Explanation Given: 1) The curve is traced by a point at a distance h from the centre of the circle. 2) The radius of circle is R which is rolling along the x - axis. 3) h < R 4) Need to prove that the generated curve has following parametric equation x = R t − h sin t , y = R − h cos t Fig: 22 Curtate cycloid Formula used: 1) Length of arc = angle * radius 2) cos θ = B a s e H y p o t e n u s e 3) sin θ = P e r p e n d i c u l a r H y p o t e n u s e 4) sin ( π − θ ) = sin θ 5) cos ( π − θ ) = − cos θ Calculation: Fig:22(i) Let’s say O be the point at the distance h from center of the circle. In fig 22(i), a is initial position of rolling circle, b is any position of the circle at angle t Referring to position b in fig 22(i). Extended line A C and a line Perpendicular to y -axis from point O which meets at point I. Evaluating y co-ordinate of point O . Referring to fig 22(i) position b we can say that value of I A is y co-ordinate of point O . And I A = I C + C A [refer fig 22(i) position b ] I A = I C + R … ( 1 ) [since C centre of circle and a is point on x -axis, C A = R ] Calculating I C from Δ O C I ∠ O C I = π − t O C = h I C O C = cos ( π − t ) [using formula cos θ = B a s e H y p o t e n u s e ] I C h = cos ( π − t ) I C = h cos ( π − t ) I C = − h cos t [using formula cos ( π − θ ) = − cos θ ] Placing value of I C in equation ( 1 ) I A = R − h cos t Hence y = R − h cos t [since I A represents y co-ordinate of point O ] Now evaluating x co-ordinate of point O ...

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Chapter 12.1, Problem 83E

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Prove that a curtate cycloid which is the curve traced by a given point on a circle has given parametric equation.

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Chapter 12.1, Problem 82E

Chapter 12.1, Problem 84E

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Chapter 12 Solutions

CALCULUS 4E (LL) W/ ACHIEVE ACCESS

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

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Prove that a curtate cycloid which is the curve traced by a given point on a circle has given parametric equation.

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Prove that a curtate cycloid which is the curve traced by a given point on a circle has given parametric equation.

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