Desperately - Hooke's law - stiffness constant

[_Greg_]

Iv done an experiment with a spring wher you add mass and record the new length. From a table of these rusults iv plotted length (m) against mass (kg), like so

Now i have to calculate the stiffness, k, from this graph with the equation:

mg = k (l - lo)

so just to varify that what iv done is correct:

mg = (x2 - x1) x g

(l - lo) = (y2 - y1)

so k = ( (x2-x1) x g ) / (y2 - y1)

im pretty sure that's correct, its almost identical from using a single value of length and mass from the table of results.

QUESTION

how do i calculate the uncertainty in k?

all my measurements of length have an uncertainty of +/-0.5mm, each of the 100g masses i use in the experiment have an uncertainty of +/- 5g and of cource there will be an uncertainty in drawing my graph.

how do i use these to get my error in k?

btw the spring has negligible mass.

iv been trying to work this out for hours to no avail and its GOT to be in TOMORROW

any help, id really appreciate it.

 

[CaptainZappo]

Depending on the sophistication of your mathematics background, error considerations can be done in different ways. If you've gone through multivariable calculus and know how to do partial derivatives, you should read up on error propagation. The link below is a good place to start, but other resources are readily available by use of Google.

http://en.wikipedia.org/wiki/Propagation_of_uncertainty

Iv done an experiment with a spring wher you add mass and record the new length. From a table of these rusults iv plotted length (m) against mass (kg), like so

View attachment 10239

Now i have to calculate the stiffness, k, from this graph with the equation:

mg = k (l - lo)

so just to varify that what iv done is correct:

mg = (x2 - x1) x g

(l - lo) = (y2 - y1)

so k = ( (x2-x1) x g ) / (y2 - y1)

im pretty sure that's correct, its almost identical from using a single value of length and mass from the table of results.

QUESTION

how do i calculate the uncertainty in k?

all my measurements of length have an uncertainty of +/-0.5mm, each of the 100g masses i use in the experiment have an uncertainty of +/- 5g and of cource there will be an uncertainty in drawing my graph.

how do i use these to get my error in k?

btw the spring has negligible mass.

iv been trying to work this out for hours to no avail and its GOT to be in TOMORROW

any help, id really appreciate it.

 

[m2003]

As a scientist, it is important to consider uncertainties and errors in any experiment or calculation. In this case, the stiffness constant, k, is a measure of how stiff the spring is, and it is important to determine the uncertainty in this value to accurately represent the results of the experiment.

To calculate the uncertainty in k, you will need to consider the uncertainties in your measurements and how they propagate through the equation. This is known as error propagation. In this case, the uncertainties in length, mass, and the graph will all contribute to the uncertainty in k.

To start, you can use the formula for error propagation to calculate the uncertainty in k:

Δk = √( (∂k/∂x)^2 * Δx^2 + (∂k/∂y)^2 * Δy^2 + (∂k/∂g)^2 * Δg^2 )

where Δk is the uncertainty in k, Δx is the uncertainty in the length measurement, Δy is the uncertainty in the mass measurement, and Δg is the uncertainty in the acceleration due to gravity (which is typically given as a constant value of 9.8 m/s^2).

To calculate the partial derivatives (∂k/∂x, ∂k/∂y, ∂k/∂g), you will need to use the equation for k that you provided:

k = ( (x2-x1) x g ) / (y2 - y1)

Taking the partial derivatives, you will get:

∂k/∂x = g / (y2 - y1)

∂k/∂y = -(x2 - x1) * g / (y2 - y1)^2

∂k/∂g = (x2 - x1) / (y2 - y1)

Substituting these values into the error propagation formula, you will get:

Δk = √( (g/(y2 - y1))^2 * Δx^2 + ((x2 - x1)*g/(y2 - y1)^2)^2 * Δy^2 + ((x2 - x1)/(y2 - y1))^2 * Δg^2 )

Now, you can plug in the values for your uncertainties and calculate the uncertainty in k. Keep in mind that the uncertainties for length and mass will need to be converted to meters and