Video Transcript
In this video, we’re looking at
dimensional analysis. The word dimensions is often used
to refer to measurements of length. For example, we can say that the
dimensions of an object are its length, width, and height. We also commonly refer to everyday
objects as being three-dimensional. But the word dimensions doesn’t
only refer to lengths. It also refers to other quantities
such as mass, time, and current. We can think of dimensions as being
fundamental or base quantities that can be combined in various ways to make more
complex quantities.
Dimensional analysis is the process
of analyzing different physical quantities by looking at the base quantities, in
other words, the dimensions, that they’re made up of. We can use dimensional analysis to
help us work out what units to express different quantities in, as well as helping
us gain insight into equations in physics. In this video, we’ll be looking at
four basic dimensions, length, mass, time, and current. As we mentioned, we refer to these
as dimensions because we consider them to be simple or base quantities. As a result, it’s possible to
express many other quantities such as acceleration, charge, and force in terms of
just these four dimensions.
In fact, any quantity can be
expressed in terms of its dimensions. This is closely related to the fact
that we can express any SI unit in terms of SI base units. Any dimension can be measured using
a single SI base unit. So, we can measure length in
meters, mass in kilograms, time in seconds, and current and amps. And more complicated units, for
example, those that we used to measure acceleration, charge, and force, can always
be expressed in terms of the SI base units. For example, in SI units, we
measure acceleration in meters per second squared or equivalently in meters times
seconds to the power of negative two. These units comprise both the SI
unit for length, the meter, and the SI unit for time, the second.
Similarly, the SI unit for charge
is the coulomb, which we represent with a capital C. But one coulomb is equivalent to
one amp second. So, we’re able to express charge in
units that comprise both the SI unit for current, the amp, and the SI unit for time,
the second. Likewise, the SI unit for force is
the newton. But newtons can equivalently be
expressed as kilogram meters per second squared, which we can also write like
this. So, we can see that force can be
expressed in units that comprise the SI unit for mass, the kilogram, the SI unit for
length, the meter, and again the SI unit for time, which is of course the
second.
The process of expressing a
quantity in terms of its dimensions is actually independent of units. So even if we were measuring length
in inches and time in years, we would still be able to express any quantity in terms
of the same dimensions. To do this, we use the symbolic
representation of each dimension. We represent the dimension of
length with a capital 𝐿, mass with a capital 𝑀, time with a capital 𝑇, and
current with a capital 𝐼. Note that these symbols don’t
represent units nor are they necessarily the symbols that we would use to represent
variables in equations. So, we use this capital 𝐿 to
represent the dimension of length regardless of the units we’re using and whether or
not that length is a constant or a variable in an equation.
We can use these symbols to
represent the ways in which dimensions are combined in more complex quantities. As a simple example, let’s consider
the dimensions of area. When we calculate an area, we
typically multiply two lengths together. For example, the area of a square
is its width multiplied by its height. And the area of a triangle is given
by half the length of its base multiplied by its perpendicular height. Because we can see that length is a
dimension and we can obtain an area by multiplying a length by another length, we
can say that the dimensions of area are length times length or equivalently length
squared.
So, even though when we calculate
the area of a shape, the two lengths that we’re multiplying together won’t
necessarily be the same. So, we’re not necessarily squaring
the length when we calculate the area. Any area still has dimensions of
length squared. And we can represent this using
symbols as 𝐿 squared. We know that area can be
represented using many different units, for example, square meters, acres, or square
inches. But the dimensions of area are
always length squared. We can work out the dimensions for
a given quantity if we have a formula that allows us to calculate that quantity. For example, for a triangle with a
base of length 𝑏 and a perpendicular height of ℎ, its area is given by the formula
half 𝑏 times ℎ.
