Question

A particle moves along the X-axis according to the equation x = 10 sin^3 (πt). The amplitudes and frequencies of components SHMs are (A) amplitude 30/4, 10/4 ; frequencies 3/2, 1/2(B) amplitude 30/4. 10/4 ; frequencies 1/2, 312 C) amplitude 10, 1O ; frequencies 1/2, 1/2(D) amplitude 30/4. 10 ; frequencies 3/2, 2

Answer 1

Amplitude 30/4,10/4 ; Frequency : 1/2,3/2

Explanation:

1) We have,

 X-coordinate of particle :

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

2) Hence, Particle have two frequencies and their corresponding amplitudes.

Corresponding Amplitudes are coefficients of respective linear term of sine function.

Here, Coefficients of sine functions are 30/4,10/4 and respective angular frequencies are $\pi \& 3\pi$

So, Frequency is $\frac{\pi}{2\pi} \&\frac{\pi}{3 \pi}  =\frac{1}{2} \&\frac{3}{2}$.

So, Option B is the correct answer .