Find length of a string, given a strange condition

[wayfarer]

Homework Statement

An apple weighs 1.11 N. When you hang it from the end of a long spring of force constant 1.53 N/m and negligible mass, it bounces up and down in SHM. If you stop the bouncing and let the apple swing from side to side through a small angle, the frequency of this simple pendulum is half the bounce frequency. (Because the angle is small, the back and forth swings do not cause any appreciable change in the length of the spring.)
What is the unstretched length of the spring (i.e., without the apple attached)?

Homework Equations

I think these are the relevant ones:
For a normal spring, f = 1/2*pi ( sqrt (k/m ) )
for a simple pendulum, T = 2*pi * sqrt (L/g)

The Attempt at a Solution

For spring with apple, if mass of the apple is m, spring constant is k, then
mg = kx (using force); combining this with f = 1/2*pi ( sqrt (k/m) ) gives:
f = 1/(2*pi) ( sqrt (g/L) )
For simple pendulum, I think we have T = 2*pi* sqrt (L/g)
So i think we should then have: 2*pi* sqrt (L/g ) = 1/2 (1/2*pi) * sqrt (g/L), i.e.
L = g/(8 * pi^2).
However this is wrong and I am not sure why. (I don't know what the right answer is either).

 

[Doc Al]

Hint: When using this formula for the swinging spring/mass pendulum:

for a simple pendulum, T = 2*pi * sqrt (L/g)

How does L relate to the unstretched length of the spring?

 

[michalll]

x is only length increment not the whole lenght

 

[wayfarer]

Dear Doc Al,
I think I understand my mistake - that I took the stretched length of the spring, and not its unstretched length now. But I am still a bit confused, and don't understand how to relate the two together - could you please tell me how I could do that?

 

[Doc Al]

Imagine that the spring has some unstretched length $L_0$. When you add the mass of the apple, by how much does it stretch?

 

[wayfarer]

I know that but I meant, I am not sure which formula is applicable here.
I thought first that we could use formula:
A = sqrt ( x_0 ^2 + (v_0x)^2/w^2 )
but it seems like v_0x = 0, so this formula doesn't seem to help much (it seems to give wrong answer in this situation).
Perhaps Hooke's Law, F = -kx would be helpful, but I seemed to try that before, by thinking that maybe kx = mg and trying along those lines - which didn't seem to work either.
Perhaps i could use COnservation of energy to try this?
Trying that, i get, if L is stretched, l is unstretched length, then
mg(L-l ) = 1/2 k (L-l)^2
L = l + (2mg)/k
Is that correct?

 

[Doc Al]

I know that but I meant, I am not sure which formula is applicable here.
I thought first that we could use formula:
A = sqrt ( x_0 ^2 + (v_0x)^2/w^2 )
but it seems like v_0x = 0, so this formula doesn't seem to help much (it seems to give wrong answer in this situation).

I don't recognize that formula or what you're trying to do. All you need are the two formulas you started out with (plus Hooke's law):

For a normal spring, f = 1/2*pi ( sqrt (k/m ) )

This one gives you the frequency of the vertical SHM. And with the information given, you now know the frequency of the pendulum motion. (How does frequency relate to period?)

for a simple pendulum, T = 2*pi * sqrt (L/g)

Use this to find the length of the spring pendulum. How does that length relate to the unstretched length?