Question

Equation of two $S.H.M,x_1=5\sin \left(2\pi t+\pi /4\right),x_2=5$ $\sqrt{2}\left(\sin 2\pi t+\cos 2\pi t\right)$. Ratio of amplitude & phase difference will be :

• A. $2:1,0$
• B. $1:2,0$
• C. $1:2,\pi /2$
• D. $2:1,\pi /2$

Answer 1

Answer: (B)

$1:2,0$

Equation of S.H.M ,

$x_1=5sin(2\pi t+\pi /4)$. . . . . . .(1)

$x_2=5\sqrt{2}(sin2\pi t+cos2\pi t)$

$x_2=10(\frac{1}{\sqrt{2}}.sin2\pi t+\frac{1}{\sqrt{2}}.cos2\pi t)$

$x_2=10(cos\frac{\pi }{4}.sin2\pi t+sin\frac{\pi }{4}.cos2\pi t)$

$x_2=10(sin2\pi t+\pi /4)$. . . . . . . . ..(2)

The ratio of amplitude, $a_1:a_2=1:2$

The phase difference is, $\varphi ={\theta }_1-{\theta }_2$

$\varphi =\frac{\pi }{4}-\frac{\pi }{4}=0$

The correct option is B.