GREPhysics.NET: GR9277 #48

GR9277 #48

Problem

GREPhysics.NET Official Solution

\prob{48}
The magnitude of the force F on an object can be determined by measuring both the mass m of an object and the magnitude of its acceleration a, where F=ma. Assume that these measurements are uncorrelated and normally distributed. if the standard deviations of the measurements of the mass and acceleration are $\sigma_m$ and $\sigma_a$ respectively, then $\sigma_F/F$ is

$(\frac{\sigma_m}{m})^2+(\frac{\sigma_a}{a})^2$
$(\frac{\sigma_m}{m}+\frac{\sigma_a}{a})^{1/2}$
$((\frac{\sigma_m}{m})^2+(\frac{\sigma_a}{a})^2)^{1/2}$
$\frac{\sigma_m \sigma_a}{ma}$
$\sigma_m/m + \sigma_a/a$

Lab Methods $\Rightarrow$ }Uncertainty

The general equation for uncertainty is given by $\delta f/f=\sqrt{\sum_i (\frac{\delta x_i}{x_i})^2}$ , where $f=\Pi_i x_i$ and $\delta x_i$ is the generalized standard deviation of quantity $x_i$ .

So, for this case, one has $f=ma$ and thus its uncertainty is given by $\delta f/f = \sqrt{(\frac{\delta m}{m})^2+(\frac{\delta a}{a})^2}$ . (Basically, one has $x_1=m$ and $x_2=a$ .)

This is choice (C).

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Comments

MuffinSpawn
2009-09-24 18:57:56

If you don't remember the product rule, you can always derive it quickly using the general function rule:

$\sigma f = \sqrt{\sum \left(\frac{\partial f}{\partial x_i} \sigma x_i\right)^2}$

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petr1243
2008-03-09 18:13:35

Just a simple application of the general rules from error analysis:

For A = B +/- C :

$\delta A$ = $\sqrt{(\delta B)^2 + (\delta C)^2}$

For A = B*C or A = B/C:

$\frac{\delta A}{A}$ = $\sqrt{(\frac{\delta B }{B})^2 + (\frac{\delta A }{A})^2}$

Just treat $\sigma$ as our $\delta$

note
2008-08-21 23:12:47

There's a typo in your last equation, I think you meant dC/C

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Just a simple application of the general rules from error analysis: For A = B +/- C : $\delta A$ = $\sqrt{(\delta B)^2 + (\delta C)^2}$ For A = B*C or A = B/C: $\frac{\delta A}{A}$ = $\sqrt{(\frac{\delta B }{B})^2 + (\frac{\delta A }{A})^2}$ Just treat $\sigma$ as our $\delta$

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