Dimensional Analysis: Units, Abbreviations, and Setup

Volume is measured in liters; length is measured in meters. Abbreviations are case sensitive, so write them with the correct capitalization.
Memorize these unit abbreviations first; they are essential for solving problems and you will need to plug them into dimensional analysis problems.
Do not use a trailing zero for whole numbers. If a value is a whole number, write it without a trailing ".0" (e.g., write 2, not 2.0).
In dimensional analysis, the unit you want the answer in goes on the top (numerator) of the first fraction, and the units you are canceling go on the bottom. The goal is to set up factors so that the undesired units cancel out and the desired units remain.
The units on top should be the opposite of the units on the bottom in each conversion factor, so they cancel appropriately.
Example context: to convert from kilograms to pounds, you will use a factor that relates kilograms to pounds and arrange it so the kg unit cancels and the result is in pounds.
Important reminder from the transcript (and a common pitfall to watch for): there was a misstatement about pounds and kilograms. The correct relationship is $1\,\text{kg} \approx 2.20462\,\text{lb}$ (i.e., kilograms per pound and pounds per kilogram are reciprocal). Use the correct form to avoid errors.

The problem should be set up so that the units on the top (desired units) remain, and the units on the bottom cancel out.
Always show the setup using conversion factors, not just the final answer. This demonstrates the dimensional reasoning and helps secure credit.
The first conversion factor to use should match any explicit unit mentioned in the problem (e.g., seconds). If a problem involves time, begin with a time-based conversion like minutes or hours to seconds.
Common structure: start with the given quantity in its unit, then multiply by a series of factors in which the numerator is the unit you want and the denominator is the unit you are canceling.
Practical rule: whenever you write a conversion factor, ensure it is of the form $\dfrac{\text{desired unit}}{\text{original unit}}$ or its reciprocal, so units cancel properly and the final unit matches the target.

Example 1: How many pounds are in six kilograms?
- Setup (show the units clearly):
- $6\,\text{kg} \times \dfrac{2.20462\,\text{lb}}{1\,\text{kg}} = 13.2277\,\text{lb}.$
- Explanation: The kg in the denominator cancels with the kg in the numerator, leaving pounds on the result.
- Alternative valid setup using the reciprocal form: $6\,\text{kg} \times \dfrac{1\,\text{lb}}{0.453592\,\text{kg}} = 13.2277\,\text{lb}.$
- Note: The exact numeric value depends on the precision used for the conversion factor; 13.23 lb is a common rounded result.
Example 2: How many inches are in four feet?
- Setup:
- $4\,\text{ft} \times \dfrac{12\,\text{in}}{1\,\text{ft}} = 48\,\text{in}.$
- Explanation: 1 ft = 12 in, so multiplying by the factor converts feet to inches and cancels the ft unit.
Time unit example (as emphasized in the transcript about choosing the first conversion factor):
- If the problem mentions seconds, start with a conversion to seconds, for example:
- $1\,\text{min} = 60\,\text{s}$
- $1\,\text{h} = 3600\,\text{s}$
- These factors allow you to convert minutes or hours into seconds and then proceed with any remaining conversions.
Example of a common conversion around the transcript’s mention of scientific notation:
- Values may appear as a number in scientific notation, e.g., $3.5\times 10^3$ , and should be handled with the same dimensional analysis approach.

Forgetting to put the final units on top or placing them incorrectly can yield an incorrect result.
Swapping the numerator and denominator in a conversion factor will give the wrong answer; always ensure the units cancel as intended.
Relying solely on mental math is discouraged in formal work; always show the setup to illustrate the dimensional reasoning.
When a statement in a source seems mathematically off (e.g., 1 pound equals 2.2 kilograms), rely on the standard, correct conversion and cite it as a correction if needed.

Convert 8 kg to pounds using the correct conversion factor.
Convert 5 feet to inches using the appropriate conversion factor.
If a problem uses minutes and seconds, convert minutes to seconds first using $1\,\text{min} = 60\,\text{s}$ and then proceed with any remaining conversions.

Dimensional analysis relies on placing the desired units on the top and arranging conversion factors so that all other units cancel.
Always show your setup, verify unit cancellation, and then compute the numerical value with appropriate significant figures.
Use correct, precise conversion factors (e.g., $1\,\text{kg} \approx 2.20462\,\text{lb}$ ) and be mindful of common misconceptions.