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June 1st, 2014, 03:28 PM   #1
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Hooke's Law Confusion

If 3 J of work are needed to stretch a spring from 8 cm to 10 cm and 5 J are needed to stretch it from 10 cm to 12 cm, what is the natural length of the spring?

So above is the problem I have been wrestling with for the past hour. I know how to find Work if I am given a force such as 40N. to stretch the spring from it's natural length of 100cm to say 105cm.

I do know then how to measure the distance from 100cm to 105cm = 5 cm which in turn = .05meters. and then how to maneuver from f(x) = k(x) and
so, f(.05) = 40
and so... k = 800

and then how to integrate our f(x) = 800x and to follow through with the integration process to obtain Work = X joules.

What I cannot seem to figure out on my own is (and for the question I posted here) we are not given a force and are not given in the spring's natural length. We are given the amount of work for "two" seperate scenarios and so I am trying to set up the equation for force and am hoping if I can solve for force that I can find when the work done = 0 (or signifies the spring is at it's natural resting place (length).

Sadly, I have been unsuccessful trying to comprehend this problem and thus I am posting it here on the forum.

*So, is my thinking correct trying to solve for the unknown variable (force) ...plugging in for WORK and how would you advance from here?


I have looked over similar Hooke's Law problems but they all seem to give the Force and Natural Length of the Spring and I have noticed that a lot of these Calculus 2 Homework problems in WebAssign have been a variation that I feel tries to get the student to "think outside of the box" , etc.

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June 1st, 2014, 08:23 PM   #2
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From memory, Hooke's Law says that $F = -kx$ where x is the extension and k is a constant that tells you how easy it is to stretch the spring. And total length $l = n + x$ where n is the natural length.

So, you have two unknowns (n and k) and two equations, so we ought to be able to get values for booth.

Can you set up the two integrals, using n and k and the information given in the question?
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June 1st, 2014, 11:44 PM   #3
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I will try
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June 2nd, 2014, 12:05 AM   #4
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actually v8 I am not sure that I understand how to set up the Integral with the two equations and two unknowns. I need help
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June 2nd, 2014, 08:53 AM   #5
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v8 I think that F= -kx leaves me with 2 unknowns in this equation alone.

I know to picture a number line ( or assign a # line ) to the spring. Thus, when we move backwards (or compress the spring) we are moving in a negative direction (hence the -k)

Thus if we have a -10 cm's then - (-10)(x) gives a positive answer of simply kx.

Now, problem I was thinking is F = kx I do not have k nor F (force) so I am thinking there is 2 unknowns in this equation and 1 unknown in the equation you gave of length l = natural length n + extension x.

The equation has given us the value for Work for each equation. i.e., one is 3J and for the other it is 5j. Both move the same distance (the first from 8cm to 10cm ...thus a distance of (.02)meters and the other 10cm to 12 cm. Thus also a distance of (.02)meters.
Although now I recongnize what you have made mention of total distance which I can clearly see as (.04)meters.

So, F, n and k I think I don't have.

Last edited by mr. jenkins; June 2nd, 2014 at 08:57 AM.
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June 2nd, 2014, 04:44 PM   #6
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Hi mr. jenkins,

Quote:
So, F, n and k I think I don't have.
Don't worry about F because you have the work which is F*displacement. When you incorporate F*displacement, you are left with only 2 unknowns, n and k, and you have 2 integral equations that you can use to solve for n.
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June 2nd, 2014, 08:36 PM   #7
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Yes, you either have $F$ or you have $-kx$, you don't have both, so $F$ and $k$ are only a single unknown in the system. Or perhaps more accurately, if you know $k$, you can find $F$ and vice versa.

You are looking to create integrals that look like $$\int_{a}^{b}{Fx}dx$$
where the limits need to be the extension, not the total length of the spring.

Try to create them, and post your efforts. We'll see if you need further assistance.
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June 3rd, 2014, 12:57 PM   #8
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Well as far as setting up the Integrals, I am still stuck with the following thinking:

F = -kx

Ok, since W = Force*Displacement I see the following and please correct me where I am wrong.

The question says 3 Joules are needed to stretch a spring from 8 cm to 10 cm
then it says:and 5 Joules are needed to stretch spring from 10 cm to 12 cm

Thus,
The first informations I would think I could set up like this:

3 = Fd

3 = F(2 cm) and for every problem in the unit thus far they have had us to change the cm to meters so... 3 = F(.02) Now, since Work = Force * Displacement I have 3 = F(.02) Thus, Force = 150

and on the other equation that looks about the same 5 = Fd = Force = 250

Now, I need to know is this correct or incorrect?

and if b and a are the limits of the extension (meaning I think not the compression but opposoite) we would still have .02 for each one so what? b =.02 and a =0 ?

Now if f(x) = -kx and I did the first part right, I now know that k(.02) = 150 and

k(.02) =250 giving us k values of 7,500 and 12,500.

Thus Integrals would look like f(x) = k(x) = 150 ... k = 7500 ... INT from a to b of 7500x dx would go (7500/2)x^2 = 3,750(.02) - 3750(0) = 75

That would be on the first Integral and second would just be 6,250(.02)-6,250(0) = 125

and if this is at all correct I am thinking I should have taken final integral value of larger subtract the smaller which would give 125-75 = 50

Ok guys
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June 3rd, 2014, 03:56 PM   #9
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W = Fd is fine for a constant force. Ours isn't constant, because as we displace the object, the spring stretches more, so the force increases.

Our integral will look like
$$W = \int_a^b Fx dx$$
We've decided that, for a spring, F=-kx where $x$ is the extension. It is also the displacement against the resisting force, the one against which we are doing work.

So $$W = \int_a^b -kx^2 dx$$

We are given two sets of conditions, the Work Donw to strecth the spring between two total lengths (NOT extensions). We can use these conditions to produce a pair of integrals on the model above. Can you fill in the blanks yet?
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June 3rd, 2014, 04:35 PM   #10
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no i don't really understand
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