Simple Harmonic Motion 5 Question 12
17. Two simple harmonic motions are represented by the equations $\quad y _1=10 \sin (3 \pi t+\pi / 4) \quad$ and $y _2=5(\sin 3 \pi t+\sqrt{3} \cos 3 \pi t)$. Their amplitudes are in the ratio of …… .
$(1986,2 M)$
Match the Columns
Show Answer
Answer:
Correct Answer: 17. $1: 1$
Solution:
- $A _1=10$ (directly)
For $A _2: y _2=5 \sin 3 \pi t+5 \sqrt{3} \cos 3 \pi t$
$$ =5 \sin 3 \pi t+5 \sqrt{3} \sin 3 \pi t+\frac{\pi}{2} $$
i.e. phase difference between two functions is $\frac{\pi}{2}$, so the resultant amplitude $A _2$ can be obtained by the vector method as under
$$ \begin{aligned} & A _2 & =\sqrt{(5)^{2}+(5 \sqrt{3})^{2}}=10 \\ \therefore \quad & \frac{A _1}{A _2} & =\frac{10}{10}=1 \end{aligned} $$
