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