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 simple pendulum made of a bob of mass *m* and a metallic wire of negligible mass has time period 2 s at T=0^{o}C. If the temperature of the wire is increased and the corresponding change in its time period is
plotted against its temperature, the resulting graph is a line of slope S. If the
coefficient of linear expansion of metal is $$\alpha $$ then the value of S is :

A

$$\alpha $$

B

$${\alpha \over 2}$$

C

2$$\alpha $$

D

$${1 \over \alpha }$$

Change of length of wire with temperature,

$$\Delta $$$$\ell $$ = $$\alpha \ell \Delta \theta $$

Time period of pendulum at temperature $$\theta $$,

T_{$$\theta $$} = 2$$\pi $$$$\sqrt {{{\ell + \Delta \ell } \over g}} $$

= 2$$\pi $$$$\sqrt {{{\ell \left( {1 + \alpha \Delta \theta } \right)} \over g}} $$

= $$2\pi \sqrt {{\ell \over g}} {\left( {1 + \alpha \Delta \theta } \right)^{{1 \over 2}}}$$

= $$2\pi \sqrt {{\ell \over g}} {\left( {1 + {{\Delta \ell } \over \ell }} \right)^{{1 \over 2}}}$$

$$ \simeq $$ T_{0} $$\left( {1 + {{\Delta \ell } \over {2\ell }}} \right)$$

Here T_{0} = time period at temperature 0^{o}C.

$$ \therefore $$ Change in time period,

$$\Delta $$T = T_{$$\theta $$} $$-$$ T_{0}

= $${{{T_0}\Delta \ell } \over {2\ell }}$$

= $${{{T_0}\left( {\alpha \ell \Delta \theta } \right)} \over {2\ell }}$$

$$ \therefore $$ $${{\Delta T} \over {\Delta \theta }}$$ = $${{{T_0}\alpha } \over 2}$$

Given that T_{0} = 2,

$$ \therefore $$ $${{\Delta T} \over {\Delta \theta }}$$ = $${{2\alpha } \over 2}$$ = $$\alpha $$

$${{\Delta T} \over {\Delta \theta }}$$ is the shape of $$\Delta $$T and $${\Delta \theta }$$

curve = S (given)

$$ \therefore $$ S = $$\alpha $$

$$\Delta $$$$\ell $$ = $$\alpha \ell \Delta \theta $$

Time period of pendulum at temperature $$\theta $$,

T

= 2$$\pi $$$$\sqrt {{{\ell \left( {1 + \alpha \Delta \theta } \right)} \over g}} $$

= $$2\pi \sqrt {{\ell \over g}} {\left( {1 + \alpha \Delta \theta } \right)^{{1 \over 2}}}$$

= $$2\pi \sqrt {{\ell \over g}} {\left( {1 + {{\Delta \ell } \over \ell }} \right)^{{1 \over 2}}}$$

$$ \simeq $$ T

Here T

$$ \therefore $$ Change in time period,

$$\Delta $$T = T

= $${{{T_0}\Delta \ell } \over {2\ell }}$$

= $${{{T_0}\left( {\alpha \ell \Delta \theta } \right)} \over {2\ell }}$$

$$ \therefore $$ $${{\Delta T} \over {\Delta \theta }}$$ = $${{{T_0}\alpha } \over 2}$$

Given that T

$$ \therefore $$ $${{\Delta T} \over {\Delta \theta }}$$ = $${{2\alpha } \over 2}$$ = $$\alpha $$

$${{\Delta T} \over {\Delta \theta }}$$ is the shape of $$\Delta $$T and $${\Delta \theta }$$

curve = S (given)

$$ \therefore $$ S = $$\alpha $$

2

Which of the following option correctly describes the variation of the speed *v* and acceleration *‘a’* of a point mass falling vertically in a viscous medium that applies a force F = − *kv,* where ‘k’ is a constant, on the body ? (Graphs are schematic and not drawn to scale)

A

B

C

D

Equation of motion for the mass,

ma = mg $$-$$ kv

$$ \Rightarrow $$ $${{dv} \over {dt}} = {{mg - kv} \over m}$$

$$ \Rightarrow $$ $$\int\limits_0^v {{{dv} \over {mg - kv}}} = {1 \over m}\int\limits_0^t {dt} $$

$$ \Rightarrow $$ $$ - {1 \over k}\left[ {\ln \left( {mg - kv} \right)} \right]_0^v = {t \over m}$$

$$ \Rightarrow $$ $$\ln \left( {{{mg - kv} \over {mg}}} \right) = - {{kt} \over m}$$

$$ \Rightarrow $$ $$1 - {{kv} \over {mg}} = {e^{ - {{kt} \over m}}}$$

$$ \Rightarrow $$ $${{kv} \over {mg}} = 1 - {e^{ - {{kt} \over m}}}$$

$$ \Rightarrow $$ $$v = {{mg} \over k}\left( {1 - {e^{ - {{kt} \over m}}}} \right)$$

ma $$=$$ mg $$-$$ k $$ \times $$ $${{mg} \over k}$$ (1 $$-$$ e$$^{ - {{kt} \over m}}$$)

