Joint Entrance Examination

Graduate Aptitude Test in Engineering

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1

MCQ (Single Correct Answer)

A particle performs simple harmonic motion with amplitude $$A.$$ Its speed is trebled at the instant that it is at a distance $${{2A} \over 3}$$ from equilibrium position. The new amplitude of the motion is:

A

$$A\sqrt 3 $$

B

$${{7A} \over 3}$$

C

$${A \over 3}\sqrt {41} $$

D

$$3A$$

We know that $$V = \omega \sqrt {{A^2} - {x^2}} $$

Initially $$v = \omega \sqrt {{A^2} - {{\left( {{{2A} \over 3}} \right)}^2}} $$

Finally $$3v = \omega \sqrt {A_{new}^2 - {{\left( {{{2A} \over 3}} \right)}^2}} $$

where $${A_{new}}$$ = final amplitude (Given at $$x = {{2A} \over 3},$$ velocity to trebled)

On dividing we get $${3 \over 1} = {{\sqrt {A_{new}^2 - {{\left( {{{2A} \over 3}} \right)}^2}} } \over {\sqrt {{A^2} - {{\left( {{{2A} \over 3}} \right)}^2}} }}$$

$$9\left[ {{A^2} - {{4{A^2}} \over 9}} \right] = A_{new}^2 - {{4{A^2}} \over 9}$$

$$\therefore$$ $$A_{new} = {{7A} \over 3}$$

Initially $$v = \omega \sqrt {{A^2} - {{\left( {{{2A} \over 3}} \right)}^2}} $$

Finally $$3v = \omega \sqrt {A_{new}^2 - {{\left( {{{2A} \over 3}} \right)}^2}} $$

where $${A_{new}}$$ = final amplitude (Given at $$x = {{2A} \over 3},$$ velocity to trebled)

On dividing we get $${3 \over 1} = {{\sqrt {A_{new}^2 - {{\left( {{{2A} \over 3}} \right)}^2}} } \over {\sqrt {{A^2} - {{\left( {{{2A} \over 3}} \right)}^2}} }}$$

$$9\left[ {{A^2} - {{4{A^2}} \over 9}} \right] = A_{new}^2 - {{4{A^2}} \over 9}$$

$$\therefore$$ $$A_{new} = {{7A} \over 3}$$

2

MCQ (Single Correct Answer)

For a simple pendulum, a graph is plotted between its kinetic energy $$(KE)$$ and potential energy $$(PE)$$ against its displacement $$d.$$ Which one of the following represents these correctly?

$$(graphs$$ $$are$$ $$schematic$$ $$and$$ $$not$$ $$drawn$$ $$to$$ $$scale)$$

$$(graphs$$ $$are$$ $$schematic$$ $$and$$ $$not$$ $$drawn$$ $$to$$ $$scale)$$

A

B

C

D

$$K.E = {1 \over 2}k\left( {{A^2} - {d^2}} \right)$$

and $$P.E. = {1 \over 2}k{d^2}$$

At mean position $$d=0.$$ At extremes positions $$d=A$$

and $$P.E. = {1 \over 2}k{d^2}$$

At mean position $$d=0.$$ At extremes positions $$d=A$$

3

MCQ (Single Correct Answer)

A pendulum made of a uniform wire of cross sectional area $$A$$ has time period $$T.$$ When an additional mass $$M$$ is added to its bob, the time period changes to $${T_{M.}}$$ If the Young's modulus of the material of the wire is $$Y$$ then $${1 \over Y}$$ is equal to :

($$g=$$ $$gravitational$$ $$acceleration$$)

($$g=$$ $$gravitational$$ $$acceleration$$)

A

$$\left[ {1 - {{\left( {{{{T_M}} \over T}} \right)}^2}} \right]{A \over {Mg}}$$

B

$$\left[ {1 - {{\left( {{T \over {{T_M}}}} \right)}^2}} \right]{A \over {Mg}}$$

C

$$\left[ {{{\left( {{{{T_M}} \over T}} \right)}^2} - 1} \right]{A \over {Mg}}$$

D

$$\left[ {{{\left( {{{{T_M}} \over T}} \right)}^2} - 1} \right]{{Mg} \over A}$$

As we know, time period, $$T = 2\pi \sqrt {{\ell \over g}} $$

When a additional mass $$M$$ is added then

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

$${{{T_M}} \over T} = \sqrt {{{\ell + \Delta \ell } \over \ell }} $$

or, $$\,\,{\left( {{{{T_M}} \over T}} \right)^2} = {{\ell + \Delta \ell } \over \ell }$$

or, $$\,\,{\left( {{{{T_M}} \over T}} \right)^2} = 1 + {{Mg} \over {Ay}}$$

$$\left[ \, \right.$$ as $$\left. {\Delta \ell = {{Mg\ell } \over {Ay}}\,} \right]$$

$$\therefore$$ $${1 \over y} = \left[ {{{\left( {{{{T_M}} \over T}} \right)}^2} - 1} \right]{A \over {Mg}}$$