Now, the factor of a half in this
equation is what we call a dimensionless number. It doesn’t have any dimensions, and
we wouldn’t represent it using units. But both the 𝑏 and the ℎ are
measurements of length. They therefore have dimensions of
length. The fact that we can calculate the
area of the triangle by multiplying two lengths together confirms that the
dimensions of area are length times length. We can use this exact same method
to work out the dimensions of more complex quantities, for example, momentum. Let’s recall that the momentum of
an object with mass 𝑚 traveling at velocity 𝑣 is given by the formula 𝑝 equals
𝑚𝑣, where 𝑝 represents momentum. We can use this formula to work out
the dimensions of momentum by considering the dimensions of each of the quantities
that we use to calculate it, in this case, mass and velocity.
The dimensions on the left-hand
side of any physical equation must be the same as the dimensions on the right-hand
side of the equation, which tells us that the dimensions for momentum must be the
same as the dimensions for mass multiplied by the dimensions for velocity. We can use square brackets to
represent when we’re talking about dimensions. For example, a lowercase 𝑝 in
square brackets represents the dimensions of momentum. As we’ve seen from this equation,
the dimensions for momentum must be equal to the dimensions for mass multiplied by
the dimensions for velocity, which we can represent like this. The first thing that we can notice
here is that mass is a dimension.
So, instead of writing square
brackets around a lowercase 𝑚 to represent the dimensions of mass, we can just
write a capital 𝑀 to represent the dimension that is mass. Having done this, we’re now one
step closer to determining the dimensions of momentum. However, to finish the job, we
still need to determine the dimensions of velocity. To do this, we can use any formula
that we might use to calculate velocity. Probably the most common is 𝑣
equals 𝑠 over 𝑡, velocity equals displacement divided by time. Once again, using the fact that the
dimensions on the left-hand side and the right-hand side of any equation must be the
same, we know that the dimensions of velocity must be equal to the dimensions of
displacement divided by the dimensions of time.
Well, because a displacement is
effectively a measurement of length and length is a dimension, we can say that
displacement has a dimension of length. And the dimensions of time are, of
course, just the dimension time. So, we can replace both of these
quantities by the symbols that represent their dimensions, showing us that the
dimensions of velocity are length divided by time. And we can now substitute this back
into our equation for the dimensions of momentum. In other words, we can replace our
lowercase 𝑣 in square brackets, representing the dimensions of velocity, with a
capital 𝐿 over capital 𝑇, representing the dimension length divided by the
dimension time. Rearranging this slightly, we can
see that our equation tells us the dimensions of momentum are mass times length
divided by time.
When we’re writing equations with
dimensions in, it’s common to use index notation rather than fractions. So instead of writing 𝑀𝐿 over 𝑇,
we would write the equivalent expression 𝑀𝐿𝑇 to the negative one. So, we’ve now seen how we can work
out the dimensions of a quantity by using a formula for that quantity and equating
the dimensions on the left- and right-hand sides. This can be made easier by
remembering the defining formulae for a few common quantities. Area has dimensions of length
squared, which we can represent 𝐿 squared. Volume has dimensions of length
cubed, which we can write 𝐿 cubed. Speed has dimensions of length
divided by time, which we can write 𝐿 over 𝑇 or 𝐿𝑇 to the negative one. And acceleration has dimensions of
length divided by time squared, which we would write 𝐿 times 𝑇 to the negative
two.
Momentum has dimensions of mass
times velocity. And since we’ve just seen that
velocity has dimensions of length divided by time, this means we can represent the
dimensions of momentum as 𝑀𝐿𝑇 to the negative one. The dimensions for force are the
same as those of momentum divided by time. Since we’ve seen that momentum has
dimensions of 𝑀𝐿𝑇 to the negative one, then we know that the dimensions of force
are given by 𝑀𝐿𝑇 to the negative one divided by 𝑇, which is equivalent to 𝑀𝐿𝑇
to the negative two. Charge has dimensions of current
times time, which we can represent with the symbol for the dimension of current,
which is 𝐼 times 𝑇. Finally, frequency has dimensions
of one over time, which we would write 𝑇 to the negative one. Now that we’ve talked about what
dimensions are and how we can calculate the dimensions of different quantities,
let’s have a go at some example questions.
