The cable of the 1900 kg elevator cab shown the figure snaps when the cab is at
ID: 2256553 • Letter: T
Question
The cable of the 1900 kg elevator cab shown the figure snaps when the cab is at rest at the first floor, where the cab bottom is a distance
above a spring of spring constant
A safety device clamps the cab against guide rails so that a constant frictional force of 4.3 kN opposes the cab's motion. (Assume that the frictional force on the cab is negligible when the cab is stationary.)
The cable of the 1900 kg elevator cab shown the figure snaps when the cab is at rest at the first floor, where the cab bottom is a distance d = 4.3 m above a spring of spring constant k = 0.18 MN/m. A safety device clamps the cab against guide rails so that a constant frictional force of 4.3 kN opposes the cab's motion. (Assume that the frictional force on the cab is negligible when the cab is stationary.) Find the speed of the cab just before it hits the spring. Find the maximum distance x that the spring is compressed (the frictional force still acts during this compression). Find the distance that the cab will bounce back up the shaft in meters. Using conservation of energy, find the approximate total distance that the cab will move before coming to rest in meters.Explanation / Answer
4400N acting upwards
18000N (1800kg * 10m/s/s) acting downwards
Resultant force downwards = 18000-4400=13600N; call this "mg". This just means instead of 10m/s/s as g, it is effectively reduced to 7.55m/s/s
Uginitial = Us + Ugfinal
"mg"hinitial = 0.5kx^2 + "mg"hfinal
13600(3.7 + natural length of spring, L) = 0.5(150000)x^2 + 13600(L - x)
50320 + 13600L = 75000x^2 + 13600L - 13600x
50320 = 75000x^2 - 13600x
0 = 75000x^2 - 13600x - 50320
x = 0.91477...m using quadratic formula (other answer is negative) <--- part 1
Us = Ug
0.5kx^2 + "mg"hinitial = "mg"hfinal
0.5*150000*0.91477^2 + 13600(L) = 13600(h + L)
62760.3 = 13600h
h = 4.614m (which seems odd, but the spring is very "propelling" - either that or I've made an error somewhere, which is likely)
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