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Mathematical Modeling

Mathematical Modeling
Homework 7 92.450 / 550 Math Modeling Due July 13
1. In class we showed that solution for the free slinky is
u(x ,t ) = 1
2
g t 2 +
1
2
f (x + c t ) + 1
2
f (x − c t ) where c = L
Æ
k/M .
(a) Find u(x ,t ) in the case f (x ) =U (x ) where U is the static solution U (x ) =
M g
k L

x −
x
2
2L

.
(b) For which x , t is your answer in (a) valid? What happens if you put x = L?
(c) If the slinky is horizontal (so g = 0) and it is stretched uniformly, then f (x
) = h x , 0 ≤ x ≤ L and
u(x ,t ) =



h c t , 0 ≤ x < c t h x , c t ≤ x ≤ L − c t h(L − c t ), L − c t < x ≤ L. Use this solution to describe what is happening to the slinky. 2. A man of mass m jumps from a bridge attached to a light bungee cord of length L and stiffness k. Under the influence of gravity g he falls to a maximum distance H below the bridge. Let x be the amount the cord is stretched and v the man’s downward velocity at time t . According to Hooke’s Law the upward force due to the spring is k x and Newton’s Second Law gives m d 2x dt 2 = mg − k x (1) where g is the acceleration due to gravity. (a) Show d 2x dt 2 = v dv dx . (b) Solve (1) after making the substitution in (a). (c) Use v = Æ 2g L when x = 0 to determine the constant of integration in (b). (d) At the lowest point, v = 0. Use this to find H . 92.550 students (extra credit for 92.450 students) Suppose at rest the man stretches a length D of cord by an amount d . Show H = L r + q r 2 − 1 where r = 1 + d /D . Hint. Hooke’s law and r are enough to give you mg /k. Bonus for all students Show the equation for the pinned slinky reduces to (1) in the case M = 0. PLACE THIS ORDER OR A SIMILAR ORDER WITH US TODAY AND GET A GOOD DISCOUNT ?

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