Question

The superposition of two SHMs of the same direction results in the oscillation of a point according to the law $x=x_0\cos \left(2.1t\right)\cos 50t$. Find the angular frequencies of the constituent oscillations and period with which they beat.

• A. $52.1s^{-1},47.9s^{-1},0.2s$
• B. $50s^{-1},2.1s^{-1},0.22s$
• C. $52.1s^{-1},47.9s^{-1},1.5s$
• D. none

Answer 1

The correct option is C $52.1s^{-1},47.9s^{-1},1.5s$

$x_0\left[\cos \left(2.1t\right)\cos \left(50t\right)\right]=\frac{x_0}{2}[\cos 52.1t+\cos 47.9t]$
We can easily see that two constituent SHM's are $y_1=\frac{x_0}{2}\cos 52.1t$ and $y_2=\frac{x_0}{2}\cos 47.9t$, So we have
${\omega }_1=52.1s^{-1}$
${\omega }_2=47.9s^{-1}$
and frequency of beats is given by $f=\frac{{\omega }_1-{\omega }_2}{2\pi }$ or $T=\frac{2\pi }{{\omega }_1-{\omega }_2}$
$\Rightarrow T=\frac{2\times \pi }{4.2}=1.49s\cong 1.5s$