a. Explanation Given: The descent time between any two points 0 ≤ a < b ≤ π on the curve is ∫ a b 1 − cos t g ( cos a − cos t ) d t . Calculation: From the given, the descent time ∫ a b 1 − cos t g ( cos a − cos t ) d t . where g is the acceleration due to gravity, t = 0 corresponds to the top of the wire, and t = π corresponds to the lowest point on the wire. Compute the integral, ∫ a b 1 − cos t g ( cos a − cos t ) d t = ∫ a b ( 1 − cos t ) ( 1 + cos t ) g ( cos a − cos t ) ( 1 + cos t ) d t = ∫ a b 1 2 − cos 2 t g ( cos a − cos t ) ( 1 + cos t ) d t = ∫ a b sin 2 t g ( cos a − cos t ) ( 1 + cos t ) d t ∫ a b 1 − cos t g ( cos a − cos t ) d t = sin t ∫ a b 1 g ( cos a − cos t ) ( 1 + cos t ) d t (1) Let u = cos t (2) Then d u = − sin t d t Obtain the lower limit, Substitute a for t in equation (2), u = cos a Thus, the lower limit of u is cos a . Obtain the upper limit, Substitute b for t in equation (2), u = cos b Thus, the upper limit of u is cos b . Substitute u = cos t and d u = − sin t d t in equation (1), ∫ a b 1 − cos t g ( cos a − cos t ) d t = − 1 g ∫ cos a cos b 1 ( cos a − u ) ( 1 + u ) d u (3) Simplify the term ( cos a − u ) ( 1 + u ) , ( cos a − u ) ( 1 + u ) = cos a + u cos a − u − u 2 = cos a + u ( cos a − 1 ) − u 2 = cos a − ( u 2 − u ( cos a − 1 ) ) = cos a − ( u 2 − u ( cos a − 1 ) − ( cos a − 1 2 ) 2 + ( cos a − 1 2 ) 2 ) That is, ( cos a − u ) ( 1 + u ) = cos a + ( cos a − 1 2 ) 2 − ( u − cos a − 1 2 ) 2 = 4 cos a + ( cos a − 1 ) 2 4 − ( u − cos a − 1 2 ) 2 = 4 cos a + ( cos 2 a + 1 − 2 cos a ) 4 − ( u − cos a − 1 2 ) 2 = ( cos a + 1 2 ) 2 − ( u − cos a − 1 2 ) 2 From equation (3), ∫ a b 1 − cos t g ( cos a − cos t ) d t = − 1 g ∫ cos a cos b 1 ( cos a − 1 2 ) 2 − ( u − cos a − 1 2 ) 2 d u (4) Let v = u − cos a − 1 2 (5) Obtain the lower limit, Substitute cos a for u in equation (5), v = cos a − cos a − 1 2 = 2 cos a − cos a + 1 2 = cos a + 1 2 Thus, the lower limit of v is cos a + 1 2 b. To determine To show: The descent time is the same for all values of a when b = π .

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ISBN: 9781323178522

Author: Briggs

Publisher: PEARSON

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Question

Chapter 7.4, Problem 83E

To determine

To find: The descent time on the interval [a,b].

To determine

To show: The descent time is the same for all values of a when b=π.

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Chapter 7.4, Problem 82E

Chapter 7.4, Problem 84E

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The descent time on the interval [ a , b ] .

Solution Summary: The author explains the descent time on the interval left[a,bright].