Question

two simple harmonic motions are represented by the following equation X1= 4 sin (2πt) X2= 2 {Sin 2πt+ cos 2πt}then the ratio of their amplitude is​

Answer 1

we know that

y = a sin(wt + @)

x1 = 4 sin(2πt)

A1 = 4

x 2 = 2 [sin2πt + cos2πt]

x2 = 2sqrt2sin(2πt + π/4)

A2 = 2sqrt2

ratio of A1/A2 = 4/2sqrt2= sqrt2 : 1

Answer 2

The ratio of their amplitude is $\sqrt 2$:1

Explanation:

$X_1=4 sin(2\pi t)$

$X_2=2(sin(2\pi t)+cos (2\pi t)$

$X_2=2\sqrt 2(sin(2\pi t)cos\frac{\pi}{4}+cos (2\pi t)sin\frac{\pi}{4})$

$sin\frac{\pi}{4}=cos \frac{\pi}{4}=\frac{1}{\sqrt 2}$

$X_2=2\sqrt 2sin(2\pi t+\frac{\pi}{4})$

$sin(x+y)= sinx cos y+cos x siny$

General SHM equation

y=$A sin(wt+\phi)$

Where A= Amplitude

By comparing with general SHM we get

$A_1=4,A_2=2\sqrt 2$

$\frac{A_1}{A_2}=\frac{4}{2\sqrt 2}=\sqrt 2$

$A_1:A_2=\sqrt 2$ :1

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