Question

So, I am trying to make parametric equations for the seconds-hand on a clock face. The clock face has a radius of 17cm. So far I have,

x=17cos(t), y=17sin(t)

What I am not sure how to do is to create a formula (period I believe) that will allow me to give xy coordinates for every second.

Answer 1

I recall that the typical period of sin/cos is 2pi, I also know that I need 60 subdivisions of the circle. Do I I just express this as 2pi/60?

Answer 2

Ok I check a few values and that seems to make sense. Since the seconds-hand should start at the 12 o clock position, I need to start at t=30pi/60, and I need it to go clockwise. Which Im not sure how to do. Any suggestions?

Answer 3

Remember that the argument of the sin and cosine have to be unitless, so you need something to multiply by t in order to cancel out the units.

You can have t as just the number of seconds, and multiply it by some angular frequency omega.

\[x=17\cos(\omega t+\delta_1) \quad ; \quad y = 17\sin (\omega t + \delta_2)\]

To get the correct placement of the hands, you need to start both x and y off with a phase shift, delta

Since the x coordinate is 0 at t=0, we can solve for the phase shift with
\[ x(0)=17\cos(\omega * 0 + \delta_1) = 0\]

similarly for y, at t=0 y is at its max displacement


\[y(0)= 17 \sin (\omega * 0 + \delta_2)=17\]


As theta increases in the positive direction, it goes anti clockwise - so there needs to be a negative angular frequency for the clock to work properly. We can solve for it knowing that every 60 seconds the clock must travel negative 2 pi radians

\[ \omega *60s = -2 \pi \]

Hope this helped ^_^ (If you even still needed help with it!)