# Question aa46e

Feb 7, 2017

0.6%

#### Explanation:

In the classical sense , the centripetal force $F$ can be measured as:

$F = \frac{m {v}^{2}}{r}$

We can take logs and say that:

$\ln F = \ln \left(\frac{m {v}^{2}}{r}\right)$

$= \ln m + \ln {v}^{2} - \ln r$

$= \ln m + 2 \ln v - \ln r$

Using differentials:

$\frac{\mathrm{dF}}{F} = \frac{\mathrm{dm}}{m} + 2 \frac{\mathrm{dv}}{v} - \frac{\mathrm{dr}}{r}$

 = 2% + 2 * 1.3% - 4% = 0.6%

Feb 8, 2017

4.66%

#### Explanation:

The formula for error propagation using standard deviations:

$\frac{\sigma F}{|} F | = \sqrt{{\left(\frac{\sigma a}{|} a |\right)}^{2} + {\left(\frac{\sigma b}{|} b |\right)}^{2} + {\left(\frac{\sigma c}{|} c |\right)}^{2} + \ldots}$

Where a, b, c, ... are parameters used to determine the uncertainty of $F$.

$\frac{\sigma F}{|} F |$ is basically a percentage in error if we multiply by 100%. Therefore,

Uncertainty in your centripetal force is sqrt(2^2+1.3^2+4^2)=4.65725%

Round it off to get 4.66%#.

There is a more robust way of measuring uncertainties using calculus and it is my go to formula.

Uncertainty of a quantity, $\sigma Q = \sqrt{{\left(\sigma a \frac{\partial Q}{\partial a}\right)}^{2} + {\left(\sigma b \frac{\partial Q}{\partial b}\right)}^{2} + {\left(\sigma c \frac{\partial Q}{\partial c}\right)}^{2} + \ldots}$

$\left(\sigma a \frac{\partial Q}{\partial a}\right)$ is basically $\sigma a$ also known as the uncertainty to be multiplied by the derivative of $Q$ with respect to parameter $a$.

Cheers.