How Do You Calculate the Spring Constant and Natural Length of a Spring?

[StudentNeedsHlp]

I'm stuck on the following application of integrals:

Question: It requires .05 joule (Newton-meter) of work to stretch a spring from a length of 8 centimeters to 9 centimeters and another .10 joule to stretch it from 9 centimeters to 10 centimeters. Evaluate the spring constant and find the natural length of the spring.

My work so far:
I know that the spring constant is the k in the equation F(x) = kx.
I know that work is: W=Integral from a to b of F(x).
I got W=FD
.05 = F(1)
F = .05

F = kx
.05 = k(1)
k = .05 ----> spring constant?

My teacher said that this isn't right...I think I'm way off and I just want to know the technique to solving these types of problems.

I don't understand how to get the natural length of the spring. In other word problems similar to this one they don't ask for the natural length, just the work or the force required to stretch or compress the spring an addition blah units. The natural length is usually given in 90% of the problems in my book.


HELP!

 

[jamesrc]

You can't go from W = \int_0^D{ \vec F\cdot d\vec s to W = FD unless the force is constant (independent of displacement). That is not the case here, since F = kΔx

W_{12} = \int_{8 \rm cm }^{9 \rm cm}{k(x-x_o)dx}
W_{12} = \frac 1 2 k(x - x_o)^2|^{9\rm cm}_{8\rm cm} = 0.05 {\rm J}

Similarly:

W_{23} = \frac 1 2 k(x - x_o)^2|^{10\rm cm}_{9\rm cm} = 0.1 {\rm J}

(which can all be written directly from the work-energy theorem if you remember the expression for the potential energy of a linear spring, but anyway...)

You see that you've got 2 equations and 2 unknowns (k and x_o), so you know you should be able to solve it. It's just a few lines of algebra from there. Let me know if you get stuck.

 

[M98Ranger]

First of all, good job on attempting to use the equation F(x) = kx to find the spring constant. However, there are a few errors in your calculations. Let's break it down step by step:

1. Find the spring constant (k)

To find the spring constant, we can use the equation F(x) = kx, where F is the force required to stretch or compress the spring and x is the displacement from the natural length of the spring. In this problem, we have two different displacements (1cm and 2cm) and the corresponding forces (0.05 joules and 0.10 joules).

So, we can set up two equations using the given information:
0.05 = k(1)
0.10 = k(2)

Solving for k in both equations, we get:
k = 0.05 (from the first equation)
k = 0.05 (from the second equation)

Since both equations give us the same value for k, we can be confident that our answer is correct. So, the spring constant in this problem is 0.05 joules/centimeter.

2. Find the natural length of the spring

The natural length of the spring is the length at which there is no external force acting on the spring and it is in equilibrium. In other words, it is the length at which the spring is not stretched or compressed.

To find the natural length, we can use the equation F(x) = kx and set the force equal to zero, since there is no external force acting on the spring at its natural length. So, we have:
0 = kx

Using the value of k that we found in the first step (k = 0.05), we can solve for x:
0 = 0.05x
x = 0

This means that the natural length of the spring is 0 centimeters. This may seem odd, but it simply means that the natural length of the spring is the same as its starting length (8 centimeters in this problem).

In summary, to solve these types of problems, you need to use the equation F(x) = kx and set up equations using the given information to find the spring constant (k) and the natural length of the spring. I hope this helps!