Problem 98 - University Physics with Modern Physics

Question

The length of a rectangle is given as \(L \pm I\) and its width as \(W \pm w .\) (a) Show that the uncertainty in its area \(A\) is \(a=L w+l W .\) Assume that the uncertainties \(l\) and \(w\) are small, so that the product \(l w\) is very small and you can ignore it. (b) Show that the fractional uncertainty im 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 \(L \pm L, W \pm w,\) and \(H \pm h .\) Find the fractional uncertainty in the volume, and show that it equals the sum of the fractional uncertainties in the length, width, and height.

Step-by-Step Solution

Answer

(a) The area uncertainty is \( a = Lw + lW \). (b) Fractional uncertainty in area is the sum of those in length and width. (c) Fractional uncertainty in volume equals the sum in length, width, and height.

Step 1: Understanding Uncertainty in Area

For a rectangle, the area is given by the formula \( A = L \cdot W \). If \( L \) and \( W \) have uncertainties \( l \) and \( w \) respectively, then the total uncertainty in the area, \( a \), is calculated as \( a = (L \pm l)(W \pm w) \). Expanding this product, we get\( A + a = LW + Lw + lW + lw \). By ignoring the product \( lw \) because it's comparatively very small, the uncertainty in area simplifies to \( a = Lw + lW \).

Step 2: Fractional Uncertainty in Area

The fractional uncertainty in a quantity is the uncertainty divided by the original value. For the area, the fractional uncertainty is \( \frac{a}{LW} \). From Step 1, the uncertainty \( a = Lw + lW \). Thus, the fractional uncertainty becomes \( \frac{Lw + lW}{LW} \), which simplifies to \( \frac{w}{W} + \frac{l}{L} \). This shows that the fractional uncertainty in the area equals the sum of the fractional uncertainties in length and width.

Step 3: Fractional Uncertainty in Volume of a Rectangular Solid

For a rectangular solid, the volume is \( V = L \cdot W \cdot H \). With respective uncertainties \( l, w, \) and \( h \) in length \( L \), width \( W \), and height \( H \), the fractional uncertainty in volume \( V \) is \( \frac{v}{LWH} \) where \( v = lhW + LHw + LWw \). Therefore, it calculates as \( \frac{v}{LWH} = \frac{l}{L} + \frac{w}{W} + \frac{h}{H} \), demonstrating that the fractional uncertainty in the volume is the sum of the fractional uncertainties in length, width, and height.

Key Concepts

Fractional Uncertainty

Fractional uncertainty is a concept used to express the precision of a measurement relative to its size. It's calculated as the ratio of the absolute uncertainty to the measured value itself. This measurement allows us to compare uncertainties across different units and scales. For example, if a rectangle's length is measured as \( L \pm l \), and its width is \( W \pm w \), the fractional uncertainty of the length is \( \frac{l}{L} \) and for the width, it's \( \frac{w}{W} \).

This fraction offers a dimensionless number that helps in understanding how significant an error is relative to the size of its measurement.
It is particularly useful when comparing different physical quantities or dealing with complex calculations, like those in areas or volumes where more than one dimension is measured with uncertainty.

Rectangular Area

The area of a rectangle is calculated using the formula \( A = L \cdot W \), where \( L \) is the length and \( W \) is the width.
The actual measurements often include small uncertainties, represented as \( L \pm l \) and \( W \pm w \).
When calculating the uncertainty in the area \( A \), you need to consider how these uncertainties propagate.

Using error propagation principles, the uncertainty \( a \) in the formula becomes \( a = Lw + lW \).
This expression is derived by expanding \( (L \pm l)(W \pm w) \) and ignoring the small product term \( lw \), which doesn’t significantly affect the result.
This simplification is crucial for accurate estimation without unnecessary complexity. It helps you understand how errors might change the calculated area from its actual value.

Rectangular Volume

To find the volume of a rectangular solid, multiply its dimensions: length, width, and height, represented as \( V = L \cdot W \cdot H \).
Similar to the area, each measurement has inherent uncertainties: \( L \pm l \), \( W \pm w \), and \( H \pm h \).
Error propagation in these contexts means you need to consider how each uncertainty affects the volume's total uncertainty.

For volume \( V \), the fractional uncertainty can be calculated with \( \frac{v}{LWH} \). Here, \( v \) represents the formula to include all dimensional errors, given by \( v = lhW + LHw + LWw \).
When simplifying, this fractional uncertainty is expressed as \( \frac{l}{L} + \frac{w}{W} + \frac{h}{H} \), showing that it is a summation of each dimension's relative error.
This sum provides a comprehensive measure of uncertainty that affects the entire volume, giving you clear insights into the degree of possible deviation.

Error Propagation

Error propagation refers to how uncertainties in measurements affect the results of calculations based on them. It's an important concept for ensuring accuracy and understanding how different errors combine in empirical formulas.
When measuring multiple factors that contribute to a final value—like area or volume—their uncertainties also affect the total calculation.

Consider a rectangle, where you determine the area and need to understand the combined uncertainty from both length and width.
Using error propagation, the formula for uncertainty becomes \( a = Lw + lW \), explainable by recognizing it involves direct additions of the partial uncertainties.
Similarly, the fractional uncertainties summed as \( \frac{l}{L} + \frac{w}{W} \) or \( \frac{l}{L} + \frac{w}{W} + \frac{h}{H} \) for volumes allow you to maintain consistency and interpret results easily.
This method ensures that calculated measurements are not misrepresentative and account for all possible deviations from true values.