Question

When two displacements represented by $y_1=asin\left(t\right)$ and $y_2=bcos\left(\omega t\right)$ are superimposed the motion is :

• A. simple harmonic with amplitude (a + b)/2
• B. not a simple harmonic
• C. simple harmonic with amplitude a/b
• D. simple harmonic with amplitude

Answer 1

The correct option is D

simple harmonic with amplitude

$y_1=asin\left(\omega t\right)y_2=bcos\left(\omega t\right)$

Let a = c cos $\left(\varphi \right)$ and b = c sin $\left(\varphi \right)$

We have,

$y_1+y_2=asin\left(\omega t\right)+bcos\left(\omega t\right)$

$=ccos\varphi sin\left(\omega t\right)+csin\varphi cos\left(\omega t\right)$

$=c\left[sin\right(\omega t+\varphi \left)\right]$

Where $c^2=a^2+b^2$

$[since{a}^2+b^2=c^{2}co{s}^2(\varphi )=c^{2}si{n}^2(\varphi )=c^2]$
$\therefore c=\sqrt{a^2+b^2}$

The superimposed motion is simple harmonic with amplitude $\sqrt{a^2+b^2}$ .