Problem 8 - College Physics

Question

Metric wrenches. (a) You have a new set of metric wrenches, but need to loosen a \(\frac{3}{8}\) inch bolt. To find out which size metric wrench to use, convert the \(\frac{3}{8}\) in. to millimeters, accurate to the nearest tenth of a millimeter. (b) If you want to tighten a 12 \(\mathrm{mm}\) bolt, what size wrench should you use in inches, accurate to the nearest hundredth of an inch? (c) English-unit wrenches often come in \(\frac{1}{8}\) inch intervals, not in decimal units. What size wrench should you use in part (b)?

Step-by-Step Solution

Answer

(a) 9.5 mm, (b) 0.47 inches, (c) 0.5 inches (i.e., 1/2 inch).

Step 1: Convert Inches to Millimeters

The conversion factor from inches to millimeters is 1 inch = 25.4 mm. To convert \(\frac{3}{8}\) inches to millimeters, multiply by this conversion factor: \[\frac{3}{8} \times 25.4 = 9.525\text{ mm}\]. Rounding to the nearest tenth of a millimeter gives us 9.5 mm.

Step 2: Convert Millimeters to Inches

To convert 12 mm to inches, use the conversion factor: 1 mm = 0.0393701 inches. Multiply 12 mm by this factor:\[12 \times 0.0393701 = 0.4724412\text{ inches}\]. Rounding to the nearest hundredth of an inch gives us 0.47 inches.

Step 3: Match Wrench Size to Nearest Fraction

English-unit wrenches commonly come in \(\frac{1}{8}\) inch increments. Check which \(\frac{1}{8}\) inch increment is closest to 0.47 inches: \[\frac{3}{8} = 0.375, \quad \frac{1}{2} = 0.5\]. Since 0.47 is closer to \(\frac{1}{2}\) inch than \(\frac{3}{8}\) inch, you would use a \(\frac{1}{2}\) inch wrench.

Key Concepts

Metric System

The metric system is a standardized system of measurement used around the globe, originating from France in the late 18th century. This system is decimal-based, meaning it uses base units increased or decreased in powers of ten. Understanding the metric system is crucial for converting between different units of measurement, such as length, mass, and volume.

When dealing with length, the base unit in the metric system is the meter (m). Other units of length are derived from the meter, such as millimeters (mm), centimeters (cm), and kilometers (km). Converting between metric units is straightforward because you move the decimal point to convert from a larger unit to a smaller one or vice versa.

For example:

• 1 meter = 1,000 millimeters
• 1 centimeter = 10 millimeters
• 1 kilometer = 1,000 meters

In the context of tools, such as wrenches, knowing the metric system allows you to choose the precise size needed for a specific task, ensuring efficiency and precision in work.

Length Measurement

Length measurement is an essential concept in everyday activities and a vital skill for various trades, such as carpentry and engineering. Different measurement units are used depending on the context and regional standards, with the two most common systems being the metric and imperial systems.

The key to converting length units across various systems lies in understanding and knowing the correct conversion factors. For example, when converting from inches to millimeters, you apply the conversion factor of 1 inch = 25.4 millimeters. This factor comes from the precise definition used globally for this specific conversion.

Performing conversions involves simple multiplication or division. For instance, converting \(rac{3}{8}\) inches to millimeters involves multiplying by 25.4, yielding approximately 9.5 mm after rounding. This precision is often necessary in mechanical work to ensure proper fit and function. Conversions often require rounding to achieve practical accuracy, especially in fields where unit precision influences outcomes significantly.

Wrench Sizes

Wrench sizes are critical in mechanical repairs and adjustments and come in metric or imperial units. Understanding these sizes is crucial for selecting the correct tool that fits properly onto fasteners, preventing damage during use. The challenge arises with bolts and nuts that have distinct dimensions and may require conversion between imperial and metric systems.

Metric wrenches are labeled according to the width of the nut they are designed to fit, measured in millimeters. Sizes typically increase incrementally, such as 10 mm, 12 mm, 14 mm, etc. On the other hand, imperial wrenches follow fractions of an inch, like \(rac{1}{4}\), \(rac{3}{8}\), \(rac{1}{2}\) inches, usually following \(rac{1}{8}\) inch intervals.

Choosing the correct wrench size often starts with identifying the bolt diameter and then converting it to the appropriate measurement system if necessary. For instance, if you need to tighten a 12 mm bolt, you first convert 12 mm to inches (approximately 0.47 inches) and then determine the closest fractional size, which would be \(rac{1}{2}\) inch, ensuring a snug fit to prevent slipping and rounding of the bolt head.