The length of a rectangle is given as ?L ± ?l and its width as ?W ± ?w?. (a) Show that the uncertainty in its area ?A is ?a = ?Lw + ?lW?. Assume that the uncertainties ?l and ?w are small, so that the product ?lw is very small and you can ignore it. (b) Show that the fractional uncertainty in the area is equal to the sum of the fractional uncertainty in length and the fractional uncertainty in width. (c) A rectangular solid has dimensions ?L ± ?l?, ?W ± ?w?, and ?H ± ?h?. Find the fractional uncertainty in the volume, and show that it equals the sum of the fractional uncertainties in the length, width, and height.
Solution 98CP a) The length of the rectangle is L ± l. The width is W ± w As we know the formula for the area of a rectangle, area = length × width = ( L ± l) × (W ± w) = LW ± Lw ± lW + lw = LW + lw ± (Lw + lW) As the uncertainties w and l are very small, we can ignore lw term. The uncertainty in area is ± (Lw + lW) . b) The fractional uncertainty or the relative uncertainty is defined as, uncertainty / measured value. The uncertainty in area is (Lw + lW). Measured value of the area is LW . Fractional uncertainty in area = (Lw + lW) / LW ----------------(1) Now the uncertainty in length is l. Measured value of the length is L. Uncertainty in width is w Measured value of width is W . Fractional uncertainty in length is l/L. Fractional uncertainty in width is w/W . The sum of fractional uncertainties in length and width is , lW+Lw l/L + w/W = LW ------------------(2) From equation (1) and (2) it is clear that, Fractional error in area = sum of fractional errors in length and width (proved) c) The same is true for volume too. There 1 more term will be added which is the height H. The error in height say ± h. The calculation is same as part b) calculation and trivial.
