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Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095

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BuyFindarrow_forward

Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095
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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Chapter 12 Solutions

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Sect-12.1 P-11ESect-12.1 P-12ESect-12.1 P-13ESect-12.1 P-14ESect-12.1 P-15ESect-12.1 P-16ESect-12.1 P-17ESect-12.1 P-18ESect-12.1 P-19ESect-12.1 P-20ESect-12.1 P-21ESect-12.1 P-22ESect-12.1 P-23ESect-12.1 P-24ESect-12.1 P-25ESect-12.1 P-26ESect-12.1 P-27ESect-12.1 P-28ESect-12.1 P-29ESect-12.1 P-30ESect-12.1 P-31ESect-12.1 P-32ESect-12.1 P-33ESect-12.1 P-34ESect-12.1 P-35ESect-12.1 P-36ESect-12.1 P-37ESect-12.1 P-38ESect-12.1 P-39ESect-12.1 P-42ESect-12.1 P-41ESect-12.1 P-40ESect-12.1 P-43ESect-12.1 P-44ESect-12.1 P-45ESect-12.1 P-46ESect-12.1 P-47ESect-12.1 P-48ESect-12.1 P-49ESect-12.1 P-50ESect-12.1 P-51ESect-12.1 P-52ESect-12.1 P-53ESect-12.1 P-54ESect-12.1 P-55ESect-12.1 P-56ESect-12.1 P-57ESect-12.1 P-58ESect-12.1 P-59ESect-12.1 P-60ESect-12.1 P-61ESect-12.1 P-62ESect-12.1 P-63ESect-12.1 P-64ESect-12.1 P-65ESect-12.1 P-66ESect-12.1 P-67ESect-12.1 P-68ESect-12.1 P-69ESect-12.1 P-70ESect-12.1 P-71ESect-12.1 P-72ESect-12.1 P-73ESect-12.1 P-74ESect-12.1 P-75ESect-12.1 P-76ESect-12.1 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P-10PSCh-12 P-11PSCh-12 P-12PSCh-12 P-13PSCh-12 P-14PS

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