ = 200$$\pi $$

$$\therefore\,\,\,$$ Speed of travelling wave

= $${\omega \over k}$$ = $${{200\pi } \over {{{5\pi } \over 4}}}$$ = 160 m/s.

y(x, t) = 2a sin kx cos $$\omega $$t

Given,

y(x, t) = 0.5 sin $$\left( {{{5\pi } \over 4}x} \right)$$ cos (200$$\pi $$t).

So, k = $${{5\pi } \over 4}$$ and $$\omega $$ = 200$$\pi $$

$$\therefore\,\,\,$$ Speed of travelling wave

= $${\omega \over k}$$ = $${{200\pi } \over {{{5\pi } \over 4}}}$$ = 160 m/s.

2

A silver atom in a solid oscillates in simple harmonic motion in some direction with a frequency of 10^{12}/sec. What is the force constant of the bonds connecting one atom with the other? (Mole wt. of silver = 108 and Avogadro number = 6.02 × 10^{23} gm mole^{–1})

A

5.5 N/m

B

6.4 N/m

C

7.1 N/m

D

2.2 N/m

6.02 $$ \times $$ 10^{23} atoms of silver = 108 gm

1 atoms of silver = $${{108 \times {{10}^{ - 3}}} \over {6.02 \times {{10}^{23}}}}$$ kg

For a harmonic oscillator

f = $${1 \over {2\pi }}$$ $$\sqrt {{k \over m}} $$

Where k = force constant

$$ \Rightarrow $$$$\,\,\,$$ f^{2} = $${1 \over {4{\pi ^2}}}$$ $$\left( {{k \over m}} \right)$$

$$ \Rightarrow $$$$\,\,\,$$ k = mf^{2} $$ \times $$ 4$$\pi $$^{2}

Given,

f = 10^{12}

m = $${{108 \times {{10}^{ - 3}}} \over {6.02 \times {{10}^{23}}}}$$

$$\therefore\,\,\,$$ k = $${{108 \times {{10}^{ - 3}}} \over {6.02 \times {{10}^{23}}}}$$ $$ \times $$ 10^{12} $$ \times $$ 4$$\pi $$^{2}

= 7.1 N/m

1 atoms of silver = $${{108 \times {{10}^{ - 3}}} \over {6.02 \times {{10}^{23}}}}$$ kg

For a harmonic oscillator

f = $${1 \over {2\pi }}$$ $$\sqrt {{k \over m}} $$

Where k = force constant

$$ \Rightarrow $$$$\,\,\,$$ f

$$ \Rightarrow $$$$\,\,\,$$ k = mf

Given,

f = 10

m = $${{108 \times {{10}^{ - 3}}} \over {6.02 \times {{10}^{23}}}}$$

$$\therefore\,\,\,$$ k = $${{108 \times {{10}^{ - 3}}} \over {6.02 \times {{10}^{23}}}}$$ $$ \times $$ 10

= 7.1 N/m

3

A granite rod of 60 cm length is clamped at its middle point and is set into longitudinal vibrations. The
density of granite is 2.7 $$\times$$ 10^{3} kg/m^{3} and its Young’s modulus is 9.27 $$\times$$ 10^{10} Pa. What will be the fundamental frequency of the longitudinal vibrations ?

A

7.5 kHz

B

5 kHz

C

2.5 kHz

D

10 kHz

As rod length = 60 cm

$$\therefore\,\,\,$$ $${\lambda \over 2}$$ = 60

$$ \Rightarrow $$$$\,\,\,$$ $$\lambda $$ = 120 cm = 1.2 m

In solid, velocity of wave,

V = $$\sqrt {{Y \over \rho }} $$

= $$\sqrt {{{9.27 \times {{10}^{10}}} \over {2.7 \times {{10}^3}}}} $$

= 5.85 $$ \times $$ 10

As we know,

v = f $$\lambda $$

$$\therefore\,\,\,$$ f = $${v \over \lambda }$$

= $${{5.85 \times {{10}^3}} \over {1.2}}$$

= 4.88 $$ \times $$ 10

$$ \simeq $$ 5 kHz

4

A tuning fork vibrates with frequency $$256$$ $$Hz$$ and gives one beat per second with the third normal mode of vibration of an open pipe. What is the length of the pipe ? (Speed of sound in air is $$340\,m{s^{ - 1}}$$)

A

$$220$$ $$cm$$

B

$$190$$ $$cm$$

C

$$180$$ $$cm$$

D

$$200$$ $$cm$$

The tuning fork vibrates with frequency 256 Hz and give one beat per second So, the organ pipe will have frequency (256 $$ \pm $$ 1) Hr.

For open organ pipe,

Frequency n = $${{N\upsilon } \over {2\ell }}$$

Here n = 255 Hz

N = 3

$$\upsilon $$ = 340 m/s

$$\therefore\,\,\,\,$$ 255 = $${{3 \times 340} \over {2 \times \ell }}$$

$$ \Rightarrow $$$$\,\,\,\,$$ $$\ell $$ = $${{3 \times 340} \over {2 \times 255}} = 2\,m$$

$$\therefore\,\,\,\,$$ $$\ell $$ = 2m or 200 cm

For open organ pipe,

Frequency n = $${{N\upsilon } \over {2\ell }}$$

Here n = 255 Hz

N = 3

$$\upsilon $$ = 340 m/s

$$\therefore\,\,\,\,$$ 255 = $${{3 \times 340} \over {2 \times \ell }}$$

$$ \Rightarrow $$$$\,\,\,\,$$ $$\ell $$ = $${{3 \times 340} \over {2 \times 255}} = 2\,m$$

$$\therefore\,\,\,\,$$ $$\ell $$ = 2m or 200 cm

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*

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*

Atoms and Nuclei *keyboard_arrow_right*

Electronic Devices *keyboard_arrow_right*

Communication Systems *keyboard_arrow_right*

Practical Physics *keyboard_arrow_right*

Dual Nature of Radiation *keyboard_arrow_right*