What are the dimensions of a
quantity that is equal to a force multiplied by a distance?
So, this question is talking about
some unknown quantity that’s equal to a force multiplied by a distance. Well, straightaway, we can make
this question easier for ourselves by representing this information in an
equation. Let’s call this unknown quantity
𝑥. And because we know that it’s equal
to force times distance, we can write down the equation 𝑥 equals 𝐹 times 𝑑, where
𝐹 represents a force and 𝑑 represents distance. The question is asking us to work
out the dimensions of this quantity that we’ve chosen to represent with the symbol
𝑥.
Let’s quickly recall that
dimensions of a set of base quantities that all quantities in physics can be made
out of, these include length, mass, time, and current. And these dimensions can be
represented by the symbols capital 𝐿, capital 𝑀, capital 𝑇, and capital 𝐼. Let’s also recall that in any
physical equation, the dimensions are the same on both sides. This means that the dimensions of
𝑥, which we’re trying to find out, must be the same as the dimensions of 𝐹 times
𝑑. And this happens to be the same as
the dimensions of 𝐹 times the dimensions of 𝑑.
We can use square brackets to
represent when we’re talking about the dimensions of a quantity. So, putting a lowercase 𝑥 in
square brackets means the dimensions of our quantity 𝑥. And this must be equal to the
dimensions of 𝐹 times the dimensions of 𝑑. So, we can now see that if we can
find the dimensions of force and the dimensions of distance, then all we need to do
is multiply them together to get the dimensions of our quantity 𝑥. It’s not too hard to convince
ourselves that the dimensions of distance are length. After all, when we measure a
distance, we’re measuring its length. So, in this equation, we can
replace the lowercase 𝑑 in square brackets with a capital 𝐿 to represent the
dimension of length. In other words, length is the
dimension of distance.
So, we’re now one step closer to
determining the dimensions of our quantity. All we need to do now is work out
the dimensions of force. Unfortunately, working out the
dimensions of force is slightly more complicated as force isn’t simply a length or a
mass or a time or a current. Instead, it’s a combination of
several of these dimensions. In order to work out which
dimensions are involved in the quantity of force and how, we can use any formula
that enables us to calculate force. For example, we can calculate the
force acting on an object by dividing its change in momentum by the time over which
its momentum changed.
Once again, we know that the
dimensions on the left side of this equation must match the dimensions on the
right-hand side. So, we can say that the dimensions
of force are equal to the dimensions of a change in momentum divided by the
dimensions of time. Now, a change in momentum has the
same dimensions as momentum. So, we can forget about the Δ
symbol and just say the dimensions of force are equal to the dimensions of momentum
divided by the dimensions of time. We can see that time is a
dimension. So, in our equation, instead of
writing a lowercase 𝑡 in brackets meaning the dimensions of time, we can write a
capital 𝑇 to represent the dimension that is time.
However, we can see that momentum
is not a dimension. So once again, we’ll have to work
out the dimensions of momentum by using a formula that we could use to calculate
momentum. So, let’s recall that momentum is
equal to mass times velocity. Because we know that the dimensions
on the left of this equation are the same as the dimensions on the right of this
equation, we know that the dimensions of momentum must be the same as the dimensions
of mass times the dimensions of velocity. So, we can replace our lowercase 𝑝
in square brackets meaning the dimensions of momentum with the dimensions of mass
multiplied by the dimensions of velocity. And because mass is a dimension, we
can replace this lowercase 𝑚 in brackets with a capital 𝑀 representing the
dimension of mass.
