Graphing is likely the most important skill you will learn in all of your science classes. Through the process of graphing and by interpreting graphs we are able to gain new insights into problem we thought we covered in depth already. Graphing also makes it easier to see connections between different variables. A picture is worth more than a thousand words. (Variables are quantities that can change from one problem to the next or sometimes also within a problem.)
So far we have tried to understand the connection between mass and volume and we have worked a great number of problems with these variables. Graphing mass and volume will help us deepen our understanding of mass, volume, density and the relationship between these variables.
Graphing in physics is the same as graphing in math, however just as we have found out before in physics we are always dealing with quantities, numbers, that stand for a distance, or a mass, or a time. We are not simply graphing numbers for their own sake. Sine we are dealing with real physical phenomena we have (most of the time) no need for three of the four quadrants of the coordinate system. We will only use the first quadrant, which is bordered by the x- and the y-axis. The point where the two lines meet is called the origin (0,0). These two axis represent two variables and each point on the graph has therefore two “coordinates”. Imagine you want to meet somebody in New York City. You would also have to give your friend two “coordinates” – “5th and 82nd Street” for instance. One coordinate is for the East-West axis (horizontal) and for the North-South axis (vertical).
The x-axis (horizontal) always shows the independent variable, that is the variable over which you have no control. This is most obvious when graphing distance and time. Time will always go on the x-axis, since it is independent of anything else. When we graph mass and volume we will pout the volume on the x-axis, for reasons that are not quite obvious yet. When picking your axis it is import to know the values you have to graph. If the largest mass you will have to graph is 290 kg, the y-axis has to go at least to 290. It does, however, not make much sense to have it go to a 1000. It would be most sensible to pick 300 as the highest vertical number and divide the y-axis into 6 equal parts: 50, 100, 150, 200, 250 and 300. The same is true for the x-axis.
The most important reason why we learn about graphing is that we are able to gain new insights into the relationships between variables. The easiest relationship to recognize is when the points that you graphed lie in a straight line, or are close to forming a straight line. Points cannot make a line, only if you were to connect them would you get a line. However, we NEVER connect the points that we plot – never. Instead what we do is find a straight line that best fits all the points. The line does not have to cross through all the points (sometimes it does not even go through any points), but is has to be a “reasonable fit” of these points. See the figure below. Drawing this straight line indicates that we believe that the two variables have a special relationship. They are said to be “proportional”. The graph is called “linear graph”.
Once we have found that the graph is linear we can then find how steep (or shallow) the line is. In math this is called “the slope”. In your workbook you have worked on several different graphs and have found the slope of these graphs. The slope is exactly what is indicated by the name, the slope of a meadow is how much it goes up or down. The slope of a roof is called pitch, that of a road is called grade.
Imagine a ladder. A ladder is steep if the bottom is very close to the wall. If you move the bottom of the ladder further away from the wall it becomes less steep. If you do that you decrease the height that the ladder actually reaches. The height is called the rise and the distance from the foot of the ladder to the wall is called the run. The slope is then defined as the ratio of the rise and the run (you have to divide these two numbers). A ladder has the same slope everywhere you climb it. If you go up one rung you go not only higher but also closer to the wall, and you will do so by the same amount. Each step on a ladder will make you rise 30 cm (about one foot) and make you move closer to the wall by 10 cm (5 inches). This is true regardless of where you are on the ladder. The relationship between how much higher you go versus how much closer you get to the wall is the same everywhere on the ladder and could be called the slope of the ladder. The slope of the ladder in the example above is 3. Remember slope = rise/run = 30 cm / 10 cm = 3. There are no units with this number. It is simply the slope of the ladder. What units could you measure this slope in anyway?
Back to our graph. You find the slope by drawing a best-fit line through your data and then picking any two points on this line. The points do not have to be data points, they only have to be two points on the line. You need two points to find the slope. If the line goes through the origin it is useful to pick the origin as one of your points. You then need to find the rise and the run between these two points. See the power point for further information.
The slope in the picture to the right would be rise/run = 3/1 = 3.
Finding the slope almost always tells us something about the variables. If the slope has a meaning (in our class it always will) we will then know that there is a relationship between the two variables and by calculating the slope we are able to write this relationship in form of an equation. In math class this is done using the equation: y = mx +b.
The slope is given the variable “m”. The variable “b” is the y-intercept, the point where the line you drew intercepts the vertical (the y-) axis. In a lot of our examples this will be zero. Try to connect what you learned about slope in math to what we are doing in physics.
When graphing mass and volume we do so to gain deeper insight into the relationship between these two variables. On a mass and volume graph for one type of material you will soon see that all points lie on one straight line. Think about what this means. Not only are both variables proportional but the relationship between them is a fixed one. The graph is linear. For every unit that you increase the mass, the volume increases by another fixed unit. In the graph at right three different materials are plooted amss against volume. You can see that for none of them the mass increses by the same unit as the volume does, although the red line (which represents ice) is close. Pay close attention to the axis, they are not the same. For the blue line, if the volume increases by one unit the mass increases by 8 units. That means that for every cubic cenitmeter that you increase the volume th amss goes up by 8g.
What does this number represent? If you think about this for a moment and remember what we talked about when we discussed proportional reasoning between mass and volume, it will be apparent that this is the density. The density can therefore be found by finding the slope on mass versus volume graph. If you flip the axi around you willl get the number that tels you by how much the volume increases if you increase the mass by one gram. This, although it has a meaning, does not have a name, as we discussed in class. If you flip the axis you will also note that now the red line will be the steepest.
Here is a problem form the workbook that you might want to work through:
Suppose we have a lump of clay represented by point P on the mass versus volume graph below. Enough mystery metal is mixed with this lump of clay so that the resulting ball of clay and metal is represented by point Q.
A. What is the density of pure clay (without the metal)? Explain your reasoning.
B. What is the density of the mystery metal? Explain your reasoning.
I am sure you will have no difficulty in findg the density of the clay. Look at point P, divide the mass (15g) by the volume (10 ccm). This will lead a density of 1.5 g/ccm. While this is the right answer you got it most likely right by accident only as part B will proof. Your answer to part B will probably be something around 2.8 g/ccm - divding 38 g by 14 ccm, as you would gte by looking at point Q.
But when solving this problem, try to picture the clay and the metal and what you are diong with them. First you have the clay. Measering its mass and volume will give you point P. Then you ADD the metal and that leads to point Q. What does point Q therefore represent? Think about it for a moment: the mass and the volume of both the clay and the metal combined. So, when you divide 38 by 14 you will get the density of both materials combined. By dividing those two numbers you actually found the slope of the line leading from 0 to point Q.
Remember that density can only be found on a graph by finding the slope. When you solved for part A, the density of the clay you found the slop of the line from 0 to point P, which by accident is the same as dividing the two numbers for point P. In order to find the density of the metal alone you need to look at where the metal shows up on the graph alone - it does so, when you add it to the clay. Thereofre you can find its density by finding the slop of the line form P to Q.