Question

maria's bicycle has a diameter of 64 cm. as she rides at the speed of 21.6 km/h, she picks up a stone in her tire. draw a graph that shows the stone's height above the ground as she continues to ride at this speed for 2 s more.

can someone please help me find the period for this graph - how long it takes to complete one cycle?
thanks!

Answer 1

i don't think this is a cycloid
because i'm only in grade 11 math and have never heard that term before
this is from the "periodic function and their properties" unit

so, is it possible to give me an answer related to that?
thank you

Answer 2

Thank you so much!

Answer 3

You are dealing with a cycloid

A good start is Wikepedia, which has a nice illustration of the movement of the stone
http://en.wikipedia.org/wiki/Cycloid

Wolfram also has a good graphic.
http://mathworld.wolfram.com/Cycloid.html

they both show the equation.
careful, you have km/h and then cm and seconds, so you will have to conversions to do

Answer 4

Well, it did ask you to draw the graph, and for that you can follow the path shown in the links I gave you.

For the period we simply have to know how long it takes for the wheel to make one rotation, that is, how long it takes for the stone to go from ground-level to ground-level.

The diameter of the wheel is 64π cm
it moves at 21.6 km/h
= 2160000/3600 cm/s
= 600 cm/s

since time = distance / rate
time to go 64π
= 64π/600 seconds
= .335 seconds

So roughly every 1/3 second it would complete what looks like a halfcircle.

since 2sec / .335 = appr 5.97
for a 2 second stretch of time, I would draw 6 of those loops.

for the first loop,
at t = 0, height = 0
at t = .335(1/4) , height = 32 , 1 quarter of rotation
at t = .335/2, height = 64 , half a rotation
at t = .335(3/4) , height = 32 , three-quarters of a rotation
at t = .335 , h = 0 , full rotation
draw a smooth curve to join the points.
etc.