Question

A particle moves along the $X-axis$ according to the equation $x={\sin }^3\left(\pi t\right)$ . The amplitudes and frequencies of components $SHMs$ are

• A. amplitude $\frac{40}{4},\frac{10}{4}$; frequencies $\frac{3}{2},\frac{1}{2}$
• B. amplitude $\frac{30}{4},\frac{10}{4}$; frequencies $\frac{1}{2},\frac{3}{2}$
• C. amplitude $10,10$; frequencies $\frac{1}{2},\frac{1}{2}$
• D. amplitude $\frac{30}{4},10$; frequencies $\frac{3}{2},2$

Answer 1

Answer: (B)

amplitude $\frac{30}{4},\frac{10}{4}$; frequencies $\frac{1}{2},\frac{3}{2}$

1) We have

x-coordinate of particle : $x=10{\sin }^3\left(\pi t\right)$

$x=10\left(\frac{3\sin \left(\pi t\right)-\sin \left(3\pi t\right)}{4}\right)$

$=\frac{30}{4}\sin \left(\pi t\right)-\frac{10}{4}\sin \left(3\pi t\right)$

2) Hence, a particle have two frequencies and their corresponding amplitudes

corresponding Amplitude are coefficients of the respective linear terms of a sine function.

$\to$ coifficients are $\frac{30}{4},\frac{10}{4}$ and respective angular frequency are $\pi$ and $3\pi$.

So frequency is $\frac{1}{2}$ and $\frac{3}{2}$