JEE Advanced PYQ Displacement, Phase, Velocity and Acceleration in SHM

Q 1 :

A point mass is subjected to two simultaneous sinusoidal displacements in $x$ -direction, $x_{1} (t) = A \sin ω t$ and $x_{2} (t) = A \sin (ω t + \frac{2 π}{3}) .$ Adding a third sinusoidal displacement $x_{3} (t) = B \sin (ω t + ϕ)$ brings the mass to a complete rest. The values of B and $ϕ$ are [2011]

$\sqrt{2} A, \frac{3 π}{4}$
$A, \frac{4 π}{3}$
$\sqrt{3} A, \frac{5 π}{6}$
$A, \frac{π}{3}$

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(2)

Two sinusoidal displacements $x_{1} (t) = A \sin ω t$ and $x_{2} (t) = A \sin (ω t + \frac{2 π}{3})$ have amplitude $A$ each, with a phase difference of $\frac{2 π}{3}$ . It is given that sinusoidal displacement $x_{3} (t) = B (\sin ω t + ϕ)$ brings the mass to a complete rest. This is possible when the amplitude of third $B = A$ and is having a phase difference of $ϕ = \frac{4 π}{3}$ with respect to $x_{1} (t)$ as shown in the figure.

[IMAGE 526]

Q 2 :

The $x - t$ graph of a particle undergoing simple harmonic motion is shown below. The acceleration of the particle at $t = \frac{4}{3} s$ is [2009]

[IMAGE 527]

$\frac{\sqrt{3}}{32} π^{2} {cm/s}^{2}$
$- \frac{π^{2}}{32} {cm/s}^{2}$
$\frac{π^{2}}{32} {cm/s}^{2}$
$- \frac{\sqrt{3}}{32} π^{2} {cm/s}^{2}$

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(4)

$The equation for the S.H.M. x = a \sin ω t$

$\Rightarrow x = a \sin (\frac{2 π}{T} \times t) = 1 \sin (\frac{2 π}{8}) t = \sin \frac{π}{4} t$ $(∵ a = 1 cm, T = 8 s from graph)$

or, velocity, $v = \frac{d x}{d t} = \frac{d}{d t} [\sin (\frac{π}{4}) t] = \frac{π}{4} \cos (\frac{π}{4}) t$

$∴$ $Acceleration a = \frac{d^{2} x}{d t^{2}} = - {(\frac{π}{4})}^{2} \sin (\frac{π}{4}) t$

At $t = \frac{4}{3} s$ acceleration

$a = \frac{d^{2} x}{d t^{2}} = - {(\frac{π}{4})}^{2} \sin (\frac{π}{4} \times \frac{4}{3})$

$= \frac{- π^{2}}{16} \sin \frac{π}{3}$

$= \frac{- \sqrt{3} π^{2}}{32} {cm/s}^{2}$

Q 3 :

The function $x = A \sin^{2} ω t + B \cos^{2} ω t + C \sin ω t \cos ω t$ represents SHM for which of the option(s) [2006]

for all value of A, B and $C (C \neq 0)$
$A = B, C = 2 B$
$A = - B, C = 2 B$
$A = B, C = 0$

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Select one or more options

(1, 2, 3)

$The given equation$

$x = A \sin^{2} ω t + B \cos^{2} ω t + C \sin ω t \cos ω t$

Rearranging the equation for SHM the sine and cosine functions should have linear power.

$∴$ $x = \frac{A}{2} (2 \sin^{2} ω t) + \frac{B}{2} (2 \cos^{2} ω t) + \frac{C}{2} (2 \sin ω t \cos ω t)$

$= \frac{A}{2} [1 - \cos 2 ω t] + \frac{B}{2} [1 + \cos 2 ω t] + \frac{C}{2} [\sin 2 ω t]$

$(1) For A = 0 and B = 0, x = \frac{C}{2} \sin (2 ω t)$

The above equation represents SHM.

$(2) If A = B and C = 2 B$ then $x = B + B \sin 2 ω t$

This is an equation of SHM.

$(3) A = - B, C = 2 B;$

$∴$ $x = B \cos 2 ω t + B \sin 2 ω t$

Two SHMs are superposed to give another SHM equation.

$(4) A = B, C = 0$ $∴$ $x = A$

This equation does not represent SHM.

Topic Question Set

Q 2 :

The $x - t$ graph of a particle undergoing simple harmonic motion is shown below. The acceleration of the particle at $t = \frac{4}{3} s$ is [2009]

[IMAGE 527]

Q 3 :

The function $x = A \sin^{2} ω t + B \cos^{2} ω t + C \sin ω t \cos ω t$ represents SHM for which of the option(s) [2006]

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Topic Question Set

Q 1 : A point mass is subjected to two simultaneous sinusoidal displacements in x-direction, x1(t)=Asinωt and x2(t)=Asin(ωt+2π3). Adding a third sinusoidal displacement x3(t)=Bsin(ωt+ϕ) brings the mass to a complete rest. The values of B and ϕ are [2011]

Q 2 : The x-t graph of a particle undergoing simple harmonic motion is shown below. The acceleration of the particle at t=43 s is [2009] [IMAGE 527]

Q 3 : The function x=Asin2ωt+Bcos2ωt+Csinωtcosωt represents SHM for which of the option(s) [2006]

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Q 2 :

The $x - t$ graph of a particle undergoing simple harmonic motion is shown below. The acceleration of the particle at $t = \frac{4}{3} s$ is [2009]

[IMAGE 527]

Q 3 :

The function $x = A \sin^{2} ω t + B \cos^{2} ω t + C \sin ω t \cos ω t$ represents SHM for which of the option(s) [2006]