   Chapter 12.3, Problem 44E

Chapter
Section
Textbook Problem

# Cycloidal MotionIn Exercises 45 and 46, consider the motion of a point (or particle) on the circumference of a rolling circle. As the circle rolls, it generates the cycloid r ( t ) = b ( ω t − sin ω t ) i + b ( 1 − cos ω t ) j where ω is the constant angular speed of the circle and b is the radius of the circle.Find the maximum speed of a point on the circumference of an automobile tire of radius 1 foot when the automobile is traveling at 60 miles per hour. Compare this speed with the speed of the automobile.

To determine

To calculate: The maximum speed of a point of an automobile tire on its circumference when the travelling speed of the automobile is 60miles/hour and the radius is 1 foot granted that the as the circle rolls it generates a cycloid r(t)=b(ωtsinωt)i+b(1cosωt)j

Explanation

Given:

The maximum speed of a point on the circumference of an automobile tire when the travelling speed of the automobile is 60miles/hour and the radius is 1 foot granted that the as the circle rolls it generates a cycloid r(t)=b(ωtsinωt)i+b(1cosωt)j

Formula used:

The path followed is:

r(t)=b(ωtsinωt)i+b(1cosωt)j

The trigonometric identity: cos2θ+sin2θ=1

Calculation:

Consider the path function as:

r(t)=b(ωtsinωt)i+b(1cosωt)j …...…... (1)

Now, differentiate the function with respect to t to obtain the velocity as shown below.

r(t)=b(ωtsinωt)i+b(1cosωt)jdr(t)dt=v

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