A common device for entertaining a toddler is a jump seat that hangs from the ho

Question

A common device for entertaining a toddler is a jump seat that hangs from the horizontal portion of a doorframe via elastic cords (the figure). Assume that only one cord is on each side in spite of the more realistic arrangement shown. When a child is placed in the seat, they both descend by a distance ds as the cords stretch (treat them as springs). Then the seat is pulled down an extra distance dm and released, so that the child oscillates vertically, like a block on the end of a spring. Suppose you are the safety engineer for the manufacturer of the seat. You do not want the magnitude of the child's acceleration to exceed 0.18g for fear of hurting the child's neck. If dm = 13 cm, what value of ds corresponds to that acceleration magnitude?

Explanation / Answer

Restoring force, F = -kx

=> ma = -kx

=> d2x/dt2 = -(k/m)x

The solution is of the form x = Asint

with A = dm = 13 cm and = (k/m)1/2

Now, kds = mg

=> k/m = 2 = g/ds

Maximum acceleration, amax = A2 = 0.18g

=> dm * (g/ds) = 0.18g

=> ds = dm / 0.18 = 13 / 0.18 = 72.2 cm

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