## Dimensional Analysis

1.4 Dimensional Analysis

### Learning Objectives

By the end of this section, you will be able to:

- Find the dimensions of a mathematical expression involving physical quantities.
- Determine whether an equation involving physical quantities is dimensionally consistent.

### Dimensional Analysis

The system of units we will use in this course in the SI system, otherwise known as the metric system. In the SI system, the base unit for length is the meter (m), for mass is the kilogram (kg), and for time is the second (s). All the other units we use in this course will be derived from these base units. For example, the units for speed come from the dimensions, which are length/time, so the derived units are meters/second or m/s.

To determine the units for speed, we just used a method called **dimensional analysis**. This is a useful tool that can be used to check your work, or determine the formula needed for a calculation. Dimensional analysis is really just a game of matching dimensions. The dimensions on the left hand side of an equation must be equal to the dimensions on the right hand side. Here’s an example: let’s say you need to calculate a distance traveled and you believe the equation you need is where ** d** is the distance,

**is the speed, and**

*v***is the time. The right hand side of the equation is a measure of .**

*t*The dimensions on the left hand side of the equation don’t match with the dimensions on the right, so this equation must be wrong. To fix the dimensions, we could take and multiply by . This means the correct equation for finding the distance would be .

**Practice!**

Practice 1.4.1 |
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Which of these equations can represent a physical equality? |

(a) 3 m = 3 s |

(b) 1 m = 1 m^{2} |

(c) 3 m = 1 m+ 2 m^{2} |

(d) 4 m^{2} = 1 m^{2} + 3 m^{2} |

(e) All of them |

(f) None of them |

Practice 1.4.2 |
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Newton’s second law of motion says that the mass of an object times its acceleration is equal to the net force on the object. Which of the following gives the correct units for force? |

(a) kg•m^{2}/s^{2} |

(b) kg/m•s^{2} |

(c) kg•m/s^{2} |

(d) kg•m^{2}/s |

(e) Other |

Practice 1.4.3 |
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The period of a swinging pendulum, which has units of seconds, depends only on the length of the pendulum P and the acceleration of gravity L. Using dimensional analysis, determine which of the following formulas for g could be correct .P |

(a) |

(b) |

(c) |