A flexible cable was hung across the Black Canyon between two points that were exactly 1 km apart and at the same elevation.
During the cool night the cable length was calculated to contract by .2 meters. The cable dip was actually measured to decrease by .2 meters!
What is the length of the cable after cooling?
clc,clearvars
for a=50:10:1000
y1=a*cosh(500/a);
y0=a*cosh(0);
dip=y1-y0;
fprintf('%17.7f ',a,dip)
fprintf('\n')
end
finds that a reasonable value for a in the formula a*cosh(x/a) is in the vicinity of a dip of 150, so I chose a dip of 150 vs 149.8 to start.
This was based on
a dip
700.0000000 186.2940554
710.0000000 183.4537072
720.0000000 180.7012800
730.0000000 178.0326639
740.0000000 175.4440048
750.0000000 172.9316850
760.0000000 170.4923051
770.0000000 168.1226673
780.0000000 165.8197610
790.0000000 163.5807488
800.0000000 161.4029544
810.0000000 159.2838509
820.0000000 157.2210508
830.0000000 155.2122961
840.0000000 153.2554495
850.0000000 151.3484869
860.0000000 149.4894891
870.0000000 147.6766360
880.0000000 145.9081991
890.0000000 144.1825370
900.0000000 142.4980887
910.0000000 140.8533697
920.0000000 139.2469667
930.0000000 137.6775335
940.0000000 136.1437872
950.0000000 134.6445043
960.0000000 133.1785172
for goal=([150 149.8])
low=850; % based on values of a for these dip values
high=870;
for t=1:50
delta=(high-low)/10;
for a=low:delta:high
y1=a*cosh(500/a);
y0=a*cosh(0);
dip=y1-y0;
disc=dip-goal;
if a>low+delta/2
if disc*prev<0
low=a-delta;
high=a;
delta=delta/10;
break
elseif disc==prev
break
end
end
prev=disc;
end
if abs(disc/goal)<1e-15
break
end
end
L=2*a*sinh(500/a);
disp([goal a disc L])
end
finding
dip Length
150 1057.67399485945
149.8 1057.52601628157
but to change the dip from 140 to 139.8:
140 1050.48422077374
139.8 1050.34481984767
or 160 vs 159.8:
160 1065.28650262173
159.8 1065.13018825188
showing differing final lengths and the length differences .148, .139, .156 meters.
After some manual searching for the appropriate difference in length that would lead to a .2 difference in dip I got
218.3 1117.44419062824
218.1 1117.24420021655
When the length shrinks from 1117.444 meters to 1117.244 meters, the dip decreases from 218.3 meters to 218.1 meters. So the length of the cable after cooling is 1117.244 meters.
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Posted by Charlie
on 2024-03-22 14:15:10 |
No Subject
