Monthly Archives: January 2011
Our second electical circuits lab was an exercise in verifying Ohm’s Law. To start, let’s look at what Ohm’s Law is and the theory behind it.
All materials have a general property of resisting the flow of electricity through themselves. This material property is quantitatively called the resistivity. (Interesting enough, resistivity is the reciprocal of conductivity.) The more difficult it is for electricity to flow through a material, the higher the resistivity value. Resistivity is purely a material property and does not depend on any particular configuration or geometry of the material. It can however be affected by temperature changes. The resistance, on the other hand, is a function of this material property, but also of the material shape. This relationship is given by:
Where R is the resistance, ρ is the resistivity, L is the length of the portion of material, and A is the uniform cross-sectional area.
The circuit element that we utilize as a model for resistance is called the resistor. The German physicist Georg Ohm investigated the relationship between the voltage across a resistor and the current through a resistor. Through this study he developed the following relationship known today as Ohm’s Law:
In order to verify this relationship for ourselves, we experimentally varied one element of the voltage, current, and resistance, kept one constant, and took readings and recorded values for the third. Then we determined the error between these recorded values and the expected values given by Ohm’s Law. Below are the tables of calculated values, measured values, and percent error for 3 different configurations.
The equipment we utilized and had set up in the first configuration consisted of a power supply (DC), a potentiometer, and the digital multimeter. The apparatus was configured as follows:
Here is the procedure for the first configuration.
- Ensure the power supply is in the off position.
- For part A we kept the voltage constant at 10 volts. Therefore, the first step is to connect the power supply to the digital multimeter with 2 leads, set the multimeter to read DC voltage, turn the power supply on, and adjust the voltage on the power supply until the multimeter reads 10 volts.
- Turn the power supply back off.
- Next connect the lead (we used red) from the positive terminal of the power supply to positive or red terminal of the potentiometer.
- Using a lead (we used black), connect the negative or black terminal of the potentiometer to the lower left black terminal of the digital multimeter.
- Using a lead (we used two leads in series for length and end plugs) Connect the lower right red terminal (for reading current) of the digital multimeter to the negative or black terminal of the power supply. At this point the circuit is complete.
- Set the digital multimeter to read milliamperes of current.
- Turn the power supply on.
- Adjust the switches on the potentiometer to obtain the required value for the current displayed on the multimeter. (See the data tables for more. The values were 10,9,8,7… and so on.)
- Add up and record the resistance supplied by the potentiometer that generated the corresponding current value.
- Compare this resistance value to the resistance value predicted by Ohm’s Law. Computer the percent error for each measurement.
- Repeat steps 9 through 11 for each required current value.
Here is a graph of our results for the first configuration. It shows the calculated values using Ohm’s Law, the measured values, and a predictive trendline with equation using the calculated values.
This first configuration gives you the idea. I have the procedure and graphs for the other implementations if anyone is interested. I have omitted them here for brevity.
Our first environmental engineering lab was concerning one method of water quality assessment called electrical conductivity. The theory behind this technique is based on the fact that pure water is not a good conductor of electricity. As the amount of inorganic, ionic elements or compounds in the water increases, the conductivity also increases. Conductivity also increases with temperature, therefore most results are standardized at 25 degrees Celsius. With this understanding we can make comparative analysis of conductivity values to assess the purity of the water. Conductivity in this context is usually in units of microsiemens per centimeter. In the lab we utilized a handheld electrical conductivity meter like the one above. Below is a table we were given of the typical value range found in different types of water.
The reason for the high conductivity in seawater is the high salt content and thus a high number of sodium and chloride ions in this aqueous solution. If you are interested in additional information, I found a good EPA website concerning water quality monitoring and electrical conductivity.
This is the second post of this series on conic sections. This time we will look at the same double-napped right circular conical surface but the intersecting plane will travel completely through one half of the surface and not through the base as seen below.
From a practical standpoint, various celestial bodies are known to travel in elliptical orbits. This includes our own planet with the sun at one of the foci. Also the reflective property of an ellipse where any light or sound emitted from one focus is reflected to the other focus is utilized in optics design. Also this phenomenon can be experienced in what are called “whispering galleries”. These are buildings with elliptical geometry such that a person at one focus can hear very easily someone speaking at the other focus due to the reflective property. One example of this that I have experienced myself is in the rotunda of the US Capital building.
