#How do I find the natural length the spring?

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fathom snow
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bro this is physics

chilly tinsel
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Energy of a stretched spring is 0.5kx², where x is stretched length.
We are given 3 lengths of spring.
If natural length of spring is l,
12 cm = l + x, x is the elongation.
We have,
12J = 0.5k(x_2² - x_1²)
x_1 is elongation when spring is 12 cm, x_2, when 16.
x_2 = x_1 + 4,
Also x_3 = x_1 + 8
Just writing x_1 as x
12J = 0.5k [(x+4)² - x²]
12 + 20 = 0.5k [(x+8)² - x²]
2 equations 2 variables. Eliminate k to get x.
That's the elongation when spring has a total length of 12.
Subtract elongation. You get natural length

thorn summit
fathom snow
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kk

thorn summit
chilly tinsel
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Divide those two equations

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Or equate k = (in terms of x ) for an equation and substitute it in other

thorn summit
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and is x_1 and x_2 4?

chilly tinsel
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a = b; equation 1
c = d ; equation 2
a/c = b/d

chilly tinsel
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X_2 is elongation at spring length 16cm
X_1 is elongation at spring length 12cm
Difference between them 4

thorn summit
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so x_1 is 4 and x_2 is 8?

chilly tinsel
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No. .

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Oh

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Let me see

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Yeah

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So spring length 12 has an elongation 4

thorn summit
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wait how is elongation calculated?

chilly tinsel
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The Energy = 0.5k(elongation)² equations?

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Difference between energies is work done.

thorn summit
chilly tinsel
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We would just plug in values given if we knew k, where it would be an equation in 1 variable (elongation)

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We have a relation between different elongations. 3rd is 1st + 8, second is 1st + 4

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So all three can be shown with just 1 variable.

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Letting 1st elongation be x, 2nd and 3rd are x + 4, x + 8

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We are given work done in moving between them.

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12 = Energy at 2nd - Energy at 1st.

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12+20 = Energy at 3rd - Energy at 1st

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(Work done = change in Energy)

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So we have 2 equations, with two variables x & k

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We need atleast two equations to solve equations with 2 variables.
We have them.