Video Transcript
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.
