Question

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

• A. simple harmonic with amplitude $\sqrt{a^2+b^2}$
• B. simple harmonic with amplitude $\frac{(a+b)}{2}$
• C. not a simple harmonic
• D. simple harmonic with amplitude $\frac{a}{b}$

Answer 1

Answer: (A)

simple harmonic with amplitude $\sqrt{a^2+b^2}$

$y_{resultant}=y_1+y_2=a\sin \left(\omega t\right)+b\cos \left(\omega t\right)=a\sin \left(\omega t\right)+b\sin (\omega t+\frac{\pi }{2})$
Thus, the displacements are at right angles to each other.
$\therefore$ amplitude of $y_{resultant}is\sqrt{a^2+b^2}$