How Far Will the Cord Stretch When a 2kg Block is Suspended?

[Ry122]

a cord with a spring constant of 100N/m has a 2.0kg block suspended from it.
the length of the cord when it is unstretched is .5m.
the block is released. determine the length of the cord when it is at the maximum length of elongation.

my attempt:
first determine the kinetic energy of the block just before the block reaches the point where
the cord begins to elongate beyond the length of .5m
it is after this point that kinetic energy then begins to be converted into potential spring energy in the cord.
it is this energy combined with the weight of the block that are going to determine the maximum length of elongation of the cord. this is because when all of the kinetic energy of the block is turned into potential spring energy the only remaining force that would cause the cord to further elongate would be the weight of the block pulling down on the cord.
First, to determine the extension of the spring due to the weight of the block
use f=kx
9.8 x 2 = 100x
x=0.196
then to determine the energy that this causes to be stored in the cord use
E=1/2kx^2
1/2(100)(.196)^2=.98 J
this energy combined with the energy that the kinetic energy provides should determine the total length of elongation of the cord
.98 + (2 x 9.8 x .5) = 1/2(100)x^2
x=.464327m
can someone tell me where I am going wrong?

 

[nickjer]

Don't use forces in an energy conservation set up. Start with initial energy and then do the final energy where the mass is stretched to its farthest...

KE = kinetic energy
GPE = gravitation potential energy
SPE = spring potential energy

Initial Energy: KE = 0 (because it starts from rest), GPE = mgh (starts from height 'h'), and SPE = 0 (spring is starting at equilibrium)

Final Energy: KE = 0 (reached max distance so speed is 0), GPE = 0 (fell as far as it will go), SPE = (1/2)*k*h^2 (now it is a distance 'h' from the equilibrium point)

You equate the energies due to energy conservation and solve for the longest displacement from equilibrium. Then you need to remember that the question is asking for the full length of the string when it is stretched this much.

 

[Ry122]

what's wrong with using forces in an energy conservation set up? why doesn't doing it that way yield the correct answer?

 

[nickjer]

You can use forces to solve for it or use conservation of energy. Both will get you the right answer. But there is no use mixing them together. One deals with vectors and the other uses scalars. You would just be doing more work than necessary.