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2004-D: Dimensional Analysis 1: Simplifying Dimensions

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2004-D: Dimensional Analysis 1: Simplifying Dimensions

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In this ladder, we are going to learn about how to use dimensional analysis to analyze and confirm physics formulas. Dimensional analysis is a difficult topic, and before we can learn it, we need to practice some mathematical techniques. Firstly, we will practice simplifying fractions made from three units: meters, seconds, and kilograms.

Simplify each of the following combinations of units.

  1. \( \frac{\text{m}}{\text{kg}}\cdot\frac{\text{s}}{\text{m}^2} \)
  2. \( \frac{\text{m}}{\text{s}^2} \cdot \frac{\text{m}^3}{\text{s}^3} \cdot \frac{\text{kg}^2}{\text{m}^2} \)
  3. \( \frac{\text{m}^2}{\text{kg}} \cdot \frac{\text{kg}^3\text{m}}{\text{s}} \cdot \frac{\text{kg}^2\text{s}^3}{\text{m}^3} \)
  4. \( \frac{\text{m}^4}{\text{kg}} \cdot \frac{\text{m}}{\text{kg}^2} \cdot \frac{\text{kg}}{\left( \text{m} \cdot \text{s} \right)^3} \)
  5. \( \frac{\text{m}^4}{\text{s}^3} \cdot \frac{\text{s}^2}{\text{m}^2} \cdot \frac{\text{kg}}{\text{s}^4} \)
  6. \( \frac{\text{m}^5}{\text{kg}^3} \cdot \frac{\text{s}^2}{\text{kg}^2} \cdot \left( \frac{\text{s}}{\text{m} } \right)^2 \)
  7. \( \frac{\text{s}}{\text{m^3}} \cdot \left( \frac{\text{s}}{\text{m}} \right)^3 \cdot \frac{\text{m}^7}{\text{s}^4} \)
  8. \( \frac{\text{s}^4}{\text{kg}^3} \cdot \left( \frac{\text{m}}{\text{kg}} \right)^4 \cdot \left( \frac{\text{s}}{\text{m}} \right)^2 \)

Fractions Made of Fractions

It can be difficult to simplify fractions of fractions. There are two methods by which you can do this:

Take whatever fraction is in the denonminator, flip it over, and multiply by it.

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