Answer
(a) $y_A(t)=sin(200\pi t)$, $y_B(t)=sin(200\pi t+ \frac{3\pi}{4})$
(b) No, $\frac{3\pi}{4}$
Work Step by Step
Identify the given quantities as $|a|=r_A=r_B=1ft, f=f_A=f_B=100Hz, y_A(0)=0, y_B(0)=\frac{\sqrt 2}{2}r=\frac{\sqrt 2}{2}$
(a) For fan A, model the height of red dot with a sine function $y_A(t)=a\cdot sin(\omega t)$ which satisfies the initial condition of $y_A(0)=0$. With $\omega=2\pi f=200\pi$, we have $y_A(t)=sin(200\pi t)$
For fan B, model the height of red dot with a sine function $y_B(t)=a\cdot sin(\omega t-b)$. To satisfy the initial condition of $ y_B(0)=\frac{\sqrt 2}{2}$ moving downwards, we have the phase $b=-\frac{3\pi}{4}$. Thus we have $y_B(t)=sin(200\pi t+ \frac{3\pi}{4})$
(b) No, the fans are not rotating in phase because they have a phase difference of $\frac{3\pi}{4}$. Clearly if we rotate fan A counterclockwise by an angle of $\frac{3\pi}{4}$, the two fans will rotate in phase.
