1.12
Dimensional Analysis
The concept of dimension is important because every mathematical equation linking physical quantities must be dimensionally consistent, implying that…
All physical quantities can be expressed using either base quantities or derived quantities and each quantity is represented by a symbol, which defines its dimensions.
For instance, the speed of a car is defined as the distance divided by time. The term distance corresponds to the quantity length, denoted with L and time with T.
Hence, we can write the dimension of the quantity speed, as L divided by T or LT to the power of minus one.
For an equation to be dimensionally correct, it should obey two rules. Number one, the expressions on each side of the equality in an equation must have the same dimensions.
Number two, the standard mathematical functions in equations must be dimensionless
For example, we know the dimension of volume is L cubed. Now, consider a cylinder with radius r and height h.
We know that the volume of a cylinder is π r squared h. The term π is a constant, and it's a dimensionless quantity. The term r corresponds to the quantity length, and we can write its dimension as L squared, and the term h also corresponds to the quantity length, which gives the dimension of the volume of the cylinder as L cubed. Hence, the equation is dimensionally correct.
As long as we know the dimensions of the individual physical quantities that appear in an equation, we can check to see whether the equation is dimensionally consistent.
Another application of dimensional analysis is to remember an equation. For example, let's say that you don't remember whether speed equals time divided by distance or distance divided by time.
The dimensions of time, distance, and speed are T, L, and LT to the power of minus one respectively. Reducing both the equations to their fundamental units on each side of the equation, we get speed equals distance divided by time.
View the full transcript and gain access to JoVE Core videos
Q1: What are dimensions in physics and how do they relate to physical quantities?
Dimensions represent the fundamental nature of physical quantities using symbols like L for length and T for time. All physical quantities, whether base quantities and derived quantities, are expressed through these dimensional symbols. For example, speed has dimensions of LT⁻¹ because it equals distance divided by time. Understanding dimensions allows you to verify equations and check calculations systematically.
Q2: What are the two rules for dimensional consistency in equations?
First, expressions on each side of an equality must have identical dimensions. Second, arguments of standard mathematical functions like trigonometric or logarithmic functions must be dimensionless. For instance, in the cylinder volume equation πr²h, the constant π is dimensionless while r² and h contribute length dimensions, yielding L³ total. Violating either rule means the equation cannot represent a correct physical law.
Q3: How can dimensional analysis help you remember physics equations?
Dimensional analysis lets you verify which form of an equation is correct by checking dimensions. If you forget whether speed equals time divided by distance or distance divided by time, you can test both using dimensions. Since speed has dimensions LT⁻¹, only distance divided by time (L/T) produces the correct dimensions, confirming the right equation form.
Q4: What happens to dimensions when you take the derivative of a physical quantity?
The derivative of a function represents a slope, which is a ratio. For physical quantities, the dimension of a derivative equals the dimension of the numerator divided by the dimension of the denominator. For example, the derivative of velocity with respect to time has dimensions of velocity divided by time, or LT⁻¹ divided by T, yielding LT⁻² for acceleration.
Q5: How do dimensions change when you integrate a physical quantity?
Since integrals represent sums of products, the dimension of an integral equals the dimension of the quantity being integrated multiplied by the dimension of the variable of integration. For instance, integrating velocity with respect to time yields dimensions of LT⁻¹ times T, which equals L, representing displacement. This dimensional relationship helps verify integral expressions in physics.
Q6: Why is dimensional consistency important for checking physics equations?
Dimensional consistency ensures equations represent valid physical laws. By reducing both sides of an equation to their fundamental dimensions, you can catch algebraic errors or typos before calculations. If dimensions don't match on both sides, the equation is incorrect. This method provides a quick, systematic way to validate formulas without performing full numerical computations.
Q7: Can dimensionless constants appear in dimensionally correct equations?
Yes, dimensionless constants like π can appear in equations without affecting dimensional consistency. In the cylinder volume formula πr²h, π is dimensionless and doesn't change the overall dimensions. The radius squared contributes L², height contributes L, and their product yields L³ for volume. Dimensionless numbers multiply quantities without altering their dimensional properties.