Units and Physical Constants

Units

Units are very important when it comes to mechanics. A unit basically tell us what we are talking about. For example, at the end of a problem we end up with the value $4.65$. However, we have no idea what $4.65$ has to do with. It could joules(energy), km/h, newtons, etc. Now if we had yielded $4.65 m/s$, then we would immediately know that it refers to speed. Thus, units basically tell what the numerical value is of, whether it be newtons, cats, dogs,etc. Constants

Constants are an important part of math and physics. Constants never change, and and the same throughout the universe. Some are calculable, such as $e$ or$\pi$, while others, like the gravitational constant, cannot be determined theoretically and must be calculated through experimentation.

Physical Quantities

Motivation

Suppose you were very hungry and your parents were making dinner. You ask, "how much longer until dinner is done?" Your parents say, "Not much longer." But you are very hungry and you want to know exactly how long it will be, so you say, "give me a number."

Now your parents can make use of numbers to give a very precise answer, but they also need a way to relate the abstract number to the real world. If they just say "five", there is no indication of how much time "five" is supposed to represent. Is it supposed to be five minutes or five hours or five seconds? A proper answer would be "five minutes." To give a meaningful answer it is necessary to give a number to specify the magnitude of the time left (in this case five), but then also to specify a physical concept in order to give the number physical meaning (in this case, a minute).

Definition of physical quantity

A physical quantity is a physical property that has both a quantitative nature, but also has physical significance so that it can't be represented by a number alone.

Examples

One example of a physical quantity is an amount of time, such as five minutes. More examples are a length, such as 3 centimeters; a speed, such as 20 kilometers per hour; or a temperature, such as $20^\circ$C.

Non-examples

Pure numbers, such as $3$, $e$, or $\pi$ are not physical quantities.

Doing math with physical quantities

Although physical quantities are not pure numbers, they can still do many of the same mathematical operations. One operation that can be performed on a physical quantity is multiplication by a pure number. For example, suppose $L$ represents the distance from your house to the store, then $2L$ represents the total amount of distance from your house to the store and back. The distance $L/2$ is half the distance to the store: if you go a distance of $L/2$ to the store and then go another $L/2$, you wind up at the store.

Another thing that can be done is to add two physical quantites. However, it is only possible to add two physical quantities of the same type. For example, you can add a length to a length or a speed to a speed, but you cannot add a speed to a length. For example, if $t_d$ is the amount of time you have to wait from now until dinner, and $t_c$ is the amount of time you have to wait after dinner until you eat cake, then $t_d+t_c$ is the amount of time you have to wait from now until when you eat cake. It corresponds to first waiting for a time $t_d$, then waiting for a time $t_c$.

The third operation that can be done on physical quantities is multiplication and division. For example, suppose you have a rectangle whose side lengths are $L_1$ and $L_2$. Then the area of the rectangle is a physical quantity with the value of $L_1 L_2$. As another example suppose you are able to go to the store, which is a distance $L$ away, and back in the time $t_d$ until dinner is ready. Then your average speed is a physical quantity, and it turns out that it is given by $2 L /t_d$.

Dimensions

When discussing physical quantities, we said that one mathematical operation that can be done is addition. However we said that you can only add two physical quantities if they have the same type. The technical name for the type of a physical constant is its dimension. It is important to understand that this usage of the word "dimension" is not the same as when saying an object is one-dimensional or two-dimensional.

Definition of dimension

Two physical quantities have the same dimension if they can be meaningfully added. If the two quantities are $x$ and $y$, we say that $x$ has dimensions of $y$ or $y$ has dimensions of $x$.

Notation

The dimension of a physical quantity $x$ is denoted $[x]$. If $x$ and $y$ have the same dimensions, we write $[x]=[y]$.

Examples of dimensions

Three important dimensions you will run into as physics student are length, time, and mass. Any physical quantity representing a length is said to have dimensions of length. It is clear such physical quantities all have the same dimension becuase any length can be added to any other by starting at one end of a ruler, moving down the ruler up to the value of the first length and then moving further by the value of the second length. The total distance traversed along the ruler is then the sum of the lengths. A similar thing can be done to add intervals of time or masses.

However, notice that time and length are different dimensions since there is no universally natural way to add a length and time.

Other examples of dimensions are area and speed. We will see how these last two can be related to the dimensions of length and time.

Non-examples of physical dimensions

Dimensions can be used outside of physics. For example if you want to specify an amount of money, it is not sufficient to just use a pure number. You must also give a denomination (you would need to specify if you mean five dollars or five cents). You could say that such a quantity of money has dimensions of money, but since money isn't an object of interest in physics, money will not be used in our system of dimensions. Many other examples of "non-physical" dimensions can be found.

In fact, sometimes quantities that can be expressed with a pure number are still thought of as dimensional quantities. For example, $x$ might refer to the amount of people in a room. Then $x$ is a pure number like five or six. However, to be clear that $x$ refers to a number of people, it is occasionally said that the $x$ has dimensions of people.

