Question

The angular velocities of three bodies in simple harmonic motion are $\omega_{1}, \omega_{2}, \omega_{3}$ with their respective amplitudes as $A_{1}, A_{2}, A_{3} .$ If all the three bodies have same mass and velocity, then:

• A. $A_{1}^{2} \omega_{1}^{2}=A_{2}^{2} \omega_{2}^{2}=A_{3}^{2} \omega_{3}^{2}$
• B. $A_{1}^{2} \omega_{1}=A_{2}^{2} \omega_{2}=A_{3}^{2} \omega_{3}$
• C. $A_{1} \omega_{1}^{2}=A_{2} \omega_{2}^{2}=A_{3} \omega_{3}^{2}$
• D. $A_{1} \omega_{1}=A_{2} \omega_{2}=A_{3} \omega_{3}$

Answer 1

Answer: (D)

Rate of change of displacement is known as velocity.
The equation for displacement $(y)$ of a body in SHM with angular velocity $\omega$ is given by
$y=a \sin \omega t$
where $a$ is amplitude.
Velocity is $v=$ rate of change of displacement
$=\frac{d y}{d t}$
$\therefore v=\frac{d y}{d t}=\frac{d}{d t}(a \sin \omega t)$
$=a \omega \cos \omega t$
Given, $v_{1}=v_{2}=v_{3}$
and $a_{1}=A_{1}, a_{2}=A_{2}$
$\therefore a_{3}=A_{3}$
$\therefore A_{1} \,\omega_{1}=A_{2} \,\omega_{2}=A_{3} \,\omega_{3}$