Question

The displacement y of a particle is given by $y=4co{s}^2\left(\frac{t}{2}\right)sin\left(1000t\right)$, This expression may be considered to be a result of the superposition of how many simple harmonic motions?

• A. two
• B. three
• C. four
• D. five

Answer 1

The correct option is B three

We can write $4co{s}^2\left(\frac{t}{2}\right)=2\times 2co{s}^2\left(\frac{t}{2}\right)=2\times (1+cost).Therefore$
$y=2\times (1+cost)\times sin\left(1000t\right)$
$=2sin\left(1000t\right)+2costsin\left(1000t\right)$
$=sin\left(1000t\right)+sin\left(1001t\right)+sin\left(999t\right)$
Thus y is a superposition of three simple harmonic motions of angular frequencies 999, 1000 and 1001 $rad{s}^{-1}$. Hence the correct choice is (b). But a superposition of two or more simple harmonic motions of different frequency does not produce a simple harmonic motion.