Question

The displacement $y$ of a 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

The correct option is B three

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

$\Rightarrow y=2\times 2{\cos }^2\left(\frac{t}{2}\right)\sin \left(1000t\right)$

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

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

$\Rightarrow y=2\sin \left(1000t\right)+\sin \left[\right(1000+1\left)t\right]+\sin \left[\right(1000-1\left)t\right]$

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

$\Rightarrow y=y_1+y_2+y_3$

As the Above expression involves three different waves, the displacement given is a result of superposition of three different waves.