Determine the force constant of the rubber band

[amanda.ka]

Homework Statement

From the experiment I did my force was equal to 0.0686N +/- 0.00196N and my x (rubber band stretch) was equal to 0.003m +/-0.002. The question asks to determine the force constant k +/- ∆k.

Homework Equations

F = kx
k = F/x

∆k = |k| ∆x/x +∆y/y

The Attempt at a Solution

k = F/x
= 0.0686N/0.003 m
= 22.8667 N/m

uncertainty: 22.8667(0.00196/0.0686 + 0.002/0.003)
final answer: 22.8667 +/- 15.8978 N/m

I am not sure if I have calculated this correctly, if someone could please check my work I'd appreciate it! Also because this is division should the final result have no more significant figures than the original value with the least number of significant digits? Thank you in advance :)

 

[Student100]

Homework Statement

From the experiment I did my force was equal to 0.0686N +/- 0.00196N and my x (rubber band stretch) was equal to 0.003m +/-0.002. The question asks to determine the force constant k +/- ∆k.

Homework Equations

F = kx
k = F/x

∆k = |k| ∆x/x +∆y/y

The Attempt at a Solution

k = F/x
= 0.0686N/0.003 m
= 22.8667 N/m

uncertainty: 22.8667(0.00196/0.0686 + 0.002/0.003)
final answer: 22.8667 +/- 15.8978 N/m

I am not sure if I have calculated this correctly, if someone could please check my work I'd appreciate it! Also because this is division should the final result have no more significant figures than the original value with the least number of significant digits? Thank you in advance :)

Hooke's law is only mildly appropriate for rubber bands and under certain constraints. I don't know your experimental set up, but your uncertainties look suspect to me. I'm curious about your method of calculating error propagation, your sig figs are also off.

 

[amanda.ka]

For my experimental setup I hung a rubber band from a support with a container tied to the bottom of the band. I measured the initial length of the rubber band (0.200 m) then added 1 coin into the bag which caused a stretch in the elastic. I measured and recorded this new length. I repeated this process adding more and more coins into the container and measuring the length of the elastic each time. Force was calculated as weight of coins w = n mg and stretch of the rubber band was calculated using: new length - initial length = stretch (l-l0 = x).

Uncertainty calculation for force:
Uncertainty of: ∆m = 0.2 g for each coin
g = 9.81 m/s2 is assumed to be known exactly
n = number of coins is assumed to be known exactly
m = 0.007 kg ± 0.0002 kg
c(A ± ∆A) = cA ± c(∆A)
(1)(9.8 m/s2)(0.007 kg ± 0.0002 kg) = 0.0686 N ± 0.00196 N

Uncertainty calculation for stretch of rubber band:
x = (l-l0)
l = 0.203
l0 = 0.200
x = l-l0
(A ± ∆A) - (B ± ∆B) = (A - B) ± (∆A + ∆B)
(0.203 ± 0.001)m – (0.200 ± 0.001)m = 0.003 ± 0.002 m

 

[haruspex]

Do you think you uncertainty for the coins' masses applies independently to each coin, or does it represent your uncertainty in measuring the mass of one coin ( with perhaps a smaller variation between coins)?
For the uncertainty in the extension, it's almost as great as the extension itself, making the results completely unreliable. Is the the extension for one coin only? I assume the uncertainty is the same with longer extensions using more coins.

 

[amanda.ka]

Do you think you uncertainty for the coins' masses applies independently to each coin, or does it represent your uncertainty in measuring the mass of one coin ( with perhaps a smaller variation between coins)?
For the uncertainty in the extension, it's almost as great as the extension itself, making the results completely unreliable. Is the the extension for one coin only? I assume the uncertainty is the same with longer extensions using more coins.

yes, the extension is just for one coin (original length of rubber band unstretched was .200 m, then it stretched to .203 m). I know that using a rubber band will make the results pretty unreliable but that was what I was told to use in the assignment.

 

[haruspex]

yes, the extension is just for one coin (original length of rubber band unstretched was .200 m, then it stretched to .203 m). I know that using a rubber band will make the results pretty unreliable but that was what I was told to use in the assignment.

It's not the rubber band that makes it unreliable. The problem is that one coin has such a small weight the amount of extension is hard to measure accurately. I don't know how many coins you had in all. If it was a sufficiently large number, you could get around the problem by considering only certain multiples of coins, like 3, 6, 9... effectively increasing the unit mass.

 

[Student100]

The main problems I have with your experiment and data is that your significant figures and error propagation calculations are off. Did you round during the propagation calculations? That's the only way I can get your value, which is a no-no. Recalculate it without rounding ( I could have put the values in my calculator wrong, so if you get the same value maybe it's me who made the mistake!). Before you do that, take a close look at your significant figures and uncertainties in your data, they're not quite right.

In addition, your large error should also tell you your experimental setup was flawed in its ability to model the modulus of the rubber bands elasticity. This isn't bad per-say, but hopefully in your report you reflect on this and what items contributed to this error. Do you have ideas about the errors that aren't immediately obvious? There's possibly a few you might not have thought of; some in which you didn't take any data on.