A greedy shift superintendent allows the coal car to be overloaded in order to i

Question

A greedy shift superintendent allows the coal car to be

overloaded in order to increase the amount of coal transferred on

each trip down the incline. he also increased the coal car speed to

5 m/s to shorten the time level.

a) how much does the coal car cable stretch if the total mass of

the coal car is 200,000 kg?

b) the cable breaks just as the the overloaded coal car passes

the empty coal car at the mid-point of the slope. The bottom edge

of the track is 9.8m above the river water level. How far out into

the river does the fully loaded coal car travel? how far out into

the river does the empty coal car travel.

c) The overloaded coal car sinks. Determine if the empty coal

car floats or sinks. assume the car is water tight.

d) what is the speed of a traveling wave on the cable just prior

to breaking? How high would a properly loaded coal float?

coal car facts- depth = 4m. width = 4m, length = 10m,

empty coal car = 58,800 N

Cable facts- young modulus = 2x10^11 N/m^2, diameter =

3cm, linear desity = 2kg/m, water density = 1000 kg/m^3

Explanation / Answer

coal car speed is v = 5 m/s

mass of coal car is M = 200,000 kg

young's modulus of cable Y = 2 * 10^11 N/m^2

length of cable l_o = [(980)^2 + (200)^2]^1/2 m

let the change in length be dl

we know that

Y = [(F/A)/(dl/l_o)]

or Y = (F/A) * (l_o/dl)

or dl = (F/A) * (l_o/Y)

where F = Mg,g = 9.8 m/s^2,A = pi * d^2/4,d = 3 cm = 3 * 10^-2 m

let a be the acceleration of the loaded coal car

we know that

v^2 - u^2 = 2a(l_o/2) = al_o

or a = (v^2 - u^2/l_o)

where u = 0

Also,

v = u + at

where t is time

or v = at

or t = (v/a)

distance travelled by loaded coal car is

S = ut + (1/2)at^2

pressure exerted by empty coal car is

P = pgh

where p is density of water(1000 kg/m^3) and h is distance travelled by coal car in water

or h = (P/pg)

where P = (w/S),w = 58,800 N,S = 4 * 10 m^2

speed of travelling wave on cable just before breaking is

v1 = (l_o/t)

height of coal which floats is

h1 = v1t + (1/2)at^2

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