Why Is the Uncertainty in a Cat's Weight Considered 1lb and Not 2lb?

[walking]

This is from Halliday et al Physics 4e, p9, sample problem 3.

Problem:

My question: Why do the authors say that the weight of the cat has 1lb uncertainty rather than 2lb? Isn't it between 127.5-119.5=8lb, and 10lb, hence 9lb+-1lb ie uncertainty of $$\frac{2}{9}\times 100 = 22\%$$ rather than 11%?

 

[etotheipi]

To be honest, I'm not sure if any of that is right. Reading to the nearest whole number of $\text{lb}$ implies that your measurements are $m_1 = (119 \pm 0.5) \text{lb}$ and $m_2 = (128 \pm 0.5) \text{lb}$. Since $m_c = m_2 - m_1$, the error in $m_c$ is$$\Delta m_c = \sqrt{(0.5 \text{lb})^2 + (0.5 \text{lb})^2} \approx 0.7 \text{lb}$$and the fractional error is then $\frac{\Delta m_c}{m_c} \approx 8\%$

 

[walking]

To be honest, I'm not sure if any of that is right. Reading to the nearest whole number of $\text{lb}$ implies that your measurements are $m_1 = (119 \pm 0.5) \text{lb}$ and $m_2 = (128 \pm 0.5) \text{lb}$. Since $m_c = m_2 - m_1$, the error in $m_c$ is$$\Delta m_c = \sqrt{(0.5 \text{lb})^2 + (0.5 \text{lb})^2} \approx 0.7 \text{lb}$$and the fractional error is then $\frac{\Delta m_c}{m_c} \approx 8\%$

Thanks etothepi.

However I was more trying to understand the author's approach (even if not technically correct, as I now understand from your post?). Basically it seems that they used their approach in a wrong way and I was just asking if my method was correct or not. They use a certain approach for the first part to get 8%, but then when I applied it to the second part (the cat's weight) I got 22% instead of the 11% they get.

At this level (beginner level undergraduate physics), would you still recommend that I ignore the author's method altogether and learn how to do it using the method you used? I am happy to dedicate some extra time on the side for learning how to deal with errors properly if you think it would be a good idea at this stage.

 

[etotheipi]

Hi @walking, error analysis can be a bit of an "art-form" and it's hard to say that something is completely wrong... but the extract you uploaded is definitely non-standard, and non-conventional.

Here's a good document about how to do it properly!

https://web.archive.org/web/2016121...ploads/2013/03/PS3_Error_Propagation_sp13.pdf

 

[walking]

Hi @walking, error analysis can be a bit of an "art-form" and it's hard to say that something is completely wrong... but the extract you uploaded is definitely non-standard, and non-conventional.

Here's a good document about how to do it properly!

https://web.archive.org/web/2016121...ploads/2013/03/PS3_Error_Propagation_sp13.pdf

Great, thank you. I have saved it and will try to find time to study it in detail at some point.

 

[jbriggs444]

One can argue that there is actually no inconsistency in the approach taken.

When you weigh yourself on the scale, you read directly from the scale and take the reported value as the measured result. However let us add one step to the procedure. Before you step on the scale, you first twirl a calibration knob so that the scale reading is zeroed.

Now the weight of the man is the difference between two readings. And the weight of the cat is the difference between two readings.

The approach of adding the errors for the two measurements in quadrature ($\sqrt{e_1^2 + e_2^2}$) is misguided in this case. It would be proper for combining independent, normally distributed errors. But in this case we have quantization errors -- independent with a bounded uniform distribution. [I've made some questionable assumptions by idealizing away all other sources of experimental error here]

Edit: I also neglected to specify the knob twirling procedure. You close your eyes, twirl the knob and see whether the result is zero. If not, then you try again. The result should then approximate a uniform distribution of calibrations such that the scale returns an unbiased measurement on average.