Question Video: Expressing Units in Terms of Fundamental Dimensions

What are the dimensions of a quantity that can be measured in kg.m²?

02:13

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.

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