The dimension of a product of physical quantities

If $x$ is a physical quantity with dimensions $[x]$, and $y$ is a physical quantity with dimensions $[y]$, then the dimensions of the product $xy$ is written $[x][y]$, and the dimensions of the quotient $x/y$ is written $[x]/[y]$.

For example, if $A$ is an area, and $t$ is a time, then the dimensions of the product $At$ is $\textrm{area}*\textrm{time}$. It might not be clear what the physical significance of this product is, so let's consider an example. Lets say it is ranining at a steady rate. Then the mass $m$ of water that falls on an area $A$ of ground over a time $t$ is proportional to both $A$ and $t$: if you double the area, twice the amount of water should be collected, and similarly if you wait twice as long, twice the amount of water should be collected. Thus it must be the case that $m=cAt$, where $c$ is some constant. We can see that the quantity $At$ naturally appears in the previous equation. Hopefully this example makes it seems plausible that any product of physical quantities might be physically significant.

Equivalence of dimensions

It was mentioned that the dimensions of area and speed can be related to the dimensions of length and time. Let's first consider the dimension of area. Specifically, lets consider the area $A$ of a circle of radius $r$. The area is given by $A=\pi r^2$. But there seems to be a problem. The left hand side, $A$, has dimensions of $\textrm{area}$, while the right hand side has dimensions of $\textrm{length}^2$, but since the two physical quantities are supposed to be equal, they are supposed to at least have the same dimensions. The resolution to this problem is to make an equivalence between the dimensions of $\textrm{area}$ and the dimensions of $\textrm{length}^2$. The equivalence goes like this: to get the area corresponding to the physical quantity $A_0$ having units of $\textrm{length}^2$, first find two lengths $L_1$ and $L_2$ whose product is $A_0$, then the area corresponding to $A_0$ is the area of the rectangle with sides of length $L_1$ and $L_2$.

Now lets see how speed is related to length and time. A speed is supposed to be a length divided by a time. Therefore, we should expect that dimensions of speed are the as dimensions of $\mathrm{length}/\mathrm{time}$. Let's suppose $v_0$ has dimensions of $\mathrm{length}/\mathrm{time}$. Then to get the speed it is supposed to represent, we find a length $L$ and a time $T$, such that $v_0=L/T$. Then $v_0$ represents the average speed of an object that moves through a total distance $L$ in the time $T$.

In certain situations, it can be useful to introduce relationships between dimensions which are usually unrelated. For example, suppose you are on a train moving at sixty miles per hour, and you are asked how far away you are from you destination. You might respond by saying you are an hour away. This might seem wrong, because you were asked for a length and you responded with a time, but what you have actually done is created an equivalence between the dimensions of length and time by saying that the length represented by a time $T$ is the distance that the train travels in the time $T$. With this equivalence in place, lengths and times actually do have the same dimensions.

Units

Motivation

Now suppose you were very hungry and you ask your parents how long until dinner, and they reply "five minutes." We have discussed how five minutes is a physical quantity and can be used to specify the amount of time with mathematical precision. The answer has two parts: "five", a number, and "minutes" which represents a physical idea that gives physical meaning to the number five.

For this answer to be useful to you, it is important that you already know how long a minutes is. For instance, if the parent had instead given the answer "exactly one amount of time until food is done," that would be a correct answer, but this answer is not useful to you because you do not know how long is one "amount of time until food is done."

In order to communicate physical quantities, it is necessary for there to be a commonly known set of "standard" physical quantities (one of these being the minute). These commonly known physical quantities are called units.

Definition

A unit is a physical quantity of known magnitude that is used together with a number (or more advanced purely mathematical object) to express a different physical quantity.

Examples

Seconds, minutes, and hours are all units of time. Feet, meters, miles, and light-years are units of length.

A kilogram is a unit of mass. A meter per second is a unit of speed.

Non-examples

Sometimes, "units" are used outside of physics. For example if someone asked you how many cookies you baked, you could say "two dozen." In this case, you can interpret "two" to be a number and "dozen" to be a unit. However, "dozen" is not a true physical unit here, because a dozen is already a purely mathematical number with no physical meaning.

Another example is the "bit" as a unit of information. Again, a bit has no physical significance on its own, so it is not a physical unit.

Even though these two examples are not units in the way we mean, you will still often hear these and others called "units." Just know there is a distinction between these non-physical units, and the units defined above.

Units are used in science and math to give a number meaning. What I mean, is that when a number is given and explained on its own, there is no context. A number such as 5 can mean anything, like 5 cats, dogs, coconuts, whatever. Units help to specify the kind of something that has a number assigned to it. If I said 5 km/hr, one would know I was referring to speed, and not have to assume units because it wasn't specified. Also, it comes in useful to know the units for a number when trying to convert a number to a form in which it is easier and more relevant to the problem. Often, if the answer to some real-world problem is 12 cookies, and you write 12 on the test, the teacher will remove percentage points. One of my teachers gave me no credit, saying that 7 cookies + 5 cookies does not equal 12 trees, which is what he assumed my answer of 12 to mean.