So, we can see that, gradually,
we’re replacing quantities with unknown dimensions with base quantities such as time
and mass. So finally, we need to work out the
dimensions of velocity. This time we can use the equation
velocity equals displacement over time. This equation shows us that the
dimensions of velocity are equal to the dimensions of displacement divided by the
dimensions of time. So, we can replace the dimensions
of velocity in this equation with the dimensions of displacement divided by the
dimensions of time. Just like distance, displacement
has dimensions of length. So, we can replace the lowercase 𝑠
in square brackets with a capital 𝐿. And once again, we can replace our
lowercase 𝑡 in square brackets with a capital 𝑇.
We can now see that with the aid of
these three formulas, we’ve managed to express the dimensions of force in terms of
mass, length, and time. We can simplify this expression to
𝑀𝐿 over 𝑇 squared, in other words, mass times length divided by time squared. And because it’s more common to use
index notation when we’re talking about dimensions, we will generally express this
as 𝑀𝐿𝑇 to the negative, two. And finally, we’re ready to
substitute this into our expression telling us the dimensions of our unknown
quantity 𝑥. This shows us that the dimensions
of our unknown quantity 𝑥 are mass times length times time to the power of negative
two times length, which we can simplify by writing 𝑀𝐿 squared 𝑇 to the negative
two.
So, there’s our answer. The dimensions of a quantity that’s
equal to a force multiplied by a distance are mass times length squared times time
to the power of negative two.
Now that we’ve answered this, let’s
take a look at a question that requires us to convert between a unit expression and
a dimension expression.
What are the dimensions of a
quantity that can be measured in kilogram meters squared?
This question requires us to make a
distinction between three very similar concepts, dimensions, quantities, and
units. Let’s quickly remind ourselves of
what each of these terms means. In physics, a quantity is a
physical property that can be expressed as a number. In other words, it’s a physical
property that can be quantified. This includes things like mass,
energy, and acceleration. A unit is a certain magnitude of a
given quantity. We often express units using
symbols. For example, we can express mass in
kilograms, energy in joules, and acceleration in meters per second squared.
Finally, dimensions are the set of
base quantities with which we can express all other quantities independent of
units. These include mass, length, time,
and current. All dimensions are examples of
quantities, but not all quantities are dimensions. However, it is possible to express
all quantities, including things like energy and acceleration, in terms of
dimensions. In this question, we’ve been given
a quantity whose units are kilogram meters squared, and we’ve been asked to find its
dimensions. Fortunately, it’s possible to
convert directly between units and dimensions. To do this, let’s look closely at
the units we’ve been given in the question.
The units here, kilogram meter
squared, are formed by multiplying different units together. In this case, we have kilograms
times meters squared or kilograms times meters times meters. Kilograms are the units that we use
to express the quantity of mass, and meters are the units that we use to express the
quantity length. As well as being quantities, we can
also see that both mass and length are dimensions. This makes it really simple to
convert from this unit expression to a dimension expression. We have the unit for mass
multiplied by the unit for length multiplied by the unit for length again. So, the corresponding dimensions
are simply mass times length times length.
We represent the quantities mass,
length, time, and current using the symbols capital 𝑀, capital 𝐿 capital 𝑇, and
capital 𝐼 using index notation wherever possible. So, mass times length times length
can be represented 𝑀𝐿 squared. And this is the final answer to our
question. The dimensions of a quantity that
can be measured in kilogram meters squared are mass times length times length or
𝑀𝐿 squared.
Let’s finish by quickly reviewing
the key points that we’ve looked at in this video. Firstly, we saw that all physical
quantities can be expressed in terms of a set of base quantities known as
dimensions. These include the quantities
length, mass, time, and current. We’ve also looked at dimension
notation, specifically the use of the symbols capital 𝐿, 𝑀, 𝑇, and 𝐼 to
represent the dimensions of length, mass, time, and current, respectively.
We can also put square brackets
around a quantity to represent the dimensions of that quantity. For example, the dimensions of
force are mass times length times time to the power of negative two. And finally, we’ve seen that we can
calculate the dimensions of different quantities by using formulas to express them
in terms of their base quantities or dimensions.
