Question

A particle is subjected to two simple harmonic motions given as $x_1=A_{1}sin\omega t$ and $x_2=A_{2}sin(\omega t+\frac{\pi }{3})$

Find A. the displacement at t = 0

B. the maximum speed of the particle and

C. the maximum acceleration of the particle

• A. $\frac{A_2\sqrt{3}}{2},\omega \sqrt{A_1^2+A_2^2+A_1{A}_2},{\omega }^2\sqrt{A_1^2+A_2^2+A_1{A}_2}$
• B. $\frac{A_1\sqrt{3}}{2},\omega \sqrt{A_1^2+A_2^2+A_1{A}_2},{\omega }^2\sqrt{A_1^2+A_2^2+A_1{A}_2}$
• C. $\frac{A_2\sqrt{3}}{2},\omega \sqrt{A_1^2+A_2^2},{\omega }^2\sqrt{A_1^2+A_2^2}$
• D. None of these

Answer 1

The correct option is A

$\frac{A_2\sqrt{3}}{2},\omega \sqrt{A_1^2+A_2^2+A_1{A}_2},{\omega }^2\sqrt{A_1^2+A_2^2+A_1{A}_2}$

(a) At t = 0, $x_1=A_{1}sin\omega t=0$

and $x_2=A_{2}sin(\omega t+\frac{\pi }{3})$

$=A_{2}sin\left(\frac{\pi }{3}\right)=\frac{A_2\sqrt{3}}{2}$

Thus, the resultant displacement at t = 0 is

(b) The resultant of the two motions is a simple harmonic motion of the same angular frequency $\omega$. The amplitude of the resultant motion is

The maximum speed is

(c) The maximum acceleration is

$a_{max}=A{\omega }^2={\omega }^2\sqrt{A_1^2+A_2^2+A_1{A}_2}$