Help? Your fishing bobber oscillates up and down from the current in the river in a harmonic motion. The bobber moves a total of 2.5 inches from its high point to its low point then returns up to the high point every 3 seconds. Write an equation modeling the motion of the bobber at its high point at time t = 0.

Question

Help? Your fishing bobber oscillates up and down from the current in the river in a harmonic motion.  The bobber moves a total of 2.5 inches from its high point to its low point then returns up to the high point every 3 seconds.  Write an equation modeling the motion of the bobber at its high point at time t = 0.

anonymous · Answer

the equation for a simple harmonic motion can be represented by a Sine function... that is
$$f(t)=A \sin \omega t$$
where
A is the amplitude of oscillation
omega w is the angular frequency of oscillation
$$\omega=2 \pi f$$
f is the frequency of oscillation

anonymous · Answer

in your problem, what do you think is A? it says, "The bobber moves a total of 2.5 inches from its high point to its low point..."

deadshot · Answer

So would A be 2.5 or 1.25?

anonymous · Answer

the amplitude is always from maximum point to minimum point...

deadshot · Answer

so then the amplitude would be 2.5

anonymous · Answer

yes...

deadshot · Answer

so that leaves us with $$f(x)= 2.5\sin wt$$

anonymous · Answer

we don't use x here, we use the time t as the abscissa...

deadshot · Answer

oh

deadshot · Answer

so then It's $$f(t)=2.5\sin wt$$

anonymous · Answer

given also the time it takes the bobber returns from low to high for time of 3 seconds, a complete period of a wave is taken from maximum to maximum (or from minimum to minimum)...
so one (1) complete period of a wave is 6 seconds...
knowing the period of a wave, we can determine the frequency of oscillation f...
$$f=\frac{ 1 }{ period }$$

anonymous · Answer

what then would be your frequency f?

deadshot · Answer

$$f=\frac{ 1 }{ 6 }$$

anonymous · Answer

after knowing f you can now compute your angular frequency w.... then complete your equation... at any time t....

anonymous · Answer

btw according to problem, "high point at time t = 0"... meaning if t=0, f(0) = 2.5 inches...
so instead of using Sine, we will use Cosine function... because the sin (0) = 0 while cos (0) =1...

anonymous · Answer

so our equation will be...
$$f(t) = 2.5 \cos 2 \pi \frac{ 1 }{ 6 }t$$

anonymous · Answer

once you set t=0; f(0) = 2.5 inches.... :)

deadshot · Answer

Wait it says that the time it takes to go from high point down to low point, and then back up to high point is 3 seconds, so shouldn't the function be $$f(t)=2.5\cos2\pi \frac{ 1 }{ 3 }t$$

anonymous · Answer

oh yes i misunderstood the phrase in the sentence... it is 3 seconds not 6 seconds... sorry... :) thanks for the correction....

deadshot · Answer

No problem. Thanks for the help!