JEE Main SHM PYQ

26.(JEE Main 2016 (Offline))

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

A. $A \sqrt{3}$

B. $\frac{7 A}{3}$

C. $\frac{A}{3} \sqrt{41}$

D. 3A

Correct Answer is Option (B)

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

Initially $v = ω \sqrt{A^{2} - {(\frac{2 A}{3})}^{2}}$

Finally $3 v = ω \sqrt{A_{n e w}^{2} - {(\frac{2 A}{3})}^{2}}$

where $A_{n e w}$ = final amplitude (Given at $x = \frac{2 A}{3},$ velocity to trebled)

On dividing we get $\frac{3}{1} = \frac{\sqrt{A_{n e w}^{2} - {(\frac{2 A}{3})}^{2}}}{\sqrt{A^{2} - {(\frac{2 A}{3})}^{2}}}$

$9 [A^{2} - \frac{4 A^{2}}{9}] = A_{n e w}^{2} - \frac{4 A^{2}}{9}$

$∴$ $A_{n e w} = \frac{7 A}{3}$

27 .(JEE Main 2014 (Offline))

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

A. amplitude of motion is $3 a$

B. time period of oscillations is $8 τ$

C. amplitude of motion is $4 a$

D. time period of oscillations is $6 τ$

Correct Answer is Option (D)

In simple harmonic motion, starting from rest,

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

$x = A \cos ω t . . . (i)$

When $t = τ, x = A - a$

When $t = 2 τ, x = A - 3 a$

From equation $(i)$

$A - a = A \cos ω τ . . . (i i)$

$A - 3 a = A \cos 2 ω τ . . . (i i i)$

As $\cos 2 ω τ = 2 \cos^{2} ω τ - 1. . . (i v)$

From equation $(i i),$ $(i i i)$ and $(i v)$

$\frac{A - 3 A}{A} = 2 {(\frac{A - a}{A})}^{2} - 1$

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

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

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

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

Now, $A - a = A$ $\cos ω τ$

$\Rightarrow \cos ω τ = \frac{A - a}{A} \Rightarrow \cos ω τ = \frac{1}{2}$

or, $\frac{2 π}{T} τ = \frac{π}{3} \Rightarrow T - 6 τ$

28.(AIEEE 2011)

Two particles are executing simple harmonic motion of the same amplitude $A$ and frequency $ω$ along the $x$ -axis. Their mean position is separated by distance $X_{0} (X_{0} > A)$ . If the maximum separation between them is $(X_{0} + A),$ the phase difference between their motion is:

A. $\frac{π}{3}$

B. $\frac{π}{4}$

C. $\frac{π}{6}$

D. $\frac{π}{2}$

Correct Answer is Option ()

29.(AIEEE 2011)

A mass $M,$ attached to a horizontal spring, executes $S . H . M .$ with amplitude $A_{1} .$ When the mass $M$ passes through its mean position then a smaller mass $m$ is placed over it and both of them move together with amplitude $A_{2} .$ The ratio of $(\frac{A_{1}}{A_{2}})$ is :

A. $\frac{M + m}{M}$

B. ${(\frac{M}{M + m})}^{\frac{1}{2}}$

C. ${(\frac{M + m}{M})}^{\frac{1}{2}}$

D. $\frac{M}{M + m}$

Correct Answer is Option (C)

The net force becomes zero at the mean point. Therefore, linear momentum must be conserved.

$∴$ $M v_{1} = (M + m) v_{2}$

$M A_{1} \sqrt{\frac{k}{M}} = (M + m) A_{2} \sqrt{\frac{k}{m + M}}$

$∴$ $(V = A \sqrt{\frac{k}{M}})$

$A_{1} \sqrt{M} = A_{2} \sqrt{M + m}$
$∴$ $\frac{A_{1}}{A_{2}} = \sqrt{\frac{m + M}{M}}$

30.(AIEEE 2009)

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. $a T + 2 π v$

C. aT/v

D. $a^{2} T^{2} + 4 π^{2} v^{2}$

Correct Answer is Option ()

Simple Harmonic Motion