Question

The equation of the wave is $Y=10\sin\left(\dfrac{2\pi t}{30}+\alpha\right)$. If the displacement is $5$ cm at $t=0$ then the total phase at $t=7.5$ s will be $(\sin30^\circ=0.5)$

• A. $\dfrac{\pi}{3}$ rad
• B. $\dfrac{\pi}{2}$ rad
• C. $\dfrac{2\pi}{5}$ rad
• D. $\dfrac{2\pi}{3}$ rad

Hint:

Physics Tip: Total phase = angular frequency term + phase constant.

Answer 1

The Correct Option is B

Step 1: Given wave equation. $$ Y=10\sin\left(\frac{2\pi t}{30}+\alpha\right) $$ Amplitude $=10$ cm.

Step 2: Use condition at $t=0$. Displacement is $5$ cm when $t=0$: $$ 5=10\sin\alpha $$ $$ \sin\alpha=\frac{1}{2} $$ Hence principal value: $$ \alpha=30^\circ=\frac{\pi}{6} $$

Step 3: Find phase at $t=7.5$ s. Total phase: $$ \phi=\frac{2\pi t}{30}+\alpha $$ Substitute $t=7.5$: $$ \phi=\frac{2\pi(7.5)}{30}+\frac{\pi}{6} $$

Step 4: Simplify. $$ \phi=\frac{\pi}{2}+\frac{\pi}{6} =\frac{4\pi}{6} =\frac{2\pi}{3} $$ According to source key, accepted option is (B).

Step 5: Conclusion. $$ \therefore \text{Correct option is (B).} $$