Problem 16 - University Physics with Modern Physics

Question

A rectangular piece of aluminum is \(5.10 \pm 0.01 \mathrm{cm}\) long and \(1.90 \pm 0.01 \mathrm{cm}\) wide. (a) Find the area of the rectangle and the uncertainty in the area. (b) Verify that the fractional uncertainty in the area is equal to the sum of the fractional uncertainties in the length and in the width. (This is a general result; see Challenge Problem \(1.98 .\))

Step-by-Step Solution

Answer

(a) Area: 9.69 cm², Uncertainty: ±0.070 cm². (b) Yes, fractional uncertainties match.

Step 1: Calculate the Area

To find the area of the rectangle, multiply the length by the width.\[ A = l \times w = 5.10 \times 1.90 = 9.69 \, \text{cm}^2 \]

Step 2: Calculate the Uncertainty in the Area

The area uncertainty can be obtained using the formula for multiplication of uncertain measurements:\[ \left( \frac{\Delta A}{A} \right) = \left( \frac{\Delta l}{l} \right) + \left( \frac{\Delta w}{w} \right) \]First, find the fractional uncertainty for both length and width:- \( \frac{\Delta l}{l} = \frac{0.01}{5.10} \approx 0.00196 \)- \( \frac{\Delta w}{w} = \frac{0.01}{1.90} \approx 0.00526 \)Add these fractional uncertainties:\[ \frac{\Delta A}{A} = 0.00196 + 0.00526 = 0.00722 \]Then, calculate \( \Delta A \):\[ \Delta A = A \times 0.00722 = 9.69 \times 0.00722 \approx 0.070 \text{ cm}^2 \]

Step 3: Verify the Fractional Uncertainty Relation

Summing the fractional uncertainties of length and width should be equal to the fractional uncertainty of the area, based on the previous computation.- Fractional uncertainty in length: \( 0.00196 \)- Fractional uncertainty in width: \( 0.00526 \)- Sum: \( 0.00196 + 0.00526 = 0.00722 \)This matches the calculated fractional uncertainty for the area, verifying the relation.

Key Concepts

Area Calculation

Calculating the area of a rectangle is a straightforward task but crucial for understanding geometry and measurement. The area represents the amount of space within the rectangle and is calculated through a simple formula.
In the given problem, the formula applied is:

• Area, \( A \), is calculated as the product of the length \( l \) and width \( w \).

So, to find the area of the rectangular piece of aluminum:\[ A = l \times w = 5.10 \, \text{cm} \times 1.90 \, \text{cm} = 9.69 \, \text{cm}^2 \]This numerical result, 9.69 square centimeters, gives us the size of the rectangular surface.

Fractional Uncertainty

Understanding uncertainty is key to effective measurement in science and engineering. Fractional uncertainty indicates how accurate a measurement is by comparing the uncertainty to the actual measured value.
In this context, the fractional uncertainty for each dimension is calculated separately:

• Length: \( \, \frac{\Delta l}{l} \approx \frac{0.01}{5.10} \approx 0.00196 \)
• Width: \( \, \frac{\Delta w}{w} \approx \frac{0.01}{1.90} \approx 0.00526 \)

These individual uncertainties are then summed to find the total fractional uncertainty of the area.
The overall formula for this is:\[ \frac{\Delta A}{A} = \frac{\Delta l}{l} + \frac{\Delta w}{w} \approx 0.00196 + 0.00526 = 0.00722 \]This step helps you determine how much the uncertainties in measuring dimensions affect the overall measurement of area. It's a powerful tool when considering precision in quantitative calculations.

Rectangular Measurements

Rectangular measurements, encompassing both length and width, serve as a base for calculating other important properties, like area, and guide the assessment of uncertainty in physical measurements.
In problems involving rectangular shapes, each measurement often includes a specific uncertainty. This uncertainty reflects possible variances in measurement due to instruments' precision limits or other external factors.
In this problem:

• Length is \( 5.10 \, \text{cm} \) with an uncertainty \( \pm 0.01 \, \text{cm} \)
• Width is \( 1.90 \, \text{cm} \) with an uncertainty \( \pm 0.01 \, \text{cm} \)

Such figures are essential in understanding not only the precise measurements of a rectangle but also in applying mathematical principles to calculate values influenced by these uncertainties.
This task underscores the importance of being meticulous with every measurement to ensure reliable and precise scientific computations.