Question

The displacement $y$ of a particle executing periodic motion is given by
$y=4{\cos }^2\left(\frac{1}{2}t\right)\sin \left(1000t\right)$. This expression may be considered to be a result of the superposition of ____ independent harmonic motions

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

Answer 1

Answer: (B)

three

$y=4{\cos }^2\left(1/2t\right)\sin \left(1000t\right)$

$=2\times 1(1+\cos t)\sin \left(1000t\right)$

$=2\sin \left(1000t\right)+2\cos t\sin \left(1000t\right)$

$=2\sin \left(1000t\right)+\sin \left(1001\right)t+\sin \left(999\right)t$

$=y_1+y_2+y_2$

As the above expression has $3$ different waves

$\therefore$ Three different waves would be superposed.