Is my solution for the Bungee Tension Problem correct?

[norcal36]

Homework Statement

A block with mass M is hanging from a rubber cord, supported such that the cord is unstretched at length L0. The mass of the cord is m which is much smaller than M. The cord has spring constant K. If the block is released: Find the tension, length of cord and speed of wave at the lowest point.

Relevant Equations

V=sq.rt T/K
Sum of the forces
Conservation of energy

Hello again,
I did a sum of the forces at the top and bottom of the motion and got:
Top...mg=ma
Bottom...T-mg=ma
Got T=2mg when I substitute the mg in for ma at bottom.Then I assumed kx=T=2mg so x(length of cord) would equal 2mg/k
Then V of wave would be ...sq.rt2mg/k
Is it that simple? Feel like I'm missing something. Thank you for your help

 

[TSny]

Homework Equations:: V=sq.rt T/K

Check your notes. I believe that K here should be replaced by the mass per unit length of the cord.
(You can use the editing toolbar to get symbols such as the square root symbol √ )

I did a sum of the forces at the top and bottom of the motion and got:
Top...mg=ma
Bottom...T-mg=ma
Got T=2mg when I substitute the mg in for ma at bottom.

It appears to me that to get this result you have assumed that the value of ma at the top is the same as the value of ma at the bottom. This might be true. But it needs to be justified.

Another approach for finding the distance x that the mass M falls is to use energy concepts.

 

[norcal36]

You are right the equation I used is for another situation. However if I use the conservation of energy I get mgx=1/2kx^2...mg=1/2kx...kx=2mg...kx=T=2mg. Again I am assuming tension will be the equivalent to kx. The x would be the displacement of the "spring" which equals length. However I am still stuck on the velocity of the wave. thanks for your time again.

 

[TSny]

You are right the equation I used is for another situation. However if I use the conservation of energy I get mgx=1/2kx^2...mg=1/2kx...kx=2mg...kx=T=2mg. Again I am assuming tension will be the equivalent to kx.

OK.

The x would be the displacement of the "spring" which equals length.

x is how much the cord is stretched from its natural length. So, how would you express the total length of the cord when M is at the bottom?

Then, can you find an expression for the mass per unit length of the cord when it is at the bottom?

 

[norcal36]

I would express the total length by L0+x...?
Then with that I could use V=√T/m/L equation which then would give me V=√2g(L0+x)...This sound correct?

 

[TSny]

I would express the total length by L0+x...?

Yes

Then with that I could use V=√T/m/L equation which then would give me V=√2g(L0+x)...This sound correct?

Be sure to distinguish between the mass of the block and the mass of the cord.