Question 360: Waves

Question

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

Accepted Answer

The given wave equation is:
$$\mathrm{Y}=10 \sin \left(\frac{2 \pi \mathrm{t}}{30}+\alpha\right)$$
At  t = 0 , the displacement  Y  is given as 5 cm. So, let's find the phase $ \alpha $ using this information :
$$5 = 10 \sin(\alpha)$$
$$\frac{5}{10} = \frac{1}{2} = \sin(\alpha)$$
Since $ \sin(30^\circ) = \frac{1}{2} $, we can say that $ \alpha = 30^\circ $ or $ \alpha = \frac{\pi}{6} $ radians, considering that sine is positive in the first and second quadrants, and seeing that $ \alpha $ represents a phase shift, it usually takes the smallest positive angle that satisfies the equation.
Now, let's calculate the total phase at $ t = 7.5 $ s :
$$\mathrm{Total \ phase} = \frac{2 \pi t}{30} + \alpha$$
Plug in the values:
$$\mathrm{Total \ phase} = \frac{2 \pi \cdot 7.5}{30} + \frac{\pi}{6}$$
$$\mathrm{Total \ phase} = \frac{2 \pi \cdot 1}{4} + \frac{\pi}{6} = \frac{\pi}{2} + \frac{\pi}{6}$$
To add these fractions, we first find a common denominator :
$$\mathrm{Total \ phase} = \frac{3\pi}{6} + \frac{\pi}{6} = \frac{4\pi}{6}$$
Now we simplify the fraction :
$$\mathrm{Total \ phase} = \frac{2\pi}{3}$$
Therefore, the total phase at $ t = 7.5 $ s is $ \frac{2\pi}{3} $ radians, which corresponds to Option D.

Answer

$$\frac{\pi}{3} \mathrm{~rad}$$

Answer

$$\frac{\pi}{2} \mathrm{~rad}$$

Answer

$$\frac{2 \pi}{5} \mathrm{~rad}$$

Answer

$$\frac{2 \pi}{3} \mathrm{~rad}$$

Waves

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

Choose the correct answer: