The function x = A sin 2 t + B cos 2 t + C sin t cos t represents SHM for which of the option(s) [2006]

Question

The function $x = A \sin^{2} ω t + B \cos^{2} ω t + C \sin ω t \cos ω t$ represents SHM for which of the option(s) [2006]

Smart Achievers · Accepted Answer

(1, 2, 3)

$The given equation$

$x = A \sin^{2} ω t + B \cos^{2} ω t + C \sin ω t \cos ω t$

Rearranging the equation for SHM the sine and cosine functions should have linear power.

$∴$ $x = \frac{A}{2} (2 \sin^{2} ω t) + \frac{B}{2} (2 \cos^{2} ω t) + \frac{C}{2} (2 \sin ω t \cos ω t)$

$= \frac{A}{2} [1 - \cos 2 ω t] + \frac{B}{2} [1 + \cos 2 ω t] + \frac{C}{2} [\sin 2 ω t]$

$(1) For A = 0 and B = 0, x = \frac{C}{2} \sin (2 ω t)$

The above equation represents SHM.

$(2) If A = B and C = 2 B$ then $x = B + B \sin 2 ω t$

This is an equation of SHM.

$(3) A = - B, C = 2 B;$

$∴$ $x = B \cos 2 ω t + B \sin 2 ω t$

Two SHMs are superposed to give another SHM equation.

$(4) A = B, C = 0$ $∴$ $x = A$

This equation does not represent SHM.

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