$$=$$ mg $$-$$ mg + mge$$^{ - {{kt} \over m}}$$

a $$=$$ g e$$^{ - {{kt} \over m}}$$

ma = mg $$-$$ kv

$$ \Rightarrow $$ $${{dv} \over {dt}} = {{mg - kv} \over m}$$

$$ \Rightarrow $$ $$\int\limits_0^v {{{dv} \over {mg - kv}}} = {1 \over m}\int\limits_0^t {dt} $$

$$ \Rightarrow $$ $$ - {1 \over k}\left[ {\ln \left( {mg - kv} \right)} \right]_0^v = {t \over m}$$

$$ \Rightarrow $$ $$\ln \left( {{{mg - kv} \over {mg}}} \right) = - {{kt} \over m}$$

$$ \Rightarrow $$ $$1 - {{kv} \over {mg}} = {e^{ - {{kt} \over m}}}$$

$$ \Rightarrow $$ $${{kv} \over {mg}} = 1 - {e^{ - {{kt} \over m}}}$$

$$ \Rightarrow $$ $$v = {{mg} \over k}\left( {1 - {e^{ - {{kt} \over m}}}} \right)$$

ma $$=$$ mg $$-$$ k $$ \times $$ $${{mg} \over k}$$ (1 $$-$$ e$$^{ - {{kt} \over m}}$$)

$$=$$ mg $$-$$ mg + mge$$^{ - {{kt} \over m}}$$

a $$=$$ g e$$^{ - {{kt} \over m}}$$

3

Consider a water jar of radius R that has water filled up to height H and is kept on astand of height h (see figure). Through a hole of radius r (r << R) at its bottom, the water leaks out and the stream of water coming down towards the ground has a shape like a funnel as shown in the figure. If the radius of the cross-section of water stream when it hits the ground is x. Then :

A

$$x = r\left( {{H \over {H + h}}} \right)$$

B

$$x = r{\left( {{H \over {H + h}}} \right)^{{1 \over 2}}}$$

C

$$x = r{\left( {{H \over {H + h}}} \right)^{{1 \over 4}}}$$

D

$$x = r{\left( {{H \over {H + h}}} \right)^{{2}}}$$

v_{1} = velocity of water when it leak from hole

v_{2} = velocity of water when it reach the ground.

From Bernoulli's principle,

$${1 \over 2}\rho {v_1}^2 + \rho gh$$ = $${1 \over 2}\rho {v_2}^2$$

$$ \Rightarrow $$ $${v_1}^2$$ + 2gh = $${v_2}^2$$

From Torricelli's theorem,

v_{1} = $$\sqrt {2gH} $$

$$ \therefore $$ $${v_2}^2$$ = 2gh + 22gH

From continuity equation,

$${A_1}{v_1}$$ = $${A_2}{v_2}$$

$$ \Rightarrow $$ $$\pi {r^2} \times \sqrt {2gH} $$ = $$\pi {x^2}\sqrt {2g\left( {h + H} \right)} $$

$$ \Rightarrow $$ x^{2} = r^{2}$$\sqrt {{H \over {H + g}}} $$

$$ \Rightarrow $$ x = r $${\left( {{H \over {H + g}}} \right)^{{1 \over 4}}}$$

v

From Bernoulli's principle,

$${1 \over 2}\rho {v_1}^2 + \rho gh$$ = $${1 \over 2}\rho {v_2}^2$$

$$ \Rightarrow $$ $${v_1}^2$$ + 2gh = $${v_2}^2$$

From Torricelli's theorem,

v

$$ \therefore $$ $${v_2}^2$$ = 2gh + 22gH

From continuity equation,

$${A_1}{v_1}$$ = $${A_2}{v_2}$$

$$ \Rightarrow $$ $$\pi {r^2} \times \sqrt {2gH} $$ = $$\pi {x^2}\sqrt {2g\left( {h + H} \right)} $$

$$ \Rightarrow $$ x

$$ \Rightarrow $$ x = r $${\left( {{H \over {H + g}}} \right)^{{1 \over 4}}}$$

4

A thin 1 m long rod has a radius of 5 mm. A force of 50 $$\pi $$kN is applied at one end to determine its Young’s modulus. Assume that the force is exactly known. If the least count in the measurement of all lengths is 0.01 mm, which of the following statements is **false** ?

A

$${{\Delta \gamma } \over \gamma }$$ gets minimum contribution
from the uncertainty in the length.

B

The figure of merit is the largest for the length of the rod.

C

The maximum value of $$\gamma $$ that can be determined is 2 $$ \times $$ 10^{14} N/m^{2}

D

$${{\Delta \gamma } \over \gamma }$$ gets its maximum contribution
from the uncertainty in strain

Number in Brackets after Paper Name Indicates No of Questions

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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*

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*

Ray & Wave Optics *keyboard_arrow_right*

Electrostatics *keyboard_arrow_right*

Current Electricity *keyboard_arrow_right*

Magnetics *keyboard_arrow_right*

Alternating Current and Electromagnetic Induction *keyboard_arrow_right*