Question

If the instantaneous displacement of a wave is $y = 2(\sin 2\pi t + \sqrt{3}\cos 2\pi t)\,cm$, then the amplitude of the wave in $cm$ is

• A. $4$
• B. $3$
• C. $5$
• D. $2$
• E. $6$

Hint:

Amplitude of $a\sin\theta + b\cos\theta$ is $\sqrt{a^2 + b^2}$.

Answer 1

The Correct Option is A

Concept:
• Expression of form: \[ a\sin\theta + b\cos\theta = R\sin(\theta + \phi) \] where $R = \sqrt{a^2 + b^2}$

Step 1: Identify coefficients
\[ a = 2,\quad b = 2\sqrt{3} \]

Step 2: Amplitude
\[ R = \sqrt{2^2 + (2\sqrt{3})^2} \] \[ = \sqrt{4 + 12} = \sqrt{16} = 4 \] Final Conclusion:
\[ = 4 \]