Dimensional Analysis Worksheet 2 Answer Key

Dimensional Analysis Worksheet 2: Answers and Comprehensive Guide

Dimensional analysis, also known as the factor-label method or unit analysis, is a powerful problem-solving technique used extensively in science and engineering. It allows you to convert units and solve complex problems by focusing on the relationships between different units. This comprehensive guide provides answers and detailed explanations for a hypothetical "Dimensional Analysis Worksheet 2," covering a range of difficulty levels. We'll explore various conversion scenarios, emphasizing the systematic approach required for accurate results.

Understanding the Fundamentals of Dimensional Analysis

Before diving into the answers, let's solidify the core principles:

1. Units as Algebraic Quantities: In dimensional analysis, units behave like algebraic variables. You can multiply, divide, and cancel them just as you would with variables like 'x' or 'y'.

2. Conversion Factors: The key to successful dimensional analysis is using conversion factors. A conversion factor is a ratio equal to 1, expressing the relationship between two different units. For example, 1 meter = 100 centimeters, so the conversion factors are 1 m/100 cm and 100 cm/1 m.

3. Setting up the Problem: The most effective approach is to start with the given quantity and its units. Then, multiply by a series of conversion factors to arrive at the desired units, ensuring that unwanted units cancel out.

Hypothetical Dimensional Analysis Worksheet 2: Problems and Solutions

This section presents a series of problems, mimicking a typical worksheet, followed by detailed solutions using dimensional analysis.

Problem 1: Converting Kilometers to Miles

Question: Convert 25 kilometers to miles, knowing that 1 mile ≈ 1.609 kilometers.
Solution:

We start with the given quantity: 25 km. We need to multiply by a conversion factor that cancels out kilometers and introduces miles.

25 km * (1 mile / 1.609 km) ≈ 15.53 miles

Therefore, 25 kilometers is approximately equal to 15.53 miles.

Problem 2: Converting Cubic Centimeters to Liters

Question: A container holds 500 cubic centimeters (cm³) of liquid. Convert this volume to liters (L). Remember that 1 L = 1000 cm³.
Solution:

Begin with the given volume: 500 cm³. We use the conversion factor 1 L / 1000 cm³ to convert cubic centimeters to liters.

500 cm³ * (1 L / 1000 cm³) = 0.5 L

Therefore, 500 cubic centimeters is equal to 0.5 liters.

Problem 3: Converting Grams to Kilograms and then to Pounds

Question: Convert 750 grams (g) to pounds (lb), given that 1 kilogram (kg) = 1000 g and 1 kg ≈ 2.205 lb.
Solution: This problem requires two conversion factors. First, convert grams to kilograms, then kilograms to pounds.

750 g * (1 kg / 1000 g) * (2.205 lb / 1 kg) ≈ 1.65 lb

Therefore, 750 grams is approximately equal to 1.65 pounds.

Problem 4: Calculating Speed with Unit Conversion

Question: A car travels 150 kilometers in 2.5 hours. Calculate its average speed in meters per second (m/s).
Solution:

First, we calculate the speed in kilometers per hour (km/hr):

150 km / 2.5 hr = 60 km/hr

Now, we need to convert this to meters per second. We'll use multiple conversion factors:

60 km/hr * (1000 m / 1 km) * (1 hr / 3600 s) ≈ 16.67 m/s

Therefore, the car's average speed is approximately 16.67 meters per second.

Problem 5: Complex Unit Conversion Involving Area

Question: Convert an area of 2 square meters (m²) to square centimeters (cm²).
Solution:

Remember that area is a square unit. We need to square the conversion factor between meters and centimeters.

2 m² * (100 cm / 1 m)² = 2 m² * (10000 cm²/1 m²) = 20000 cm²

Therefore, 2 square meters is equal to 20000 square centimeters.

Problem 6: Density and Unit Conversion

Question: The density of a substance is 2.7 g/cm³. What is its density in kg/m³?
Solution:

We need to convert grams to kilograms and cubic centimeters to cubic meters. Again, we must cube the linear conversion factor.

2.7 g/cm³ * (1 kg / 1000 g) * (100 cm / 1 m)³ = 2.7 g/cm³ * (1 kg / 1000 g) * (1000000 cm³/1 m³) = 2700 kg/m³

Therefore, the density of the substance is 2700 kg/m³.

Problem 7: Volume and Unit Conversion with Irregular Shapes

Question: An irregularly shaped object displaces 350 mL of water. Convert this volume to cubic meters (m³). Remember 1L = 1000 mL and 1L = 1000 cm³
Solution:

First convert mL to L, then L to cm³, and finally cm³ to m³:

350 mL * (1 L / 1000 mL) * (1000 cm³/1 L) * (1 m/100 cm)³ = 0.00035 m³

Therefore, the volume of the irregularly shaped object is 0.00035 cubic meters.

Problem 8: A Real-World Application: Fuel Efficiency

Question: A car achieves a fuel efficiency of 25 miles per gallon (mpg). Convert this to kilometers per liter (km/L). Use the following conversions: 1 mile ≈ 1.609 km and 1 gallon ≈ 3.785 liters.
Solution:

25 miles/gallon * (1.609 km/1 mile) * (1 gallon/3.785 liters) ≈ 10.7 km/L

Therefore, the car's fuel efficiency is approximately 10.7 kilometers per liter.

Advanced Dimensional Analysis Problems

The following problems introduce more complex scenarios requiring a deeper understanding of dimensional analysis principles.

Problem 9: Combined Units and Multiple Conversions

Question: Convert a speed of 50 miles per hour (mph) to meters per second (m/s).
Solution:

This requires several conversion steps: miles to kilometers, kilometers to meters, hours to minutes, and minutes to seconds.

50 miles/hr * (1.609 km/1 mile) * (1000 m/1 km) * (1 hr/60 min) * (1 min/60 s) ≈ 22.35 m/s

The speed is approximately 22.35 meters per second.

Problem 10: Dealing with Exponents in Units

Question: Convert a volume flow rate of 100 cubic feet per minute (ft³/min) to cubic meters per second (m³/s). (1 ft ≈ 0.3048 m)
Solution:

Remember to cube the linear conversion factor for volume:

100 ft³/min * (0.3048 m/1 ft)³ * (1 min/60 s) ≈ 0.0472 m³/s

The volume flow rate is approximately 0.0472 cubic meters per second.

Tips for Mastering Dimensional Analysis

Organize your work: Write out each step clearly, showing all units and conversion factors. This helps prevent errors and makes it easier to check your work.
Cancel units systematically: Make sure that units cancel out correctly. If they don't, you've likely made a mistake in your setup.
Use appropriate significant figures: Pay attention to the number of significant figures in your measurements and round your final answer accordingly.
Practice regularly: The more you practice dimensional analysis, the better you'll become at identifying the correct conversion factors and setting up the problems efficiently.
Break down complex problems: If a problem seems overwhelming, break it down into smaller, more manageable steps.

By understanding the fundamental principles and practicing with a variety of problems, you can develop a strong command of dimensional analysis. This skill will be invaluable in your scientific and engineering endeavors, allowing you to confidently approach complex unit conversions and problem-solving scenarios. Remember, practice is key to mastering this essential technique. Consistent effort will lead to greater proficiency and accuracy in applying dimensional analysis to diverse calculations.