When a additional mass $$M$$ is added then

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

$${{{T_M}} \over T} = \sqrt {{{\ell + \Delta \ell } \over \ell }} $$

or, $$\,\,{\left( {{{{T_M}} \over T}} \right)^2} = {{\ell + \Delta \ell } \over \ell }$$

or, $$\,\,{\left( {{{{T_M}} \over T}} \right)^2} = 1 + {{Mg} \over {Ay}}$$

$$\left[ \, \right.$$ as $$\left. {\Delta \ell = {{Mg\ell } \over {Ay}}\,} \right]$$

$$\therefore$$ $${1 \over y} = \left[ {{{\left( {{{{T_M}} \over T}} \right)}^2} - 1} \right]{A \over {Mg}}$$

4

MCQ (Single Correct Answer)

A particle moves with simple harmonic motion in a straight line. In first $$\tau s,$$ after starting from rest it travels a distance $$a,$$ and in next $$\tau s$$ it travels $$2a,$$ in same direction, then:

A

amplitude of motion is $$3a$$

B

time period of oscillations is $$8\tau $$

C

amplitude of motion is $$4a$$

D

time period of oscillations is $$6\tau $$

In simple harmonic motion, starting from rest,

At $$t=0,$$ $$x=A$$

$$x = A\cos \omega t\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( i \right)$$

When $$t = \tau ,\,\,x = A - a$$

When $$t = 2\,\tau ,\,x = A - 3a$$

From equation $$(i)$$

$$A - a = A\cos \omega \,\tau \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( {ii} \right)$$

$$A - 3a = A\cos 2\omega \,\tau \,\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( {iii} \right)$$

As $$\cos 2\omega \,\tau = 2{\cos ^2}\omega \tau - 1...\left( {iv} \right)$$

From equation $$(ii),$$ $$(iii)$$ and $$(iv)$$

$${{A - 3A} \over A} = 2{\left( {{{A - a} \over A}} \right)^2} - 1$$

$$ \Rightarrow {{A - 3a} \over A} = {{2{A^2} + 2{a^2} - 4Aa - {A^2}} \over {{A^2}}}$$

$$ \Rightarrow {A^2} - 3aA = {A^2} + 2{a^2} - 4Aa$$

$$ \Rightarrow 2{a^2} = aA \Rightarrow \,\,\,\,\,\,\,A = 2a$$

$$ \Rightarrow {a \over A} = {1 \over 2}$$

Now, $$A-a=A$$ $$\cos \omega \tau $$

$$ \Rightarrow \cos \omega \tau = {{A - a} \over A} \Rightarrow \,\,\cos \omega \tau = {1 \over 2}$$

or, $${{2\pi } \over T}\tau = {\pi \over 3} \Rightarrow \,\,\,T - 6\,\tau $$

At $$t=0,$$ $$x=A$$

$$x = A\cos \omega t\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( i \right)$$

When $$t = \tau ,\,\,x = A - a$$

When $$t = 2\,\tau ,\,x = A - 3a$$

From equation $$(i)$$

$$A - a = A\cos \omega \,\tau \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( {ii} \right)$$

$$A - 3a = A\cos 2\omega \,\tau \,\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( {iii} \right)$$

As $$\cos 2\omega \,\tau = 2{\cos ^2}\omega \tau - 1...\left( {iv} \right)$$

From equation $$(ii),$$ $$(iii)$$ and $$(iv)$$

$${{A - 3A} \over A} = 2{\left( {{{A - a} \over A}} \right)^2} - 1$$

$$ \Rightarrow {{A - 3a} \over A} = {{2{A^2} + 2{a^2} - 4Aa - {A^2}} \over {{A^2}}}$$

$$ \Rightarrow {A^2} - 3aA = {A^2} + 2{a^2} - 4Aa$$

$$ \Rightarrow 2{a^2} = aA \Rightarrow \,\,\,\,\,\,\,A = 2a$$

$$ \Rightarrow {a \over A} = {1 \over 2}$$

Now, $$A-a=A$$ $$\cos \omega \tau $$

$$ \Rightarrow \cos \omega \tau = {{A - a} \over A} \Rightarrow \,\,\cos \omega \tau = {1 \over 2}$$

or, $${{2\pi } \over T}\tau = {\pi \over 3} \Rightarrow \,\,\,T - 6\,\tau $$

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