Question Video: Determining the Dimensions of Physical Quantities Physics

What are the dimensions of a quantity that is equal to a force multiplied by a distance?

05:13

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 thatโ€™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. Letโ€™s call this unknown quantity ๐‘ฅ. And because we know that itโ€™s equal to force times distance, we can write down the equation ๐‘ฅ equals ๐น times ๐‘‘, where ๐น represents a force and ๐‘‘ represents distance. The question is asking us to work out the dimensions of this quantity that weโ€™ve chosen to represent with the symbol ๐‘ฅ.

Letโ€™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 ๐ฟ, capital ๐‘€, capital ๐‘‡, and capital ๐ผ. Letโ€™s also recall that in any physical equation, the dimensions are the same on both sides. This means that the dimensions of ๐‘ฅ, which weโ€™re trying to find out, must be the same as the dimensions of ๐น times ๐‘‘. And this happens to be the same as the dimensions of ๐น times the dimensions of ๐‘‘.

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

So, weโ€™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 isnโ€™t simply a length or a mass or a time or a current. Instead, itโ€™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 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 ๐‘ก 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, weโ€™ll have to work out the dimensions of momentum by using a formula that we could use to calculate momentum. So, letโ€™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 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‘š in brackets with a capital ๐‘€ representing the dimension of mass.

So, we can see that, gradually, weโ€™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 ๐‘  in square brackets with a capital ๐ฟ. And once again, we can replace our lowercase ๐‘ก in square brackets with a capital ๐‘‡.

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

So, thereโ€™s our answer. The dimensions of a quantity thatโ€™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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