Question

The displacement $y$ of a particle is given by $y = 4\cos^2(t/2) \sin (1000 t)$. This expression may be considered to be a result of the superposition of how many simple harmonic motions?

• A. 4
• B. 3
• C. 2
• D. 5
• E. 6

Hint:

Any complex periodic wave can be broken down into a sum of simple harmonic waves (Sines/Cosines). The number of unique frequency terms equals the number of SHMs.

Answer 1

The Correct Option is B

Concept:
To find the number of SHMs, we need to expand the trigonometric product into a sum of individual sine or cosine functions using trigonometric identities. [itemsep=4pt]
• Identity: $\cos^2 \theta = \frac{1 + \cos 2\theta}{2}$
• Identity: $2\sin A \cos B = \sin(A+B) + \sin(A-B)$

Step 1: Expand the $\cos^2$ term.
\[ y = 4 \left( \frac{1 + \cos(t)}{2} \right) \sin(1000t) \] \[ y = 2(1 + \cos t) \sin(1000t) \] \[ y = 2\sin(1000t) + 2\sin(1000t)\cos(t) \]

Step 2: Expand the product term.
Using $2\sin(1000t)\cos(t) = \sin(1001t) + \sin(999t)$: \[ y = 2\sin(1000t) + \sin(1001t) + \sin(999t) \]

Step 3: Count the components.
The expression is now the sum of three distinct sine functions ($\omega = 1000, 1001, 999$). Each represents one SHM.