Question

Statement 1: A $S.H.M$ may be assumed as composition of many $S.H.Ms$.
Statement 2: Superposition of many $S.H.Ms$ (along same line) of same frequency will be a $S.H.M$

• A. Statement 1 is false, Statement 2 is true
• B. Statement 1 is true, Statement 2 is true; Statement 2 is the correct explanation for Statement 1
• C. Statement 1 is true, Statement 2 is true; Statement 2 is not the correct explanation for Statement 1
• D. Statement 1 is true, Statement 2 is false

Answer 1

The correct option is B Statement 1 is true, Statement 2 is true; Statement 2 is the correct explanation for Statement 1

Let the two $S.H.M.s$ are given as $y_1=Asin\left(wt\right)$ and $y_2=Acos\left(wt\right)$ which are superimposed on each other.

Thus the resultant motion is $y=y_1+y_2$

$y=Asin\left(wt\right)+Acos\left(wt\right)$

$y=\sqrt{2}A[\frac{1}{\sqrt{2}}\times sin\left(wt\right)+\frac{1}{\sqrt{2}}\times cos\left(wt\right)]$

$y=\sqrt{2}A[cos\left(\frac{\pi }{4}\right)sin\left(wt\right)+sin\left(\frac{\pi }{4}\right)cos\left(wt\right)]$

$⟹y=\sqrt{2}A\left[sin\right(wt+\pi /4\left)\right]$ which also represents a $S.H.M.$

Thus superposition of many $S.H.M.s$ along same line and frequency will be a $S.H.M.$