Problem 14 - University Physics with Modern Physics

Question

With a wooden ruler you measure the length of a rectangular piece of sheet metal to be 12 \(\mathrm{mm}\) . You use micrometer calipers to measure the width of the rectangle and obtain the value 5.98 \(\mathrm{mm}\). Give your answers to the following questions to the correct number of significant figures. (a) What is the area of the rectangle? (b) What is the ratio of the rectangle's width to its length? (c) What is the perimeter of the rectangle? (d) What is the difference between the length and width? (e) What is the ratio of the length to the width?

Step-by-Step Solution

Answer

Area: 72 mm²; Width/Length Ratio: 0.50; Perimeter: 36 mm; Difference: 6.0 mm; Length/Width Ratio: 2.0.

Step 1: Determine the Area of the Rectangle

To find the area of the rectangle, use the formula: \[ A = ext{length} \times ext{width} \]Substitute the given values:\[ A = 12 \, \mathrm{mm} \times 5.98 \, \mathrm{mm} \]Calculating the above gives:\[ A = 71.76 \, \mathrm{mm}^2 \]Since the least number of significant figures among the measurements is 2 (from the length, 12 mm), we round the area to 72 mm².

Step 2: Calculate the Ratio of Width to Length

The ratio of the rectangle's width to its length is given by:\[ \text{Ratio} = \frac{\text{width}}{\text{length}} = \frac{5.98 \, \mathrm{mm}}{12 \, \mathrm{mm}} \]Compute the division:\[ \text{Ratio} \approx 0.4983 \]Since the length in the division has 2 significant figures, we round the ratio to 0.50.

Step 3: Find the Perimeter of the Rectangle

The formula for the perimeter of a rectangle is:\[ P = 2(\text{length} + \text{width}) \]Plug in the given measurements:\[ P = 2(12 \, \mathrm{mm} + 5.98 \, \mathrm{mm}) \]Calculate:\[ P = 2(17.98 \, \mathrm{mm}) = 35.96 \, \mathrm{mm} \]Rounding to the least number of significant figures, 36 mm.

Step 4: Determine the Difference Between Length and Width

The difference is found using:\[ \text{Difference} = \text{length} - \text{width} = 12 \, \mathrm{mm} - 5.98 \, \mathrm{mm} \]Perform the subtraction:\[ \text{Difference} = 6.02 \, \mathrm{mm} \]Round this to 6.0 mm using 2 significant figures.

Step 5: Calculate the Ratio of Length to Width

The ratio of the rectangle's length to its width is:\[ \text{Ratio} = \frac{\text{length}}{\text{width}} = \frac{12 \, \mathrm{mm}}{5.98 \, \mathrm{mm}} \]Compute the division:\[ \text{Ratio} \approx 2.0067 \]Rounding to the least number of significant figures, the ratio is 2.0.

Key Concepts

Measurement Accuracy

Measurement accuracy is crucial when determining values that will be multiplied or divided. It refers to how close a measured value is to the true value or an accepted standard, and is often expressed in terms of significant figures. Significant figures are the digits in a number that contribute to its precision. They include all non-zero digits, any zeros between significant digits, and any trailing zeros in the decimal part. In the exercise, the length measurement is 12 mm, which contains two significant figures. The width is measured at 5.98 mm with three significant figures. When performing calculations such as area, perimeter, or ratios, the result should be rounded to the least number of significant figures. This ensures the accuracy of the measurements is preserved throughout any calculations.

Unit Conversion

Unit conversion is the process of converting a measure to an equivalent value in a different unit. While the exercise focuses on measurements in millimeters, understanding unit conversions is vital for comprehensive learning. Let's say you needed to convert millimeters to centimeters. Since 1 cm equals 10 mm, you would divide a millimeter measure by 10 to convert it into centimeters. For instance, converting the length of 12 mm would result in 1.2 cm. Converting units also helps in understanding quantities in different contexts, which is especially useful in fields ranging from science and engineering to everyday situations. Remember to ensure consistency in units across all measurements when performing calculations.

Rectangular Area Calculation

Calculating the area of a rectangle involves multiplying its length by its width. The formula is straightforward and given by \( A = ext{length} \times ext{width} \). In our exercise, substituting the given values, the calculation becomes: \( A = 12 \, \text{mm} \times 5.98 \, \text{mm} \). This results in an unrounded area of 71.76 mm². However, due to the limitation of significant figures, we round the area to 72 mm² as it must conform to the smallest significant figure count, in this case, two from the length measurement. Understanding this concept aids in carrying out precise and accurate measurements in practical applications like engineering and carpentry.

Perimeter Calculation

The perimeter of a rectangle is calculated by summing twice its length and twice its width, formulated as \( P = 2( ext{length} + ext{width}) \). Using our exercise's measurements, the perimeter would be calculated as \( P = 2(12 \, \text{mm} + 5.98 \, \text{mm}) = 35.96 \, \text{mm} \). As perimeter must be communicated with the correct significant figures, we round it to 36 mm. Perimeter calculations are vital in situations where the boundary length is significant, such as fencing a garden or planning the layout for a floor. Accurate perimeter measurement allows for optimal planning and resource management, ensuring no wastage or shortage of materials.