Joint Entrance Examination

Graduate Aptitude Test in Engineering

Strength of Materials Or Solid Mechanics

Structural Analysis

Construction Material and Management

Reinforced Cement Concrete

Steel Structures

Geotechnical Engineering

Fluid Mechanics and Hydraulic Machines

Hydrology

Irrigation

Geomatics Engineering Or Surveying

Environmental Engineering

Transportation Engineering

Engineering Mathematics

General Aptitude

1

A closed organ pipe has a fundamental frequency of 1.5 kHz. The number of overtones that can be distinctly heard by a person with this organ pipe will be (Assume that the highest frequency a person can hear is 20,000 Hz)

A

4

B

7

C

6

D

5

For closed organ pipe, resonate frequency is odd multiple of fundamental frequency.

$$ \therefore $$ (2n + 1) f_{0} $$ \le $$ 20,000

(f_{0} is fundamental frequency = 1.5 KHz)

$$ \therefore $$ n = 6

$$ \therefore $$ Total number of overtone that can be heared is 7. (0 to 6)

$$ \therefore $$ (2n + 1) f

(f

$$ \therefore $$ n = 6

$$ \therefore $$ Total number of overtone that can be heared is 7. (0 to 6)

2

A cylindrical plastic bottle of negligible mass is filled with 310 ml of water and left floating in a pond with still water. If pressed downward slightly and released, it starts performing simple harmonic motion at angular frequency $$\omega $$. If the radius of the bottle is 2.5 cm then $$\omega $$ is close to – (density of water = 10^{3} kg/m^{3}).

A

2.50 rad s^{$$-$$1}

B

3.75 rad s^{$$-$$1}

C

5.00 rad s^{$$-$$1}

D

7.90 rad s^{$$-$$1}

Restoring force due to pressing the bottle with small
amount x,

F = $$ - \left( {\rho Ax} \right)g$$

$$ \Rightarrow $$ ma = $$ - \left( {\rho Ax} \right)g$$

$$ \Rightarrow $$ a = $$ - \left( {{{\rho Ag} \over m}} \right)x$$

$$ \therefore $$ $${{\omega ^2} = {{\rho Ag} \over m}}$$ = $${{{\rho \left( {\pi {r^2}} \right)g} \over m}}$$

$$ \Rightarrow $$ $$\omega $$ = $$\sqrt {{{{{10}^3} \times \pi \times {{\left( {2.5 \times {{10}^{ - 2}}} \right)}^2} \times 10} \over {310 \times {{10}^{ - 3}}}}} $$ = 7.90 rad/s

F = $$ - \left( {\rho Ax} \right)g$$

$$ \Rightarrow $$ ma = $$ - \left( {\rho Ax} \right)g$$

$$ \Rightarrow $$ a = $$ - \left( {{{\rho Ag} \over m}} \right)x$$

$$ \therefore $$ $${{\omega ^2} = {{\rho Ag} \over m}}$$ = $${{{\rho \left( {\pi {r^2}} \right)g} \over m}}$$

$$ \Rightarrow $$ $$\omega $$ = $$\sqrt {{{{{10}^3} \times \pi \times {{\left( {2.5 \times {{10}^{ - 2}}} \right)}^2} \times 10} \over {310 \times {{10}^{ - 3}}}}} $$ = 7.90 rad/s

3

A particle executes simple harmonic motion with an amplitude of 5 cm. When the particle is at 4 cm from the mean position, the magnitude of its velocity in SI units is equal to that of its acceleration. Then, its periodic time in seconds is -

A

$${{4\pi } \over 3}$$

B

$${3 \over 8}\pi $$

C

$${7 \over 3}\pi $$

D

$${{8\pi } \over 3}$$

$$v = \omega \sqrt {{A^2} - {x^2}} \,\,$$ . . .(1)

$$a = - {\omega ^2}x$$ . . .(2)

$$\left| v \right| = \left| a \right|$$ . . .(3)

$$\omega \sqrt {{A^2} - {x^2}} = {\omega ^2}x$$

$${A^2} - {x^2} = {\omega ^2}{x^2}$$

$${5^2} - {4^2} = {\omega ^2}\left( {{4^2}} \right)$$

$$ \Rightarrow \,\,\,3 = \omega \times 4$$

$$T = 2\pi /\omega $$

$$a = - {\omega ^2}x$$ . . .(2)

$$\left| v \right| = \left| a \right|$$ . . .(3)

$$\omega \sqrt {{A^2} - {x^2}} = {\omega ^2}x$$

$${A^2} - {x^2} = {\omega ^2}{x^2}$$

$${5^2} - {4^2} = {\omega ^2}\left( {{4^2}} \right)$$

$$ \Rightarrow \,\,\,3 = \omega \times 4$$

$$T = 2\pi /\omega $$

4

A particle undergoing simple harmonic motion has time dependent displacement given by x(t) = Asin$${{\pi t} \over {90}}$$. The ratio of kinetic to potential energy of this particle at t = 210 s will be:

A

$${1 \over 9}$$

B

3

C

2

D

1

K = $${1 \over 2}$$m$${\omega ^2}$$A^{2}cos^{2}$$\omega $$t

U = $${1 \over 2}m{\omega ^2}$$ A^{2} sin^{2} $$\omega $$t

$${k \over U}$$ = cot^{2} $$\omega $$t = cot^{2} $${\pi \over {90}}$$(210) = $${1 \over 3}$$

Hence ratio is 3 (most appropriate)

U = $${1 \over 2}m{\omega ^2}$$ A

$${k \over U}$$ = cot

Hence ratio is 3 (most appropriate)

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