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
$=\frac{dy}{dt}$
$\therefore v=\frac{dy}{dt}=\frac{d}{dt}(a\sin \omega t)$
$=a\omega \cos \omega t$
Given, $v_1=v_2=v_3$
and $a_1=A_1,a_2=A_2$
$\therefore a_3=A_3$
$\therefore A_1{\omega }_1=A_2{\omega }_2=A_3{\omega }_3$