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 mm. With micrometer calipers, you measure the width of the rectangle to be 5.98 mm. Use the correct number of significant figures: What is (a) the area of the rectangle; (b) the ratio of the rectangle's width to its length; (c) the perimeter of the rectangle; (d) the difference between the length and the width; and (e) the ratio of the length to the width?

Step-by-Step Solution

Answer

(a) Area = 72 mm²; (b) Width to Length Ratio = 0.50; (c) Perimeter = 36 mm; (d) Difference = 6.0 mm; (e) Length to Width Ratio = 2.0.

Step 1: Identifying Given Numbers

We are given two measurements: the length of the rectangle is 12 mm, and the width is 5.98 mm. It's important to note the significant figures in these measurements: 12 has two significant figures, while 5.98 has three.

Step 2: Calculating the Area

The formula for the area of a rectangle is \( \text{Area} = \text{length} \times \text{width} \). Substituting in the given values, we get \( \text{Area} = 12 \times 5.98 = 71.76 \text{ mm}^2 \). However, the result needs to reflect the least number of significant figures in the given numbers (which is 2). Hence, the area is 72 mm².

Step 3: Calculating the Ratio of Width to Length

The ratio of the width to the length is given by \( \text{Ratio} = \frac{\text{width}}{\text{length}} \). Substituting the values, \( \text{Ratio} = \frac{5.98}{12} \approx 0.4983 \). Applying significant figures, this result should have 2 significant figures, as per the least in given measures. So, the ratio is 0.50.

Step 4: Calculating the Perimeter

The perimeter of a rectangle is calculated as \( \text{Perimeter} = 2(\text{length} + \text{width}) \). Substituting the given values, \( \text{Perimeter} = 2(12 + 5.98) = 2 \times 17.98 = 35.96 \text{ mm} \). Adjust this to 2 significant figures, resulting in a perimeter of 36 mm.

Step 5: Calculating the Difference between Length and Width

The difference between the length and the width is calculated by \( \text{Difference} = \text{length} - \text{width} = 12 - 5.98 = 6.02 \text{ mm} \). With 2 significant figures, the difference is 6.0 mm.

Step 6: Calculating the Ratio of Length to Width

Determine the ratio of the length to the width using \( \text{Ratio} = \frac{\text{length}}{\text{width}} \). Substituting the values, \( \text{Ratio} = \frac{12}{5.98} \approx 2.004 \). This should be rounded to 2 significant figures, so the ratio is 2.0.

Key Concepts

Measurement Accuracy

When making measurements, it is crucial to consider the concept of significant figures. Significant figures are the digits in a number that contribute to its precision. These figures include all non-zero digits, any zeros between significant digits, and any trailing zeros in a decimal number.

Using the correct number of significant figures is essential. It reflects the precision of the instrument used for the measurement. For instance, measuring 12 mm with a wooden ruler may only be accurate to two significant figures. Contrast this with a micrometer caliper measuring 5.98 mm which is precise to three significant figures.

This accuracy influences calculations like multiplication or division, where the result must have the same number of significant figures as the least in any of the original numbers. Whether calculating area, ratios, or differences, dimension accuracy ensures meaningful and reliable data.

Geometry Formulas

Geometry formulas are mathematical expressions used to calculate different properties of shapes. For rectangles, the core formulas include:

• Area: Calculated as \( \text{Area} = \text{length} \times \text{width} \)
• Perimeter: Given by \( \text{Perimeter} = 2(\text{length} + \text{width}) \)
• Ratios and differences: Use simple division and subtraction to relate different dimensions of the shape.

Understanding these formulas helps in solving problems accurately. Once you know the measurements, you can easily compute the desired properties by substituting directly into these formulas to find area, perimeter, and other necessary metrics.

Rectangular Area Calculation

The calculation of a rectangular area is an application of basic geometry. First, identify the length and width of the rectangle. Use the formula \( \text{Area} = \text{length} \times \text{width} \) to calculate.

In our example, using given measurements of 12 mm and 5.98 mm, the calculated area initially appears as 71.76 mm². However, due to the concept of significant figures, and because the least precision here is two significant figures, we round the area to 72 mm².

This result not only abides by the rules of arithmetic but also provides a true reflection of the measurement accuracy of the tools used to obtain the dimensions.

Physics Problem Solving

Solving physics problems often involves applying accurate mathematical concepts like those in geometry to real-world scenarios. In this problem, understanding geometry, precision, and measurement ductility are key.

Here are some general steps when solving these kinds of problems:

• Identify the values provided clearly and the required significant figures.
• Choose and apply the relevant formulas or equations.
• Perform calculations carefully, respecting arithmetic rules such as significant figures.

The physics of rectangular objects often also involves studying properties like ratios and difference measures. Correct application of these principles ensures that calculations mirror physical reality accurately. Whether it’s determining area, comparing dimensions, or computing ratios, clear comprehension of underlying concepts and the precision of your measurement tools directly affects your solution's validity.