Tricki

## Dimensional correctness and scaling

### Quick description

Physical quantities and models need to be dimensionally consistent. See also dimensional analysis.

### Prerequisites

Calculus, basic algebra

### Example 1

From Newton's second law and basic assumptions, we might have derived the equation for a spring acting under gravity:

where is the height of the mass, its mass, the spring constant of the spring, is the gravitational acceleration at the Earth's surface, and is (of course) time.

Only quantities with the same dimensional units can be added. The dimensional units of a product is the product of the dimensional units: .

### General discussion

The quantity must have the units of acceleration. If we use to denote the physical units of a quantity , then ( representing the units of time), (representing the units of mass), and (representing the units of length). Thus the dimensional units of are .

The spring constant must have units .

If we are looking for the period of oscillation of the spring-mass system, then we need to look for something that has units . Since is an a priori unknown function of , our period must depend only on the other quantities: , , . Now means that . Since , and are independent units, this comes down to the equations for : , for : , and for : . This has the solution , and . Thus we should be looking for a dimensionless multiple of . Indeed the period is .