### Video Transcript

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 thatโs 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. Letโs call this unknown quantity
๐ฅ. And because we know that itโs equal
to force times distance, we can write down the equation ๐ฅ equals ๐น times ๐, where
๐น represents a force and ๐ represents distance. The question is asking us to work
out the dimensions of this quantity that weโve chosen to represent with the symbol
๐ฅ.

Letโs 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 ๐ฟ, capital ๐, capital ๐, and capital ๐ผ. Letโs also recall that in any
physical equation, the dimensions are the same on both sides. This means that the dimensions of
๐ฅ, which weโre trying to find out, must be the same as the dimensions of ๐น times
๐. And this happens to be the same as
the dimensions of ๐น times the dimensions of ๐.

We can use square brackets to
represent when weโre talking about the dimensions of a quantity. So, putting a lowercase ๐ฅ in
square brackets means the dimensions of our quantity ๐ฅ. And this must be equal to the
dimensions of ๐น times the dimensions of ๐. 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 ๐ฅ. Itโs not too hard to convince
ourselves that the dimensions of distance are length. After all, when we measure a
distance, weโre measuring its length. So, in this equation, we can
replace the lowercase ๐ in square brackets with a capital ๐ฟ to represent the
dimension of length. In other words, length is the
dimension of distance.

So, weโre 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 isnโt simply a length or a
mass or a time or a current. Instead, itโs 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 ฮ
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 ๐ก in brackets meaning the dimensions of time, we can write a
capital ๐ to represent the dimension that is time.

However, we can see that momentum
is not a dimension. So once again, weโll have to work
out the dimensions of momentum by using a formula that we could use to calculate
momentum. So, letโs 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 ๐
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 ๐ in brackets with a capital ๐ representing the
dimension of mass.

So, we can see that, gradually,
weโre 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 ๐
in square brackets with a capital ๐ฟ. And once again, we can replace our
lowercase ๐ก in square brackets with a capital ๐.

We can now see that with the aid of
these three formulas, weโve managed to express the dimensions of force in terms of
mass, length, and time. We can simplify this expression to
๐๐ฟ over ๐ squared, in other words, mass times length divided by time squared. And because itโs more common to use
index notation when weโre talking about dimensions, we will generally express this
as ๐๐ฟ๐ to the negative, two. And finally, weโre ready to
substitute this into our expression telling us the dimensions of our unknown
quantity ๐ฅ. This shows us that the dimensions
of our unknown quantity ๐ฅ are mass times length times time to the power of negative
two times length, which we can simplify by writing ๐๐ฟ squared ๐ to the negative
two.

So, thereโs our answer. The dimensions of a quantity thatโs
equal to a force multiplied by a distance are mass times length squared times time
to the power of negative two.