Question

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

• A. Amplitude of motion is $3a$
• B. Time-period of oscillation is $8t$
• C. Amplitude of motion is $4a$
• D. Time-period of oscillation is $6t$

Answer 1

The correct option is D Time-period of oscillation is $6t$

Consider a general equation,

$x=Acos\omega t$

Apply for OP,

$A-a=Acos\omega t$

similarly,

$A-(a+2a)=Acos2\omega t$

[$\therefore$ time doubled]

from here,

$cos\omega t=\frac{A-a}{A}$

$cos2\omega t=\frac{A-3a}{A}cos2\theta =2{cos}^2\theta -1{\left(\frac{A-a}{A}\right)}^2=\frac{A-30+A}{2A}$

from here,

$A=2a$

So, amplitude of the motion is $2a$.

$2a-a=2acos\omega t\frac{\pi }{3}=\omega t\frac{\pi }{3}=\frac{2\pi }{T}tT=6t$

Time period$=6t$