Question

Functions/Trig help please!

The piston in a lawn mower motor moves up and down 30 times in 1 s. Its total travel from the top position to the bottom position is 10cm. Vertical position y, in centimetres, versus time, t, in seconds can be modelled using a sinusoidal function.

a) Write an equation to model the motion of the piston. (I have 5 for amplitude, which do I use Cos/Sin?, How do I get the compression/stretch factor?)

b) Is this the only possible equation? Justify your answer.

Answer 1

y=5cos(1/30*t)

Answer 2

You could have a bunch of equations, because you could start the piston at t=0 at many different places ie top bottom middle

Answer 3

The answer given is y = 5Sin(60*pi*t) . How did they get that?

Answer 4

In particular, how'd they get 60*pi from the equation?

Answer 5

30*2*pi
from equation of omega or w

Answer 6

w=2*pi*f
f=(30 times per second)
w=60*pi

Answer 7

5 for the amplitude is correct,
it does 30 cycles/periods in 1sec, so it takes 1/30 of a second to do 1 period

keep in mind that regular period for sine will be \(\bf 2\pi\) so for the new period will be \(\bf \cfrac{1}{30}=\cfrac{2\pi}{\textit{new period}}\implies \textit{new period}=60\cdot 2\pi\)

Answer 8

bah, more typos

Answer 9

\(\bf \cfrac{1}{30}=\cfrac{2\pi}{\textit{new period}}\implies \textit{new period}=30\cdot 2\pi\)

Answer 10

I haven't learned the equation of omega before... Clears things up for me now, thanks so much!

Answer 11

and yes, you could use sine or cosine equally, depending on where the piston starts off

Answer 12

\(\bf y = A \quad sin( \color{red}{B}x + C ) + D\\
\cfrac{1}{30}=\cfrac{2\pi}{\color{red}{\textit{B}}}\implies \color{red}{\textit{B}}=30\cdot 2\pi\)

Answer 13

since we know the period is 1/30, I meant, thus the factor B will be that