1.13:
Problem Solving: Dimensional Analysis
Suppose a simple pendulum of mass m is attached to a string of length L, oscillating under the influence of gravity g. What is the form of the equation for the pendulum's time period?
Initially, identify and list the variables involved in the problem. The time period T can be expressed as the product of these variables, each raised to an unknown exponent. Here, k is a dimensionless constant.
Excluding the dimensionless constant, an equation relating the dimensions of the variables with the time period is obtained.
Now, by equating the exponents of the dimensions on both sides and solving the equations, the values of the unknown exponents are determined.
On substituting the exponents, the final expression for the time period is obtained, which is a product of the constant k and the square root of the length over gravitational acceleration.
One of the limitations of dimensional analysis is that it does not allow us to find the value of the dimensionless constant k.
1.13:
Problem Solving: Dimensional Analysis
Every mathematical equation that connects separate distinct physical quantities must be dimensionally consistent, which implies it must abide by two rules. For this reason, the concept of dimension is crucial. The first rule is that an equation's expressions on either side of an equality must have the exact same dimension, i.e., quantities of the same dimension can be added or removed. The second rule stipulates that all popular mathematical functions, such as exponential, logarithmic, and trigonometric functions, must have dimensionless arguments in an equation.
It is dimensionally inconsistent for an equation to break either of these two rules, so an equation cannot be a representation of any physical law's accurate assertion. Dimensional analysis can help to remember the different laws of physics, check for algebraic errors or typos, and even speculate on the shape that future laws of physics might take.
The base quantities can be used to create any desired physical quantities. A quantity is stated as the product of various powers of the base quantities when it is expressed in terms of the base quantities. The dimension of the quantity in that base is the exponent of a base quantity that appears in the equation.
Consider the physical quantity force, which is defined as mass multiplied by acceleration. Acceleration is calculated as the change of velocity divided by a time interval, while the length divided by the time interval equals velocity. As a result, force has the following dimensions: one in mass, one in length, and minus two in time.
Suggested Reading
- OpenStax. (2019). University Physics Vol. 1. [Web version]. Retrieved from https://openstax.org/books/university-physics-volume-1/pages/1-4-dimensional-analysis; section 1.4; pages 22-25.