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JEE Mains Previous Years Questions with Solutions

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1

AIEEE 2007

MCQ (Single Correct Answer)
Two springs, of force constant $${k_1}$$ and $${k_2}$$ are connected to a mass $$m$$ as shown. The frequency of oscillation of the mass is $$f.$$ If both $${k_1}$$ and $${k_2}$$ are made four times their original values, the frequency of oscillation becomes
A
$$2f$$
B
$$f/2$$
C
$$f/4$$
D
$$4f$$

Explanation

The two springs are in parallel.

$$f = {1 \over {2\pi }}\sqrt {{{{K_1} + {K_2}} \over m}} \,\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( i \right)$$

$$f' = {1 \over {2\pi }}\sqrt {{{4{K_1} + 4{K_2}} \over m}} $$

$$ = {1 \over {2\pi }}\sqrt {{{4\left( {{K_1} + 4{K_2}} \right)} \over m}} $$

$$ = 2\left( {{1 \over {2\pi }}\sqrt {{{{K_1} + {K_2}} \over m}} } \right)$$

$$ = 2f\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ from eqn. $$\left( i \right)$$
2

AIEEE 2007

MCQ (Single Correct Answer)
A particle of mass $$m$$ executes simple harmonic motion with amplitude a and frequency $$v.$$ The average kinetic energy during its motion from the position of equilibrium to the end is
A
$$2{\pi ^2}\,m{a^2}{v^2}$$
B
$${\pi ^2}\,m{a^2}{v^2}$$
C
$${1 \over 4}\,m{a^2}{v^2}$$
D
$$4{\pi ^2}m{a^2}{v^2}$$

Explanation

KEY CONCEPT : The instantaneous kinetic energy of a particle executing $$S.H.M.$$ is given by

$$K = {1 \over 2}m{a^2}{\omega ^2}{\sin ^2}\omega t$$

$$\therefore$$ average $$K.E. = < K > = < {1 \over 2}m{\omega ^2}{a^2}{\sin ^2}\omega t > $$

$$ = {1 \over 2}m\omega {}^2{a^2} < {\sin ^2}\omega t > $$

$$ = {1 \over 2}m{\omega ^2}{a^2}\left( {{1 \over 2}} \right)$$

$$\left( \, \right.$$ as $$\left. { < {{\sin }^2}\theta > = {1 \over 2}} \right)$$

$$ = {1 \over 4}m{\omega ^2}{a^2} = {1 \over 4}m{a^2}{\left( {2\pi v} \right)^2}$$

$$\left( \, \right.$$ $$\left. {\omega = 2\pi v} \right)$$

or, $$\,\,\,\,\, < K > = {\pi ^2}m{a^2}{v^2}$$
3

AIEEE 2006

MCQ (Single Correct Answer)
The maximum velocity of a particle, executing simple harmonic motion with an amplitude $$7$$ $$mm,$$ is $$4.4$$ $$m/s.$$ The period of oscillation is
A
$$0.01$$ $$s$$
B
$$10$$ $$s$$
C
$$0.1$$ $$s$$
D
$$100$$ $$s$$

Explanation

Maximum velocity,

$${v_{\max }} = a\omega ,\,\,\,\,\,{v_{\max }} = a \times {{2\pi } \over T}$$

$$ \Rightarrow T = {{2\pi a} \over {{v_{\max }}}} = {{2 \times 3.14 \times 7 \times {{10}^{ - 3}}} \over {4.4}} \approx 0.01\,s$$
4

AIEEE 2006

MCQ (Single Correct Answer)
Starting from the origin a body oscillates simple harmonically with a period of $$2$$ $$s.$$ After what time will its kinetic energy be $$75\% $$ of the total energy?
A
$${1 \over 6}s$$
B
$${1 \over 4}s$$
C
$${1 \over 3}s$$
D
$${1 \over 12}s$$

Explanation

$$K.E.\,$$ of a body undergoing $$SHM$$ is given by,

$$K.E. = {1 \over 2}m{a^2}{\omega ^2}{\cos ^2}\,\omega t,$$

$$T.E. = {1 \over 2}m{a^2}{\omega ^2}$$

Given $$K.E.=0.75T.E.$$

$$ \Rightarrow 0.75 = {\cos ^2}\omega t \Rightarrow \omega t = {\pi \over 6}$$

$$ \Rightarrow t = {\pi \over {6 \times \omega }} \Rightarrow t = {{\pi \times 2} \over {6 \times 2\pi }} \Rightarrow t = {1 \over 6}s$$

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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