Explanation Given: 1) Generating circle of cycloid has radius 1. 2) Need to prove tangent line at a point P on cycloid passes through top point of rolling circle. Formula used: 1) Point-slope formula states that at point ( x 1 , y 1 ) having slope m , equation of the line can be written as ( y − y 1 ) = m ( x − x 1 ) 2) Slope of the tangent line = d y d x 3) Chain rule states that d y d x = d y d t . d t d x 4) d d x ( k ) = 0 where k is any constant value 5) d d x ( cos x ) = − sin x 6) d d x ( sin x ) = cos x 7) d d x ( x ) = 1 8) ( a − b ) 2 = a 2 − 2 a b + b 2 9) sin 2 θ + cos 2 θ = 1 Calculation: Parametric equation of cycloid is represented by following equation x = r ( t − sin t ) and y = r ( 1 − cos t ) Where r = radius of circle generating cycloid t = angular displacement of the generating circle Let’s first evaluate co-ordinates of point P at t x P = r ( t − sin t ) and y P = r ( 1 − cos t ) Slope of tangent at P will be d y d x = d y d t . d t d x d x d t = d d t ( t − sin t ) d x d t = d d t ( t ) − d d t ( sin t ) d x d t = 1 − cos t [using formula d d x ( x ) = 1 and d d x ( sin x ) = cos x ] d y d t = d d t ( 1 − cos t ) d y d t = d d t ( 1 ) − d d t ( cos t ) d y d t = 0 − ( − sin t ) [using formula d d x ( k ) = 0 and d d x ( cos x ) = − sin x ] d y d t = sin t Slope of tangent at P is d y d x = sin t 1 − cos t Calculating co-ordinates of point O Assuming that circle started rolling at origin where centre of circle is ( 0 , ...

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

See similar textbooks

Concept explainers

Angles in Circles

Angles within a circle are feasible to create with the help of different properties of the circle such as radii, tangents, and chords. The radius is the distance from the center of the circle to the circumference of the circle. A tangent is a line made p…

Arcs in Circles

A circular arc is the arc of a circle formed by two distinct points. It is a section or segment of the circumference of a circle. A straight line passing through the center connecting the two distinct ends of the arc is termed a semi-circular arc.

Videos

Question

Chapter 12.1, Problem 82E

To determine

Need to prove that the tangent line at given point on the cycloid always passes through the top point on the unit circle generating the cycloid, as indicated in given fig.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 12.1, Problem 81E

Chapter 12.1, Problem 83E

Students have asked these similar questions

For the system consisting of the lines: and 71 = (-8,5,6) + t(4, −5,3) 72 = (0, −24,9) + u(−1, 6, −3) a) State whether the two lines are parallel or not and justify your answer. b) Find the point of intersection, if possible, and classify the system based on the number of points of intersection and how the lines are related. Show a complete solution process.

3. [-/2 Points] DETAILS MY NOTES SESSCALCET2 7.4.013. Find the exact length of the curve. y = In(sec x), 0 ≤ x ≤ π/4

H.w WI M Wz A Sindax Sind dy max Утах at 0.75m from A w=6KN/M L=2 W2=9 KN/m P= 10 KN B Make the solution handwritten and not artificial intelligence because I will give a bad rating if you solve it with artificial intelligence

Chapter 12 Solutions

CALCULUS (CLOTH)

Ch. 12.1 - Prob. 1PQ Ch. 12.1 - Prob. 2PQ Ch. 12.1 - Prob. 3PQ Ch. 12.1 - Prob. 4PQ Ch. 12.1 - Prob. 5PQ Ch. 12.1 - Prob. 6PQ Ch. 12.1 - Prob. 7PQ Ch. 12.1 - Prob. 1E Ch. 12.1 - Prob. 2E Ch. 12.1 - Prob. 3E

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

Need to prove that the tangent line at given point on the cycloid always passes through the top point on the unit circle generating the cycloid, as indicated in given fig.

Solution Summary: The author explains that the tangent line at a point P on cycloid always passes through top point of rolling circle.

Concept explainers

Videos

Need to prove that the tangent line at given point on the cycloid always passes through the top point on the unit circle generating the cycloid, as indicated in given fig.

Want to see the full answer?

Chapter 12 Solutions