Dimensional Analysis Practice Problems Worksheet Answers

Dimensional Analysis Practice Problems Worksheet Answers: A Comprehensive Guide

Dimensional analysis, also known as the factor-label method or unit analysis, is a powerful technique used to convert units and solve problems in physics, chemistry, and engineering. It's based on the principle that physical quantities have both a magnitude and a unit, and these units must be consistent throughout a calculation. Mastering dimensional analysis is crucial for success in STEM fields. This comprehensive guide provides a detailed explanation of dimensional analysis, accompanied by practice problems and their solutions.

Understanding Dimensional Analysis

Dimensional analysis relies on the systematic cancellation of units using conversion factors. A conversion factor is a ratio of two equivalent quantities expressed in different units. For example, 1 meter = 100 centimeters, so the conversion factors are 1 m/100 cm and 100 cm/1 m. By carefully selecting and multiplying these factors, we can change the units of a quantity without altering its value.

The key to success in dimensional analysis is to:

• Identify the starting quantity and its units.
• Identify the desired units.
• Find appropriate conversion factors to link the starting and desired units.
• Set up the calculation to ensure that unwanted units cancel.
• Perform the arithmetic to obtain the final answer with the correct units.

Practice Problems and Solutions

Let's dive into some practice problems to solidify your understanding of dimensional analysis. These problems span various complexities, from simple unit conversions to more involved multi-step calculations.

Problem 1: Simple Unit Conversion

Question: Convert 1500 meters to kilometers.

Solution:

We know that 1 kilometer (km) = 1000 meters (m). Therefore, our conversion factor is 1 km/1000 m.

1500 m * (1 km / 1000 m) = 1.5 km

Answer: 1500 meters is equal to 1.5 kilometers.

Problem 2: Multi-Step Unit Conversion

Question: A car travels at a speed of 60 miles per hour (mph). Convert this speed to meters per second (m/s). Use the following conversion factors: 1 mile = 1609 meters, 1 hour = 3600 seconds.

Solution:

We need to convert miles to meters and hours to seconds. We'll use a chain of conversion factors:

60 mph * (1609 m / 1 mile) * (1 hour / 3600 s) = 26.82 m/s

Answer: 60 mph is approximately equal to 26.82 meters per second.

Problem 3: Area Conversion

Question: Convert 10 square feet (ft²) to square meters (m²). Use the conversion factor 1 foot = 0.3048 meters.

Solution:

Remember that area is a two-dimensional quantity. We need to square the conversion factor:

10 ft² * (0.3048 m / 1 ft)² = 0.929 m²

Answer: 10 square feet is approximately equal to 0.929 square meters.

Problem 4: Volume Conversion

Question: Convert 5 gallons of water to liters. Use the conversion factor 1 gallon = 3.785 liters.

Solution:

This is a simple unit conversion.

5 gallons * (3.785 liters / 1 gallon) = 18.925 liters

Answer: 5 gallons is equal to 18.925 liters.

Problem 5: Density Calculation

Question: A block of metal has a mass of 150 grams and a volume of 20 cubic centimeters. Calculate its density in grams per cubic centimeter (g/cm³).

Solution:

Density is defined as mass per unit volume: Density = Mass / Volume

Density = 150 g / 20 cm³ = 7.5 g/cm³

Answer: The density of the metal block is 7.5 grams per cubic centimeter.

Problem 6: More Complex Unit Conversion with Multiple Steps

Question: A river flows at a rate of 25 cubic feet per minute (ft³/min). Convert this flow rate to cubic meters per hour (m³/hr). Use the conversion factors: 1 foot = 0.3048 meters, 1 minute = 1/60 hour.

Solution:

This problem requires converting cubic feet to cubic meters and minutes to hours. Remember to cube the linear conversion factor for volume:

25 ft³/min * (0.3048 m / 1 ft)³ * (60 min / 1 hr) ≈ 42.5 m³/hr

Answer: The river flow rate is approximately 42.5 cubic meters per hour.

Problem 7: Combined Units

Question: Convert a speed of 50 kilometers per hour (km/hr) to meters per second (m/s).

Solution:

This involves converting kilometers to meters and hours to seconds.

50 km/hr * (1000 m/1 km) * (1 hr/3600 s) ≈ 13.89 m/s

Answer: 50 km/hr is approximately 13.89 m/s.

Problem 8: Using Multiple Conversion Factors

Question: A rectangular field measures 100 yards by 50 yards. What is the area of the field in square meters? (1 yard = 0.9144 meters)

Solution:

First, find the area in square yards: 100 yards * 50 yards = 5000 square yards. Then, convert square yards to square meters:

5000 yd² * (0.9144 m / 1 yd)² ≈ 4180.64 m²

Answer: The area of the field is approximately 4180.64 square meters.

Problem 9: Problem Solving with Units

Question: A cylindrical tank has a radius of 2 meters and a height of 5 meters. What is its volume in liters? (1 cubic meter = 1000 liters)

Solution:

First, calculate the volume of the cylinder using the formula V = πr²h:

V = π * (2 m)² * 5 m ≈ 62.83 m³

Then convert cubic meters to liters:

62.83 m³ * (1000 L / 1 m³) = 62830 L

Answer: The volume of the tank is approximately 62,830 liters.

Problem 10: Advanced Problem – Fuel Efficiency

Question: A car travels 300 miles on 10 gallons of gasoline. What is its fuel efficiency in kilometers per liter? Use the following conversion factors: 1 mile = 1.609 kilometers, 1 gallon = 3.785 liters.

Solution:

First, calculate miles per gallon: 300 miles / 10 gallons = 30 miles/gallon. Then convert miles to kilometers and gallons to liters:

30 miles/gallon * (1.609 km / 1 mile) * (1 gallon / 3.785 L) ≈ 12.74 km/L

Answer: The fuel efficiency of the car is approximately 12.74 kilometers per liter.

Tips for Success in Dimensional Analysis

• Write out all units: This helps you track units and ensure proper cancellation.
• Be meticulous: Pay close attention to detail to avoid errors.
• Use consistent units: Make sure all units are consistent within a problem.
• Check your answer: Always check if your answer makes sense and if the units are correct.
• Practice regularly: The more you practice, the more comfortable and proficient you will become.

This extensive worksheet provides a range of dimensional analysis problems, progressing in complexity. By working through these examples and understanding the underlying principles, you’ll significantly improve your problem-solving skills in science and engineering. Remember that consistent practice is key to mastering dimensional analysis. Don’t hesitate to revisit these problems and try similar ones to reinforce your learning.