Dimensions in Physics: physicsthisweek.com
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Greetings this is Dr. Johnson-Steigelman
from physicsthisweek.com. Let’s start out our first unit in physics by
first talking about dimensions. It turns out that in physics, there are two ways that we used the term dimension. The first of these ways that we use it is the way that you’re probably more familiar with, when we talk about the dimensions
that we use in geometry. So an object that has zero dimension would be just a
point, where as a line has one dimension. If you have a surface, in other words, you
take that line and spread it out in a direction perpendicular to the original
line, you get a two-dimensional plane. That is literally a flat surface. Now if
we expand it in the other direction we get a three-dimensional object. Now in
physics, we usually use three dimensions X,Y, and Z. The orientation of those
will depend on the problem that we’re working on, but more often than that we
actually limit ourselves to two dimensions and typically that is the x
and y directions, although we can manipulate those as necessary depending
on the particular problem. The other way that we use dimensions in physics is
when we talk about different types of measurement. So this is a sign in a small
town in out west somewhere. The funny thing about this sign is (if you’re not
seeing it already) is the fact that they’re adding three different types of
measurements to get one group of numbers. They’ve got population, which would be
measured in humans, feet above sea level obviously measured in feet, and a year.
They’re adding those all together. What turns out in physics, and just in
life in general, you can only add these numbers when they have the same
dimension. Whenever we’re talking about dimensions we use brackets to denote what dimension we’re talking about. So for example, L would represent
the dimension of length, I put the L in a set of brackets. Now this leads us to the SI or the International System of Units, also known as the metric system. In this system of measurement, there are actually one, two, three, four, five, six, seven base measurements. Those base measurements
are length, time, mass, temperature, amount (and typically when we talk about amounts we’re talking about number of molecules or numbers of atoms those subatomic
particles or those atomic particles that are very very small). We also talk about
current and light intensity. Notice that each of these dimensions has a
particular unit that we’re more interested in whenever we’re using what
we call the base units. So for example, the unit for length is the meter. The
unit for time is the second, although we can have other units related to that
like hours and days and weeks and so on but the base unit is second. The
third that will use this semester in particular is the mass of an object. Okay,
whenever we have a set of measurements we can really only subtract or add it if
they have the same dimension. So for example I’m about five foot 10, or five
foot 10 inches if we say it with the actual units all in there and notice
that five feet and ten inches in each of those things are measured in dimension
of length so feet is a length unit, inches or a length unit, and I’m safe to
add those through things together. Now I realize it’s early in the semester to be
talking about this type of a thing, but if we take the end of the semester:
Graduation, this particular semester when I’m recording this, is on May 19th 2018
and that graduation happens at 2 p.m. Well, my first class of the semester is
on January 29th and it starts at 9:00 a.m. Well, if I want to subtract those two it turns out that we’ve got physics for the
next 111 days five hours and about zero minutes, of course depending on exactly
when you’re watching this video lecture. But notice I’ve taken two time things
and subtracted those in each of those times was measured in months, days, years, hours, and minutes and when I subtract them I also get a unit or a dimension of
time with mixed units of time. If we could only use the base
measurements or the base units or the base types of measurements – the base
dimensions, then there’s not a whole lot we could do, but it turns out that we can
actually use multiples of these things or divisions of these things.
Multiplying or dividing is perfectly legitimate. So whenever we talk about the
speed or velocity and we’ll talk about the specifics for those, we end up taking
a length measurement divided by a time measurement. So in the metric system
whenever we talked about accelerations we also use length measurements divided
by two time units so when in the metric system we use meters per second for
speed or velocity, and meters per second squared for acceleration. Now we can get even more complicated than this. We can take a base unit multiplied it by a
derived unit like acceleration so when we talk about the force Newton’s second
law turns out to be the sum of the forces is the mass times the
acceleration. That means that, if I take a mass dimension multiply it by a length
dimension and divide it by two time dimensions or time squared dimension, then I’ve got a unit that is called a kilogram meter per second squared. That has a special name in the metric system we name that newton, of course
after Sir Isaac, one of the first not necessarily the first but one of the
most famous physicists of all times. The beauty of using dimensions is when
we look at equations the dimensions, or the units give us a way to check to make
sure that we’re doing things correctly. So in this case the x or Delta x is a
unit of length: That’s how far the object has moved and I can find that by
multiplying the velocity with dimensions of length over time times a time unit.
And then the half has no units that’s just a number then my acceleration has
those lengths per time squared dimension and I’m multiplying that by time squared
which has dimensions of time squared. Now it turns out I can treat these a lot like I
do numbers when I was back in algebra class and so I can cancel out the T’s on
the top and the T’s on the bottom and I’m left with units of length in the
first term and units of length in the second term which tells me that I’ve put
this equation together correctly. Now as the semester goes on, we’ll mostly
work with units kind of as proxies for the dimensions. But they behave exactly
the same way so if I have a distance traveled in units of meters, I can get
that by multiplying meters per second times seconds and again that one half is
just a number. So I can use meters per second squared
for my acceleration second squared for my time squared and these units nicely
cancel out. Just the way I expect them to. So I’m adding meters equals meters plus
meters. Everything is good. Now, if I do happen to make a mistake I can
usually catch that if I do my units. So one thing I should mention here is
oftentimes I will put in the numbers as well, but kind of towards the end of
the problem, I’ll just do a quick check of the units
like I’m doing here. So let’s say I had gone through and used m/s times time and
then I had done my acceleration but I forgot to square so my time I just put
it in is some number in seconds and I forget to square it. Well when I do the
algebra that allows me to cancel out some units I end up with a meter plus a
meter per second and that’s a very good sign that I’ve screwed up somewhere. That I will need to go back and recheck my work and hopefully catch that before
I submit my answer. This is one of the neat tricks of physics. It’s often
self-healing, so that you can fix this type of mistake as necessary. Okay, as a review, remember dimension has two meanings in physics. You have your
spatial dimensions or your directions. Most of the time in physics, we will
use two dimensions at once and usually denote those by x and y. We can also use
it as a type of measurement that’s what the other meaning of dimension. The
base units that we’ll use mostly this semester are length time and mass, that
have units in the metric system as you can see there the Newton can turn into
the combination of kilogram meters per second squared. And hopefully this has
been helpful so throughout the semester I’m going to be using the Open Stax
textbooks as references. If you happen to be taking my Physics of Sound class then
you might actually use that third book our fourth book that’s fifth book that’s
listed there called the Physics of Music and occasionally I’ll use that in all my
classes, just as a quick reference. If you need any other help and you’d like to
watch videos similar to this one you can visit me at physicsthisweek.com, and
then go through the menus there to figure out the specific topic that
you’re looking at. Okay, have a good week and let’s do well this semester in
physics.

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