String waves on a cable car and free falling.

[Gravitino22]

Homework Statement

You're riding a cable car from Bogota up to Monser-
rate. During your ride there is a small earthquake that
sends vertical, transverse waves propagating along the
cable. The cable's tension is T, its linear mass den-
sity is u, and the wavelength of the wave is \lambda. If the
amplitude of the wave is large enough, the motion of
the cable car will momentarily leave you in free fall.
Show that in order for this to happen the amplitude
must be given by

Y => \frac{g\lambda^{2}u}{4\pi^{2}T}

Homework Equations

position along a wave: y=Ysin(kx-wt)

string waves speed: v = \sqrt{\frac{T}{u}}

The Attempt at a Solution

My train of thought is that for free fall to happen the acceleration of the wave must be greater than the acceleration of gravity.

I tried taking the second partial derivate with respect to time of the waves position equation to obtain its acceleration which gave me:

a_{y}=-w^{2}Ysin(kx-wt) that should be => then g

but i don't know where the tension would or the linear mass density would come in.


Any help pointing me in the right direction would be appriciated! thanks :)

 

[willem2]

a_{y}=-w^{2}Ysin(kx-wt) that should be => then g

but i don't know where the tension would or the linear mass density would come in.

try to compute k and w.

 

[Gravitino22]

w = 2\pif where f is the angular frequency and k =\frac{2\pi}{\lambda}

that would give you sin(2\pi(\frac{x}{\lambda} - ft)) by factoring a 2pii inside... still not seeing how the sine term is somehow related to the tension and linear mass density =/

 

[willem2]

the frequency depends on the wave speed and the wavelength

 

[Gravitino22]

ok i made the following substitutions:

f=\frac{v}{\lambda} and made v=\frac{x}{t} (iam unsure about this one)


so far i have this

g<=-\frac{4\pi^{2}x^{2}Y}{\lambda^{2}t^{2}}sin(2\pi) the sin term is 1.

 

[Gravitino22]

wait wait! i have it just substitute back for v^{2} and which equals T/u and viola!


tho there's a small problem...the negative sign...


Thanks a lot tho man!

 

[Gravitino22]

nm! i don't have it now...iam stupid sine of 2pii is not 1 lol that sine term still bothering me...somehow i know i have to make the stuff inside the sine equal to pii/2.

 

[willem2]

nm! i don't have it now...iam stupid sine of 2pii is not 1 lol that sine term still bothering me...somehow i know i have to make the stuff inside the sine equal to pii/2.

you already gave an equation for v.

For the amplitude you can just take the maximum value of sin() which is 1

 

[Gravitino22]

hmm yea that's what i was thinking...Thanks a lot :)