JEE Main 2015 — Physics Waves & Oscillations

2015mcqhard

A simple harmonic oscillator of angular frequency $2\mathrm{rad}{s}^{-1}$ is acted upon by an external force $F=\mathrm{sin}⁡t N.$ If the oscillator is at rest in its equilibrium position at $t=0,$ its position at later times is proportional to:

Official previous-year question

Held on 10 Apr 2015 · Verified 6 Jul 2026.

Options

A
$\mathrm{sin}t+\frac{1}{2}\mathrm{cos}2t$
B
$cost-\frac{1}{2}\mathrm{sin}2t$
C
$\mathrm{sin}t-\frac{1}{2}\mathrm{sin}2t$
D
$\mathrm{sin}t+\frac{1}{2}\mathrm{sin}2t$

Did you get this right?

Solution

It is given that oscillator at rest at $t=0$ i.e. at $t=0$ , $v=0$ . So, in option, we can check by putting $v=\frac{dx}{dt}=0$ .

(1) $If x\propto \mathrm{sin}⁡t+\frac{1}{2}\mathrm{cos}⁡2t$

$\Longrightarrow v\propto \mathrm{cos}⁡t+\frac{1}{2}\times 2 (-\mathrm{sin}⁡2t)$

$\Longrightarrow at t=0, v\propto 1-0\neq 0$

(2) $If x \propto \mathrm{cos}⁡t-\frac{1}{2}\mathrm{sin}⁡t$

$\Longrightarrow v\propto -\mathrm{sin}⁡t-\frac{1}{2}\mathrm{cos}⁡t$

$\Longrightarrow at t=0, v\propto -\frac{1}{2}\neq 0$

(3) $If x\propto \mathrm{sin}⁡t-\frac{1}{2} sin 2t$

$then \upsilon \propto \mathrm{cos}⁡t-\frac{1}{2}\times 2\mathrm{cos}⁡2t$

$\Longrightarrow at t=0, v\propto 1-1=0$

(4) $If x\propto \mathrm{sin}⁡t+\frac{1}{2}\mathrm{sin}⁡2t$

$\Longrightarrow v\propto \mathrm{cos}⁡t+\frac{1}{2}\times 2\mathrm{cos}⁡2t$

$\Longrightarrow at t=0, v\propto 1+1$

$\Longrightarrow v\propto 2\neq 0$

$\therefore in option (3) v=0 at t=0$

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Solution

It is given that oscillator at rest at $t=0$ i.e. at $t=0$ , $v=0$ . So, in option, we can check by putting $v=\frac{dx}{dt}=0$ .

(1) $If x\propto \mathrm{sin}⁡t+\frac{1}{2}\mathrm{cos}⁡2t$

$\Longrightarrow v\propto \mathrm{cos}⁡t+\frac{1}{2}\times 2 (-\mathrm{sin}⁡2t)$

$\Longrightarrow at t=0, v\propto 1-0\neq 0$

(2) $If x \propto \mathrm{cos}⁡t-\frac{1}{2}\mathrm{sin}⁡t$

$\Longrightarrow v\propto -\mathrm{sin}⁡t-\frac{1}{2}\mathrm{cos}⁡t$

$\Longrightarrow at t=0, v\propto -\frac{1}{2}\neq 0$

(3) $If x\propto \mathrm{sin}⁡t-\frac{1}{2} sin 2t$

$then \upsilon \propto \mathrm{cos}⁡t-\frac{1}{2}\times 2\mathrm{cos}⁡2t$

$\Longrightarrow at t=0, v\propto 1-1=0$

(4) $If x\propto \mathrm{sin}⁡t+\frac{1}{2}\mathrm{sin}⁡2t$

$\Longrightarrow v\propto \mathrm{cos}⁡t+\frac{1}{2}\times 2\mathrm{cos}⁡2t$

$\Longrightarrow at t=0, v\propto 1+1$

$\Longrightarrow v\propto 2\neq 0$

$\therefore in option (3) v=0 at t=0$

JEE Main 2015 — Physics Waves & Oscillations

Options

Did you get this right?

Solution

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JEE Main 2015 — Physics Waves & Oscillations

Options

Did you get this right?

Solution

Your note

JEE Main Waves & Oscillations in other years

More JEE Main Waves & Oscillations questions

Other Physics topics for JEE Main

Mechanics

Electromagnetism

Modern Physics

Optics

Thermodynamics

Keep practicing Waves & Oscillations