Sinusoidal Functions: a pebble stuck in a tire's tread

J

Jaclyn

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As you stop your car at a traffic light, a pebble becomes wedged between the tire treads. When you start off, the distance of the pebble from the pavement varies sinusoidally with the distance you have traveld. The period is, of course, the circumference of the wheel. Assume that the diameter of the wheel is 24 inches
a. Sketch a graph of the function.
b. Write an equation of this function.
c. Predict the distance from the pavement when you have gone 15 inches.
The computatuin will be easier if you do not transform the radian to degree measures, since the constant B in the equation is not a multiple of pie.
 


MY EDIT: I misunderstood the excercise (even when goofing off), so I deleted my contribution.

 
Assuming the pebble begins at the origin and the tire rolls along a straight line, we have a cycloid with parametric equations [B}: x= 12[?-sin(?)] and y = 12{1-cos(?)]

[C]. when x = 15 , ?-sin(?) = 5/4, ? = 2.10876 radians = 120.823°.
Hence y (the distance from the pavement) = 12[1-cos(2.10876)] = 18.1487in.
 
I assume the instructor made a mistake by defining the path to be sine curve.
 


I do not assume a constant linear velocity. We are told that the height function is sinusoidal with a period of 24 Pi. To acheive this function, the linear velocity must be non-constant.

Heh, heh. :wink:

 
Re:

mmm4444bot said:


I do not assume a constant linear velocity. We are told that the height function is sinusoidal with a period of 24 Pi. To acheive this function, the linear velocity must be non-constant.

Heh, heh. :wink:

Click to expand...

For a cycloid - only assumption needed is there is no slip - that is -

\(\displaystyle \frac{d\theta}{dt} \, = \, \omega \, = \, \frac{v}{r}\)

there is no need to assume 'v' to be constant.
 
The velocity of the car is immaterial as when the wheel has traveled 15in., regardless of its velocity, the height of the pebble above the pavement is 18.1487 in.
 


OIC. I was thinking with time as the independent variable in the height function. DOH!

It seems to me that the velocity could be made to produce a sinusoidal height function with respect to time.

(Another joke gone bad.) :oops:

 
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