In order to derive an algebraic expression for the ellipse we begin with the definition that an ellipse is the set of points in the plane such that the sum of the distances from the point to each of the two foci is constant for every point. A good way to visualize this is to imagine you have a piece of string, two pushpins, a piece of paper or cardboard, and a pencil. If you use the two pushpins to pin down each end of the string such that the distance between the two pushpins is less than the length of the string. Then use the pencil to pull the string taut in every direction and make a mark with the pencil along this perimeter as seen below you will have constructed an ellipse.
This way of looking at the ellipse probably makes it more intuitively clear what is meant by the above definition. You can see that every point on the ellipse has the same distance from one pin to the pencil and back to the other pin. This is the constant length of the string.
We begin with an ellipse centered on the origin and the two foci on the x-axis. One focus at (c,0) and the other at (-c,0). Then the distance from a point (x,y) in the plane to each focus is given by:
Now by the above definition we know that the sum of these distances should equal a constant for every point of the ellipse. Although it might seem strange we will use the constant 2a. The reason for the 2 will become more evident once we reach the end of our derivation which goes as follows.
The reason for the change in the last step is due to the fact that we know that the length of the string, 2a, must be greater than the distance between the foci, 2c. If 2a is greater than 2c, then a is greater than c and a squared must be greater than c squared. Typically a substitution is made at this point of,
Also notice from the substituting expression that a is greater than b. From this equation we can find the location of the major axis (longer axis) vertices and the endpoints of the minor axis (shorter axis). Simply set y=0 in the above expression and we get:
Since as we said before that a is greater than b, then the minor axis is along the y axis and has a length of 2b and the major axis is along the x axis and has a length of 2a. Conversely the equation for an ellipse with the major axis along the y axis is:
Just as with the parabola we can use the same substitutions for translation of axes and rotation of axes to develop expressions for other more general ellipses.
One final point of interest concerning the relationship between an ellipse, a circle, and a parabola. If the distance between the two foci is reduced to zero (i.e. foci are at the same point) the ellipse becomes a circle. If this same interfocal distance goes to infinity, the ellipse becomes a parabola.
The next post in this series soon to follow on hyperbolas.
My first soil mechanics lab was an exercise in analyzing the moisture content of a soil mixture. Soil is an aggregate consisting of what are called “phases”. The 3 phases are solids (tiny rocks and minerals), water, and air. The water and air make up what is called the void space. In evaluating moisture content (w), we are only interested in the water and solids portion. In fact, the moisture content is defined as the ratio of the weight of the water content (Ww) to the weight of the solid content (Ws). The weight of the air is considered negligible.
The procedure we followed to obtain values for these calculations was to use 3 small metal tins or canisters. First we used a digital scale to measure the weight of each canister. This weight we called W1. Next we partially filled each canister with a small sample of soil. Next we measured the canister and the soil. This weight we called W2. Finally we placed the canisters in a soil drying oven to dry out the soil and remove the water from the samples. After 24 hours we returned to the lab and used the digital scale to measure the dried soil samples in the canisters. This weight we called W3. To find the weight of the water (Ww) we used:
To find the weight of the solids (Ws) we used:
Here is a tabulation of the results we obtained:
This semester I have the pleasure of engaging in a little electrical engineering in my circuits lab. The first area we touched on in here and the first lab revolved around resistors. (Hence the corny title.) Below I have constructed a diagram of the typical two-terminal resistors that we are using and the typical 4 band color coding scheme that they employ.
So take as an example the above resistor diagram. The first step in determining the resistance provided by this particular resistor is to place it in a horizontal orientation as seen above with the metallic-colored band to the right. (In the example, the band is a metallic gold color.) Next you read the bands from left to right like reading this text.
The first band in our example is brown. Using the table, we see brown corresponds to the first digit of the resistance, 1. The second band is black. By the table we see the second digit in the resistance is a zero. The third band which is red tells us what the multiplier is. In this case the multiplier is 100. So with this information we can now calculate the resistance. We multiply the first two digits, 10, times the multiplier, 100, and obtain the resistance in ohms, 1000. Usually this is called 1 kilohm. The final band tells us the tolerance of the resistor. In our example the final band is gold, so the tolerance is +/- 5%. This means that the resistor could have a value from 950 ohms to 1050 ohms.
Besides learning how to determine the resistance we used a digital multimeter to measure the actual value and verify it falls within the range of the tolerance.
Any questions? More to come soon.
