p>Dimensional analysis, also known as factor-label method or unit conversion, is a powerful problem-solving technique used across various scientific disciplines, including physics, chemistry, engineering, and even everyday life. It’s a systematic way to convert units of measurement by multiplying by conversion factors. A conversion factor is a ratio that expresses how many of one unit are equal to another unit (e.g., 1 inch = 2.54 centimeters). By carefully tracking and canceling units, dimensional analysis helps ensure accuracy in calculations and provides a built-in error-checking mechanism. A Dimensional Analysis Worksheet is designed to give you practice applying this technique, reinforcing your understanding of units and how they relate to each other. This post will provide a key to a typical dimensional analysis worksheet. This key is intended to allow you to check your work and identify any areas where you might need additional review.
Dimensional Analysis Worksheet Key
This key provides answers and explanations for common dimensional analysis problems. Remember that showing your work, including the proper units and cancellation steps, is just as important as arriving at the correct final numerical answer. Pay close attention to the conversion factors used and how they’re arranged to ensure units cancel correctly. The goal is not just to memorize answers but to understand the process. A solid understanding of dimensional analysis is crucial for success in many scientific and technical fields.
Example Problems and Solutions
Here is the Dimensional Analysis Worksheet Key, formatted in HTML to be easily readable. Note that the specific problems on *your* worksheet might vary, but the principles and methods demonstrated here will apply.
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Problem 1: Converting meters to inches. Given: 5.0 meters. Conversion factor: 1 inch = 2.54 centimeters; 1 meter = 100 centimeters.
Solution:
To convert meters to inches, we need two conversion factors. First, we convert meters to centimeters, and then centimeters to inches.- 5.0 meters * (100 centimeters / 1 meter) * (1 inch / 2.54 centimeters) = 196.85 inches (approximately)
Answer: 197 inches (approximately, rounded to three significant figures)
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Problem 2: Converting miles per hour to meters per second. Given: 60 miles per hour. Conversion factors: 1 mile = 1609 meters; 1 hour = 3600 seconds.
Solution:
We need to convert both miles to meters and hours to seconds. This requires multiplying by two conversion factors.- 60 miles / 1 hour * (1609 meters / 1 mile) * (1 hour / 3600 seconds) = 26.82 meters/second (approximately)
Answer: 27 m/s (approximately, rounded to two significant figures)
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Problem 3: Calculating the volume of a rectangular prism in cubic centimeters. Given: Length = 2.0 inches, Width = 3.0 inches, Height = 4.0 inches. Conversion factor: 1 inch = 2.54 centimeters.
Solution:
First, calculate the volume in cubic inches: Volume = Length * Width * Height = 2.0 in * 3.0 in * 4.0 in = 24 cubic inches. Then, convert cubic inches to cubic centimeters. Important note: because the volume is *cubic* inches, the conversion factor also needs to be cubed to ensure the correct unit cancellation. (1 inch)^3 = (2.54 cm)^3- Volume = 24 in3 * (2.54 cm / 1 in)3 = 24 in3 * (16.387 cm3 / 1 in3) = 393.29 cm3 (approximately)
Answer: 390 cm3 (approximately, rounded to two significant figures)
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Problem 4: Converting grams per milliliter to kilograms per liter. Given: Density = 1.5 g/mL. Conversion factors: 1 kilogram = 1000 grams; 1 liter = 1000 milliliters.
Solution:
We need to convert grams to kilograms and milliliters to liters.- 1.5 grams / 1 milliliter * (1 kilogram / 1000 grams) * (1000 milliliters / 1 liter) = 1.5 kilograms/liter
Answer: 1.5 kg/L
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Problem 5: Converting a flow rate from gallons per minute to liters per second. Given: 10 gallons per minute. Conversion factors: 1 gallon = 3.785 liters; 1 minute = 60 seconds.
Solution:
This problem requires converting gallons to liters and minutes to seconds.- 10 gallons / 1 minute * (3.785 liters / 1 gallon) * (1 minute / 60 seconds) = 0.6308 liters/second (approximately)
Answer: 0.63 L/s (approximately, rounded to two significant figures)
Remember to pay close attention to significant figures and rounding rules in your final answers. The key is understanding how to correctly apply the conversion factors to arrive at the desired units. Practice is key to mastering dimensional analysis. Good luck!
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