# Mechanics

calculus-based physics for scientists and engineers

## Dimensions

### Dimension of a Physical Quantity

We (the royal we, meaning physicists) have established seven quantities to serve as fundamental quantities. The dimension of these quantities is simply the name of the quantity! Since only three are used in mechanics, I've limited our discussion to those, which are shown below.

•   Quantity
• Length
• Mass
• Time
•   Dimension Name
• Length
• Mass
• Time
•   Dimension Symbol
• L
• M
• T

Other quantities don't get their own dimension name. Their dimension is written in terms of the fundamental dimensions.

example: The quantity of force has a dimension of Mass·Length/Time2, symbolically given by M·L/T2.

In the short video below, I describe what this means and the notation one uses to work with dimensions.

### Three Rules for Manipulating Dimensions

As you might have heard, in physics we do math with physical quantities! There are three rules to remember about dimensions when it comes to math.

1. You can only add quantities with the same dimension. This means that both qualities on either side of an equals sign have the same dimension.
2. Dimensions multiply like scalars.
3. Exponents and results of trig functions and logarithms are dimensionless. The arguments of trig functions are also dimensionless.

To learn the notation and how to work with dimensions, see the short video below.

### Dimensional Analysis

Dimensional analysis is a way of using the above rules to both determine the dimensions of physical quantities and check your work when you solve problems. The best way to learn this is to see and do examples. The video below walks you through it.

### Solution

1. If the position of a particle, $$x(t)$$ as a function of time, $$t$$ is given by $$x(t) = \alpha t^2 + \beta t^3 + C \sin (\omega t)$$ what are the dimensions of $$\alpha$$, $$\beta$$, $$C$$, and $$\omega$$?

### Solution

2. To keep an object moving in circular motion, a force must be applied to the object that is equal to some dimensionless constant times a product of the mass of the object, $$m$$, the radius of circular motion, $$r$$ (a length), and the speed of the object, $$v$$, each to some unknown power. We would write such an expression as: $$F = \alpha m^q r^p v^n$$ where $$\alpha$$ is the dimensionless constant. Find the unknown powers using dimensional analysis.

### Solution

1. According to Newton's universal law of gravity, the magnitude of the force of gravity a particle with mass, $$m_1$$ exerts on a mass, $$m_2$$ is given by $$F = \frac{Gm_1m_2}{r^2}$$ where $$r$$ is the distance between them and $$G$$ is a constant. What is the dimension of $$G$$?