The parabola is one of four conic sections that I intend to provide a discourse on. The conic sections are so named because they can be obtained by passing a plane at varying angles through a double-napped right circular conical surface. The intersection of the plane and the surface is the conic section. In the case of the parabola, the plane passes through one half of the surface at an angle such that the plane is parallel to a generating straight line on the conical surface as seen below.
In the real world parabolas are typically seen when any projectile object travels through the air and is subjected to the pull of gravity (if air resistance is negligible). If you have ever seen a long exposure photograph of a bouncing ball that is moving laterally, you have seen a series of parabolas. Or if you have observed a water fountain flow as the water goes up and then down due to gravity. Also, the golden McDonald’s arches are a pair of parabolas. Paraboloids, the three- dimensional cousin of the parabola, is a common shape used in satellite dishes and headlight beam reflective surfaces. In highway engineering, the parabola is used to design vertical curves or hillcrests.
We can derive an algebraic expression for the parabola using the definition that a parabola is the set of all ordered pairs or points that lie in a plane that are equidistant from a fixed point called the focus and a fixed line called the directrix that also lie in the plane. (Provided the focus does not lie on the directrix) A line in the plane that is perpendicular or orthogonal to the directrix that passes through the focus is called the axis of symmetry or simply the axis. In layman’s terms it divides the parabola into two halves that are mirror images of one another. The point on the parabola that the axis intersects is called the vertex.
Consider a parabola whose vertex is at the origin (0,0) and has a focus at the point (0,f). Intuitively then we know that the distance from the vertex to the focus is f. (This is also called the focal distance.) We also see that this parabola will open along the y-axis. Since the vertex lies on the parabola it must be equidistant from the focus and directrix. Consequently we know that the directrix must be the line y = -f. Next using the distance formula, we can say the distance from a random point in the plane (x,y) to the focus (0,f) is given by:
Next realize that the point on our directrix that falls vertically below the point (x,y) is given by (x,-f). Then using the same methodology as above we can say that the vertical distance from the directrix to our point (x,y) is given by:
By the definition of a parabola we can set these two values equal to one another and resolve to standard form:
Through similar reasoning we could derive the expression for a parabola with a vertex at the origin that opens along the x-axis using a focus point of (f,0) and a directrix of x = -f. As might be expected this will yield:
Next let us consider how we can develop expressions for a parabola with a vertex that is not at the origin. The methodology utilized to consider a vertex at any general point in the plane is usually called translation of axes.
Translation of Axes
Consider the above two axes. The base x and y axes are shifted to a new general location where the new origin, O’, is at the base coordinate position (h,k). The point P can be described in both coordinate systems and the two systems can be related as follows:
Using these short equations in a substitutionary fashion we can go through the same procedure as above and derive an expression for a parabola with a vertical axis with a vertex at O’ or base coordinate (h,k). This yields:
Likewise, for a parabola with a horizontal axis we obtain:
Rotation of Axes
The other transformation we can do is called rotation of axes and it allows us to generate an expression for a parabola with an axis at an angle other than vertical or horizontal.
As with translation we want to develop a relationship between the two coordinate systems as they describe the same point in the plane. Observing the diagram above we will define the length of the blue line from the origin to the point P as having the magnitude d. Next consider the triangle OPQ’. Using this right triangle we can say:
Then look at triangle OPQ. Similarly as above we can say:
Using the trigonometry identities:
We can write an expressions for x and y and substitute using x’ and y’ to obtain expressions of the relationship between the two systems.
Finally using these last two expressions we will solve for x’ and y’. This will allow us to generate a parabola on the X’ and Y’ axes and then make a substitution in order to find the expression for the parabola in the base coordinate system. The derivation is done by solving each of the last two expressions for x’ and y’ respectively. This will result in four equations:
Next we will set x’ from the one equation equal to x’ from the other and solve for y’. Likewise we will set y’ equal to y’ and then solve for x’.
The last step in each of the above derivations is accomplished by the following identity:
In a similar fashion as above with translation we can substitute the two boxed equations into the general parabola equation to derive a general equation for a rotated parabola. An important caveat to remember with regard to this formula is that the angle can only be an acute angle. However this does not pose a problem since any parabola in any quadrant can be obtained by an acute rotation from an adjacent axis. I will not here go through each step to derive the general formula as it is rather lengthy and cumbersome for this format. However here is the end result in implicit form for a parabola whose original axis was vertical:
I hope that this has been helpful and I hope the next time you are using the water fountain you notice the lowly parabola at work.
I would love to hear your questions or comments.