Math Bicycle Question HARD STUFF?

Question

Really needed help with this one...

The pedals of a bicycle are mounted on a bracket that is 29.0cm above the ground. Each pedal is 16.5 cm from the bracket. Assume that the bicycle is pedaled at a rate of 12 cycles per minute.

a) Draw a graph showing the height of a pedal above the ground for a few cycles. Assume that the pedal starts at the topmost position at t=0

b) Write an equation for the function in part A

c) Use your equation to determine the heigh of the pedal after 5 seconds, 12 seconds, and 18 seconds

Answer 1

The motion of the pedals plotted on a graph with the x-axis marked as time will give a Sine wave.

h - final height.

H - height of amplitude of pedals.

Sin - Sine in rad/s.

f - frequency.

t - time of cycle.

Equation:

h = H Sin 2πft

Since you are starting at a maximum height add π/2.  Now the equation becomes:

Equation:

h = H Sin (2πft + π/2)

f = 12 cyc./ min.

f = 12 cyc./ 60 sec.    (÷ 60)

f = 0∙2 cyc. / 1 sec.

f = 0∙2 Hz.

Now the equation becomes:

h = H Sin (2π(0∙2)t + π/2)

h = H Sin (0∙4πt + π/2)

The maximum amplitude of the pedals is 16∙5 cm and the displacement of the height is 29∙0cm.

So now the equation becomes:

h = H Sin (0∙4πt + π/2)

h = 16∙5 Sin (0∙4πt + π/2) + 29

(a)

Since you are starting at a maximum value draw a Sine wave at it's maximum point (Cosine wave) at t = 0. Remember the curve is displaces by 29cm.

One complete cycle will have a time span of 0∙2 sec.

(b)

h = 16∙5 Sin (0∙4πt + π/2) + 29

(c)

After 5 seconds.

h = 16∙5 Sin (0∙4πt + π/2) + 29

h = 16∙5 Sin (0∙4π(5) + π/2) + 29

h = 16∙5 Sin (5/2 π) + 29

h = 16∙5 + 29

h = 45∙5 cm.  (For top pedal).

h = 45∙5 - 2(16∙5)

h = 12∙5 cm   (For bottom pedal).

After 12 seconds.

h = 16∙5 Sin (0∙4πt + π/2) + 29

h = 16∙5 Sin (0∙4π(12) + π/2) + 29

h = 16∙5 Sin (5∙3 π) + 29

h = -13∙34878... + 29

h = 15∙65121... cm.  (For top pedal).

The distance the top pedal comes down from the max. height will be the same distance the bottom pedal comes up by.

Height of bottom pedal = Lowest pedal heigth + Max.height - top pedal height.

Height of bottom pedal = 12∙5 + 45∙5 - 15∙65121...

Height of bottom pedal = 42∙34879...

After 18 seconds.

h = 16∙5 Sin (0∙4πt + π/2) + 29

h = 16∙5 Sin (0∙4π(18) + π/2) + 29

h = 16∙5 Sin (7∙7 π) + 29

h = -13∙34878041... + 29

h = 15∙65121959... cm.  (For top pedal).

Height of bottom pedal = Lowest pedal heigth + Max.height - top pedal height.

Height of bottom pedal = 12∙5 + 45∙5 - 15∙65121959......

Height of bottom pedal = 42∙348......cm.

Answer 2

qb

h(t) = 29.0 + 16.5cos (0.4tπ)

qc

h(5) = 29.0 + 16.5cos (0.4*5*π)

= 29 + 16.5cos (2π)

= 29 + 16.5*1

= 45.50 cm

h(12) = 29.0 + 16.5cos (0.4*12*π)

= 29 + 16.5cos (4.8π)

= 29 + 16.5*(-0.809017)

= 15.65 cm

h(18) = 29.0 + 16.5cos (0.4*18*π)

= 29 + 16.5cos (7.2π)

= 29 + 16.5*(-0.809017)

= 15.65 cm