Calculating Strain Energy in a Hanging Spring

[Nemo's]

Homework Statement

A spring hangs vertically from a fixed point. When a mass of 94g is suspended from the spring the spring stretches by 2.6 cm before temporarily coming to rest. Determine the extension of the spring at which the strain energy is half its maximum value.

Homework Equations

F=0.5kx^2

The Attempt at a Solution

Loss in Gravitational Potential Energy = Gain in Strain Energy
So Half Strain energy = Half Gravitational Potential Energy
0.5k(.026*10^-2)^2=0.012 so k=6000/169
and since k=F/x using F=0.094*9.8 to get x

 

[haruspex]

0.5k(.026*10^-2)^2=0.012 so k=6000/169

I don't understand your working there. Pls post full details. .026*10^-2 doesn't look right - do you mean .026? For k, I get about 70N/m.

 

[Nemo's]

Oops sorry. Yes I meant 0.026 but my value for k is 6000/169 (approx. 35.5)
Then I used the Weight force and my k in F=kx (hook's law) to get x so (0.094*9.8)/(35.5)=x

 

[haruspex]

Oops sorry. Yes I meant 0.026 but my value for k is 6000/169 (approx. 35.5)

OK, but I asked you to explain how you got that equation. Where does the .012 come from? That looks to me like half the strain energy at max extension, but to compute k you need to be using the whole strain energy at max extension.

 

[Nemo's]

edited post

 

[Nemo's]

Where does the .012 come from? That looks to me like half the strain energy at max extension

Yes the .012 is half the gravitational potential energy lost (mgh) which is equal to half the strain energy gained(1/2 k x^2) (Using conservation of energy)

Loss in Gravitational Potential Energy = Gain in Strain Energy
So Half Strain energy = Half Gravitational Potential Energy
0.5k(.026)^2=0.012

but to compute k you need to be using the whole strain energy at max extension.

So I wrongly equated the whole strain energy to half the strain energy. I corrected it now:
.25k(.026)^2=.012
Now I get k=71

For k, I get about 70N/m.

Is that how you got it ?

 

[haruspex]

.25k(.026)^2=.012
Now I get k=71

Is that how you got it ?

No. You are told:

A spring hangs vertically from a fixed point. When a mass of 94g is suspended from the spring the spring stretches by 2.6 cm before temporarily coming to rest.

That is enough information to calculate k. No mention is made of any half amounts of energy yet.

 

[Nemo's]

O.K but in the beginning I tried using F=ke to get k. for F= .094*9.8 and e=.026, k=35.4
Why didn't I get the true value for k ?

 

[haruspex]

O.K but in the beginning I tried using F=ke to get k. for F= .094*9.8 and e=.026, k=35.4
Why didn't I get the true value for k ?

Because that equation assumes the extension is to the equilibrium point (zero acceleration), but the extension given is to zero KE point.

 

[Nemo's]

Thank you so much :D