Answer to Question #145152 in Classical Mechanics for Ali Omer

Question

You are sitting on a surfboard that is riding up and down on some swells. The board’s vertical displacement is given by

Y = (1.2m)cos((t)(1/2s) +π/6)

Find amplitude, angular frequency, phase constant, frequency, and period of the motion.
Where is the surfboard at t = 1.0s?
Find the velocity and acceleration as functions of time t.
Find the initial values of the position, velocity, and acceleration of the surfboard.

Answer 1

given data

$y=1.2cos(\frac{1}{2}t+\frac{\pi}{6})$

comparing with

$y=Acos(\omega t+\phi)$

we get

amplitude =1.2 m

angular frequency=0.5 rad/s

phase=$\frac{\pi}{6}$

time period=$\frac{2\pi}{\omega}=4\pi second$

frequency=$\frac{1}{4\pi}=0.08HZ$

at t=1 s

$y=1.2cos(\frac{1}{2}+\frac{\pi}{6})=0.62m$

$v=\frac{dy}{dt}=-0.6sin(\frac{1}{2}t+\frac{\pi}{6})$

at t=0

$v=-0.3m/s^2$

and

$a=\frac{dv}{dt}=-0.3cos(\frac{1}{2}t+\frac{\pi}{6})\\at\space t=0s\\a=-0.25m/s^2$