# Error estimation using differentials

1. May 15, 2014

### bawbag

1. The problem statement, all variables and given/known data

A force of 500N is measured with a possible error of 1N. Its component in a direction 60° away from its line of action is required, where the angle is subject to an error of 0.5°. What (approximately) is the largest possible error in the component?

2. Relevant equations

3. The attempt at a solution

The component force is $$F_x = F cos \theta$$

so $$lnF_x~=~lnF~+~lncos\theta$$

applying differentials: $$\frac{dF_x}{F_x} = \frac{dF}{F} + \frac{d~cos\theta}{cos\theta} (-sin\theta)$$$$=\frac{dF}{F} + \frac{sin^{2} \theta}{cos\theta}d\theta$$

plugging in values $$\frac{dF_x}{F_x} = \frac{1}{500} + \frac{3}{4} \frac{2}{1} \frac{1}{2} \frac{\pi}{180} = 0.002 + 0.013 = 0.015$$
so the error is $(0.015)(500)cos(60) = 3.75N$

The solution says 4.28N, however, which I confirmed by checking each error combination. Where am I going wrong here?

2. May 15, 2014

### vela

Staff Emeritus
You differentiated incorrectly. You should have
$$\frac{dF_x}{F_x} = \frac{dF}{F} + \frac{d\theta}{\cos\theta}(-\sin\theta).$$

3. May 15, 2014

### bawbag

Gotcha, I figured it would be something simple like that! Thanks a lot!