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Hendrick
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Homework Statement
“Geologist A” at the bottom of a cave signals to his colleague “Geologist B” at the surface by pushing a 11.0 kg box of samples from side to side. This causes a transverse wave to propagate up the 77.0 m rope. The total mass of the rope is 14.0 kg. Take g = 9.8 m/s².
How long does it take for the wave to travel from the bottom of the cave to the surface?[Hint: Find an analytic expression v(z) for the wave speed as a function of distance. Then use the fact that at any given point on the rope the time dt taken to travel a small distance dz is given by: dt=dz/v(z). Then integrate to obtain the total travel time. ]
http://oasis.phy.auckland.ac.nz/oasis/a/question/187416/image.gif
Homework Equations
u = mR/z
T(z) = u.z.g + mB.g
v(z) = (T(z)/u)^(1/2)
dt=dz/v(z)
z = the length of the rope = L (used for integrating)
The Attempt at a Solution
v(z) = (T(z)/u)^(1/2)
v(z) = ((mR/z).z.g + mB.g/(mR/z))^(1/2)
v(z) = ((mR.g + mB.g)/(mR/z))^(1/2)
v(z) = ([(mR.g)/(mR/z)] + [(mB.g)/(mR/z)])^(1/2)
v(z) = ([(mR.g.z)/mR] + [(mB.g.z)/mR])^(1/2)
v(z) = ([mR.g.z] + [(mB.g.z)/mR])^(1/2)
dt=dz/v(z)
dt=dz/([mR.g.z] + [(mB.g.z)/mR])^(1/2)
Integration:
...f L
t= | (1/([mR.g.z] + [(mB.g.z)/mR])^(1/2)).dz
...j 0
...f L
t= | 2.([mR.g.z^2/2] + [(mB.g.z^2)/mR.2])^(1/2).dz
...j 0
I think I integrated it properly but when substituted the values
mB = mass of box
mR = mass of rope
g = 9.8 ms^2
z = 77.0 m
I didn't get the correct answer of t = 2.52s
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