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JEE Mains Previous Years Questions with Solutions

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1

JEE Main 2013 (Offline)

MCQ (Single Correct Answer)
The amplitude of a damped oscillator decreases to $$0.9$$ times its original magnitude in $$5s$$. In another $$10s$$ it will decrease to $$\alpha $$ times its original magnitude, where $$\alpha $$ equals
A
$$0.7$$
B
$$0.81$$
C
$$0.729$$
D
$$0.6$$

Explanation

as $$\,\,A = {A_0}{e^{{{bt} \over {2m}}}}$$ (where, $${A_0} = $$ maximum amplitude)

According to the questions, after $$5$$ second,

$$0.9{A_0} = {A_0}{e^{ - {{b\left( 5 \right)} \over {2m}}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( i \right)$$

After $$10$$ more second,

$$A = {A_0}{e^{ - {{b\left( {15} \right)} \over {2m}}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( {ii} \right)$$

From eqns $$(i)$$ and $$(ii)$$

$$A = 0.729\,{A_0}$$

$$\therefore$$ $$\alpha = 0.729$$
2

JEE Main 2013 (Offline)

MCQ (Single Correct Answer)
An ideal gas enclosed in a vertical cylindrical container supports a freely moving piston of mass $$M.$$ The piston and the cylinder have equal cross sectional area $$A$$. When the piston is in equilibrium, the volume of the gas is $${V_0}$$ and its pressure is $${P_0}.$$ The piston is slightly displaced from the equilibrium position and released,. Assuming that the system is completely isolated from its surrounding, the piston executes a simple harmonic motion with frquency
A
$${1 \over {2\pi }}\,{{A\gamma {P_0}} \over {{V_0}M}}$$
B
$${1 \over {2\pi }}\,{{{V_0}M{P_0}} \over {{A^2}\gamma }}$$
C
$${1 \over {2\pi }}\,\sqrt {{{A\gamma {P_0}} \over {{V_0}M}}} $$
D
$${1 \over {2\pi }}\,\sqrt {{{M{V_0}} \over {A\gamma {P_0}}}} $$

Explanation

$${{Mg} \over A} = {P_0}$$

$$Mg = {P_0}A\,\,\,\,...\left( 1 \right)$$

$${P_0}V_0^\gamma = P{V^\gamma }$$

$$P = {{{P_0}x_0^\gamma } \over {{{\left( {{x_0} - x} \right)}^y}}}$$

Let piston is displaced by distance $$x$$

$$Mg - \left( {{{{P_0}x_0^\gamma } \over {{{\left( {{x_0} - x} \right)}^\gamma }}}} \right)A = {F_{restoring}}$$



$${P_0}A\left( {1 - {{x_0^\gamma } \over {{{\left( {{x_0} - x} \right)}^\gamma }}}} \right) = {F_{restoring}}$$

$$\left[ {{x_0} - x \approx {x_0}} \right]$$

$$F = - {{\gamma {P_0}Ax} \over {{x_0}}}$$

$$\therefore$$ Frequency with which piston executes $$SHM.$$

$$f = {1 \over {2\pi }}\sqrt {{{\gamma {P_0}A} \over {{x_0}M}}} = {1 \over {2\pi }}\sqrt {{{\gamma {P_0}{A^2}} \over {M{V_0}}}} $$
3

AIEEE 2012

MCQ (Single Correct Answer)
If a simple pendulum has significant amplitude (up to a factor of $$1/e$$ of original ) only in the period between $$t = 0s\,\,to\,\,t = \tau \,s,$$ then $$\tau \,$$ may be called the average life of the pendulum When the spherical bob of the pendulum suffers a retardation (due to viscous drag) proportional to its velocity with $$b$$ as the constant of proportionality, the average life time of the pendulum is (assuming damping is small) in seconds :
A
$${{0.693} \over b}$$
B
$$b$$
C
$${1 \over b}$$
D
$${2 \over b}$$

Explanation

The equation of motion for the pendulum, suffering retardation

$$I\alpha = - mg\left( {\ell \sin \theta } \right) - mbv\left( \ell \right)$$ where $$I = m{\ell ^2}$$

and $$\alpha = {d^2}\theta /d{t^2}$$

$$\therefore$$ $${{{d^2}\theta } \over {d{t^2}}} = - {g \over \ell }\tan \theta + {{bv} \over \ell }$$

on solving we get $$\theta = {\theta _0}\,{e^{{{bt} \over 2}\sin \left( {\omega t + \phi } \right)}}$$

According to questions $${{{\theta _0}} \over e} = {\theta _0}{e^{{{ - b\tau } \over 2}}}$$

$$\therefore$$ $$\tau = {2 \over b}$$
4

AIEEE 2011

MCQ (Single Correct Answer)
Two particles are executing simple harmonic motion of the same amplitude $$A$$ and frequency $$\omega $$ along the $$x$$-axis. Their mean position is separated by distance $${X_0}\left( {{X_0} > A} \right)$$. If the maximum separation between them is $$\left( {{X_0} + A} \right),$$ the phase difference between their motion is:
A
$${\pi \over 3}$$
B
$${\pi \over 4}$$
C
$${\pi \over 6}$$
D
$${\pi \over 2}$$

Explanation

For $${X_0} + A$$ to be the maximum separation $$y$$ one body is at the mean position, the other should be at the extreme.

Questions Asked from Simple Harmonic Motion

On those following papers in MCQ (Single Correct Answer)
Number in Brackets after Paper Indicates No. of Questions
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