Dimensional Analysis
Related Examples and Practice Problems
Additional Worked Out Examples/ Practice
Density calculations: Practice using the density formula
Density comparisons: Comparing the densities to determine which will float or sink
Density and volume relationships: Changes in mass or volume affect density
Density and temperature relationships: Impact of temperature on density
Topic Summary & Highlights
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Core Concept
Dimensional analysis (also called factor-label method or picket fence) is a method used to convert between units, perform unit conversions, and check the correctness of mathematical equations.
It relies on the principles of dimensional consistency, which states that physical quantities being added, subtracted, multiplied, or divided must have the same dimensions or units.
Practice Tips
Set Up Conversion Factors: Use conversion factors written as fractions (e.g., 1 km=1000 m\text{e.g., } 1 \, \text{km} = 1000 \, \text{m}e.g., 1km=1000m) to cancel out units and guide the calculation.
Write Units Explicitly: Always include units throughout calculations to track what cancels out and ensure the desired units remain.
Check for Unit Consistency: Confirm that the final units match the target measurement (e.g., speed in m/s\text{m/s}m/s, volume in cm3\text{cm}^3cm3).
Use a Step-by-Step Approach: Break complex conversions into smaller steps, such as converting time from hours to seconds before applying it to velocity.
Topic Overview Podcast
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Core Concept
The process of dimensional analysis involves using conversion factors, which are ratios of equivalent quantities expressed in different units, to convert from one set of units to another.
Here's a step-by-step approach to using dimensional analysis:
Identify the given quantity: Identify the quantity you have, along with its unit.
Determine the desired unit: Determine the unit you want to convert the given quantity to.
Set up conversion factors: Find or derive the appropriate conversion factors that relate the given unit to the desired unit. Conversion factors often come from conversion tables, unit equivalencies, or relationships derived from mathematical formulas.
Construct conversion factor chains: Use multiple conversion factors as needed to create a chain of ratios that cancel out unwanted units and leave you with the desired unit. Each conversion factor should be chosen in a way that the units cancel out appropriately.
Perform the calculation: Multiply the given quantity by the conversion factors, making sure that units cancel out correctly. The final result will be the desired quantity expressed in the desired unit.
Metric Conversions
A base unit is the basis of measurement in the sciences. The most commonly used base units are liters (L) for liquids, grams (g) for mass, and meters (m) for distance. Metric base units can be converted to other useful quantities by adding a prefix (see the common metric conversions table below). The base unit does not have a prefix, and these prefixes are not used in any other measuring system.
Single Step Conversion
Example: How many meters are in 735 kilometers?
$735 \, \text{km} \times \frac{1000 \, \text{m}}{1 \, \text{km}} = 735 \times 1000 \, \text{m} = 735,000 \, \text{meters}$
Multi-Step Conversions
Multi-step conversions are needed when a direct conversion is not known or when multiple units are present that need to be converted. Multi-step conversions are solved the same way as one-step conversions. However, because there are multiple conversion factors, the steps are repeated as many times as needed.
Example: Convert 67 miles per hour into meters per second.
$\frac{67 \, \text{miles}}{\text{hour}} \times \frac{1609.34 \, \text{meters}}{1 \, \text{mile}} \times \frac{1 \, \text{hour}}{3600 \, \text{seconds}} = \frac{67 \times 1609.34}{3600} \, \frac{\text{meters}}{\text{second}} = 29.94 \, \frac{\text{meters}}{\text{second}}$
