Question

A particle moves along the $x$-axis according to the equation $x=10sin3\left(t\right)$. The amplitudes and frequencies of component SHMs are

• A. amplitude $\frac{30}{4},\frac{10}{4}$ ; 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

The correct option is B amplitude $\frac{30}{4},\frac{10}{4}$ ; frequencies $\frac{1}{2},\frac{3}{2}$

$x=10{sin}^{3}tsin\theta =3sin\theta -4{sin}^3\theta {sin}^3\theta =\frac{3sin\theta -sin3\theta }{4}x=10\left[\frac{3sint-sin3t}{4}\right]x=\frac{30}{4}sint-\frac{5}{2}sin3t$

From here, we can clearly see that its the combination of two SHM's

$x_1=\frac{30}{4}sintA=\frac{30}{4}\omega =1$

[comparing with $x=Asin\omega t$]

$2\pi =1f=\frac{1}{2\pi }{x}_2=\frac{5}{2}sin3tA=\frac{5}{2}\omega =32\pi f=3f=\frac{3}{2\pi }$