Joint Entrance Examination

Graduate Aptitude Test in Engineering

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1

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

$${\pi \over 2}$$

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

2

MCQ (Single Correct Answer)

If $$x,$$ $$v$$ and $$a$$ denote the displacement, the velocity and the acceleration of a particle executing simple harmonic motion of time period $$T,$$ then, which of the following does not change with time?

A

$$aT/x$$

B

$$aT + 2\pi v$$

C

$$aT/v$$

D

$${a^2}{T^2} + 4{\pi ^2}{v^2}$$

For an $$SHM,$$ the acceleration $$a = - {\omega ^2}x$$ where $${\omega ^2}$$ is a constant. Therefore $${a \over x}$$ is a constant. The time period $$T$$ is also constant. Therefore $${{aT} \over x}$$ is a constant.

3

MCQ (Single Correct Answer)

The displacement of an object attached to a spring and executing simple harmonic motion is given by $$x = 2 \times {10^{ - 2}}$$ $$cos$$ $$\pi t$$ metre. The time at which the maximum speed first occurs is

A

$$0.25$$ $$s$$

B

$$0.5$$ $$s$$

C

$$0.75$$ $$s$$

D

$$0.125$$ $$s$$

Here, $$x = 2 \times {10^{ - 2}}\cos \,\pi \,t$$

$$\therefore$$ $$v = {{dx} \over {dt}} = 2 \times {10^{ - 2}}\,\pi \sin \pi t$$

For the first time, the speed to be maximum,

$$\sin \pi t = 1$$ or, $$\sin \pi t = \sin {\pi \over 2}$$

$$ \Rightarrow \pi t = {\pi \over 2}\,\,\,$$ or, $$\,\,\,\,t = {1 \over 2} = 0.5\,\sec .$$

$$\therefore$$ $$v = {{dx} \over {dt}} = 2 \times {10^{ - 2}}\,\pi \sin \pi t$$

For the first time, the speed to be maximum,

$$\sin \pi t = 1$$ or, $$\sin \pi t = \sin {\pi \over 2}$$

$$ \Rightarrow \pi t = {\pi \over 2}\,\,\,$$ or, $$\,\,\,\,t = {1 \over 2} = 0.5\,\sec .$$

4

MCQ (Single Correct Answer)

A point mass oscillates along the $$x$$-axis according to the law $$x = {x_0}\,\cos \left( {\omega t - \pi /4} \right).$$ If the acceleration of the particle is written as $$a = A\,\cos \left( {\omega t + \delta } \right),$$ then

A

$$A = {x_0}{\omega ^2},\,\,\delta = 3\pi /4$$

B

$$A = {x_0},\,\,\delta = - \pi /4$$

C

$$A = {x_0}{\omega ^2},\,\,\delta = \pi /4$$

D

$$A = {x_0}{\omega ^2},\,\,\delta = - \pi /4$$

Here,

$$x = {x_0}\cos \left( {\omega t - \pi /4} \right)$$

$$\therefore$$ Velocity, $$v = {{dx} \over {dt}} = - {x_0}\omega \sin \left( {\omega t - {\pi \over 4}} \right)$$

Acceleration,

$$a = {{dv} \over {dt}} = - {x_0}{\omega ^2}\cos \left( {\omega t - {\pi \over 4}} \right)$$

$$ = {x_0}{\omega ^2}\cos \left[ {\pi + \left( {\omega t - {\pi \over 4}} \right)} \right]$$

$$ = {x_0}{\omega ^2}\cos \left( {\omega t + {{3\pi } \over 4}} \right)$$ $$\,\,\,\,\,\,\,\,\,\,\,...\left( 1 \right)$$

Acceleration, $$a = A\cos \left( {\omega t + \delta } \right)$$ $$\,\,\,\,\,\,\,\,\,\,\,...\left( 2 \right)$$

Comparing the two equations, we get

$$A = {x_0}{\omega ^2}$$ and $$\delta = {{3\pi } \over 4}.$$

$$x = {x_0}\cos \left( {\omega t - \pi /4} \right)$$

$$\therefore$$ Velocity, $$v = {{dx} \over {dt}} = - {x_0}\omega \sin \left( {\omega t - {\pi \over 4}} \right)$$

Acceleration,

$$a = {{dv} \over {dt}} = - {x_0}{\omega ^2}\cos \left( {\omega t - {\pi \over 4}} \right)$$

$$ = {x_0}{\omega ^2}\cos \left[ {\pi + \left( {\omega t - {\pi \over 4}} \right)} \right]$$

$$ = {x_0}{\omega ^2}\cos \left( {\omega t + {{3\pi } \over 4}} \right)$$ $$\,\,\,\,\,\,\,\,\,\,\,...\left( 1 \right)$$

Acceleration, $$a = A\cos \left( {\omega t + \delta } \right)$$ $$\,\,\,\,\,\,\,\,\,\,\,...\left( 2 \right)$$

Comparing the two equations, we get

$$A = {x_0}{\omega ^2}$$ and $$\delta = {{3\pi } \over 4}.$$

On those following papers in MCQ (Single Correct Answer)

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Units & Measurements *keyboard_arrow_right*

Motion *keyboard_arrow_right*

Laws of Motion *keyboard_arrow_right*

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