Dimensional Analysis: Newtonian Physics Lecture Series Lecture 1: Chapter 1 Section 3

This is a natural continuation of Chapter 1 Section 2.  When we make models, we need to make sure that, when we run the numbers, the dimensions of the solution naturally make sense.  The word "dimension" in physics means something more general than the definition we are used to here in every day life.  We typically think of dimensions as length, width, and height; we do live in a three dimensional world, after all.  The word dimension in science means the units that we are working with.  This could be Distance (like length of a box), or is could be time (duration of a trip), or it could be mass (I weigh 15 stone).  The dimensions of density are kilograms per cubic meter (kg/m3).

When these dimensions are used in mathematical equations, they are treated as if they are variables we've seen in Algebra; they can be canceled and squared like any other value in mathematics.  For example, if we have a room that is 20ft.×30ft.×15ft, then we multiply the numbers together, but we also multiply the three units together to get a volume of 9,000ft3 (nine thousand cubic feet). Division works the same way.  When we are looking for the square footage of an apartment, and only know it's volume of 4,500ft3 (for some reason) and it's height of 12 ft. (assuming that it's a bottom story, so flat cieling), we can divide the two numbers and divide ft3 by ft, and vuala! we have 375ft2.

When we're adding and subtracting, the dimensions do not increase or decrease in their exponent for the same reason variables don't increase or decrease in their exponents when we're combining like terms; we are not going from one dimension to two dimensions, we are merely staying within the same dimension (adding one lines length to another line) or going from one dimension to a completely different dimension (even if they have the same unit, like when we're applying the Pythagorean Theorem).

Next time I will cover conversion of units.  Until then, stay curious.

←Previous                  Chapter 1                  Physics                  Other Lecture Series                 Next→

Comments

Post a Comment

Popular posts from this blog

General Chemistry 1 Section 1.3: Properties of Matter

Image
Hello internet, and welcome to the Science of Life General Chemistry lecture series.  In this session, I'll cover some of the properties of the materials in which chemists are interested. Properties There are two types of properties which we deal with in General Chemistry.  There are more types, but there are specifically two that are important for this course.  There are physical properties (properties which do not describe the chemistry of the substance, such as the phase) and chemical properties (which do describe the chemistry of the substance, such as whether or not it's an acid).  Properties can also be classed as either extrinsic (depends on the quantity of the substance, such as the mass) or intrinsic (does not depend on the quantity, such as melting point).  Boiling point is an intrinsic physical property; it doesn't describe the chemistry of the substance (alas the physical part) and the boiling point at that pressure doesn't change with quantity of sub
Read more

Systems of Equations in Two Variables - Chapter 3 Section 1

Typically, problems involving at least two unknowns are solved by translating the problem to two or more equations, such as the slope-intercept form and/or the standard form of the line. Developing Equations from Sentence Form When deciphering word problems, we typically try to take a word or phrase which explains what a variable is and represent that word or phrase as a single letter instead. For instance, the energy consumed by your household so far this billing cycle can be represented by the letter E, and the total cost so far in that pay cycle can be represented by the letter C. It is also useful to translate commonly used words into mathematical operations. For example, the product of two things m and c represents multiplication, while the square of c represents squaring the letter c. So the product of the two values of m and the square of c would be $m \times c^2$. I will leave a table of some common and uncommon phrases and their mathematical translations here . Outs
Read more

Algebra Chapter 1 Section 7: Scientific Notation

In science, there are many numbers which are either exceedingly large (for example, the number of water molecules in a liter of water) or exceedingly small (for example, the charge, in Coulombs, of a single electron).  This is the nature of the reality we live in.  This is why, in some cases, it is easier to read and right these extremely small and large values in a more compact notation.  Yes, this does somewhat reduce accuracy, but the loss of accuracy has very little baring on the accuracy of the final answer.  The most common compact notation in Social Sciences, Life Sciences, and Physical Sciences is called Scientific Notation . The notation for Scientific Notation is $N \times 10^{n}$, where N is a number which is at least 1 but less than 10 $[1 \le N < 10]$ and n is the magnitude of the number.  There are a couple steps to convert to scientific notation: Place a decimal point in between the first two significant digits of the number. Find the magnitude by determining
Read more