The main focus of this experiment is to determine the coefficient of linear expansion of a metal rod and also to determine factors affecting the change in length in thermal expansion. To do so, two metal rods made from different materials were examined and tested for that principle. The length and the temperature of the rods were initially recorded using the resistance of the metal rods. Afterwards, the rod is heated by passing a steam in it using a steam generator. The change in length is measured using a micrometer. The post resistance and temperature were also recorded. After experimentation, we found out that temperature affects the dimension of an object. We also found out that aluminum has larger numerical value of coefficient of linear expansion than copper which means aluminum expands greater than copper. Apart from this, the obtained coefficient of linear expansion is compared to the actual value of each rod. We had obtained an error of 37% and 38% for aluminum and copper metal, respectively.
ABSTRACT.
INTRODUCTION Perhaps, in examining the strength of a bridge, civil engineers consider the temperature of the surrounding where bridge is to be constructed. For some time, its length might be shorter or longer than its actual length. Temperature can affect materials’ dimension. With relatively few exceptions, all objects expand when they are heated and contract when they are cooled, including the materials used in bridges. Once it is not considered, the project may lead to accidents and disaster especially when it is done in a extreme hot or cold place. This is the reason for putting expansion joints in the bridge. One of the basic properties of matter is the temperature. In lay man’s term, it is the measure of the hotness or coldness of a certain body and is somewhat in line with the sense of touch. But by formally defining it, it is related to the thermal energy held by an object, which is the kinetic energy of the random motion of the particles composing it. Among other physical property of a material, temperature is said to be one of the most fundamental. Temperature affects the dimensions of a certain material. When an object is heated or cooled, it expands or contracts in all dimensions. It is known as thermal expansion. However, for practical reasons, scientists and engineers often focus on two different kinds of expansion – on one and two dimensions. For one dimension, it is linear expansion while for two and three
dimensions, that is area and volume expansion. This expansion or contraction (negative counterpart of expansion) is caused by a change in the mean interatomic spacing due to a change in the amplitude of vibration of the atoms. For this experiment we are focusing on the one dimension expansion for simplification purposes. In the three first states of matter, gases are the simplest and the most easily affected by temperature. It is easily expands and compressed in a small variance in temperature unlike on liquids and solids upon comparison. It is due to the orientation of the molecules containing the matter. Gases can be easily affected because its molecules are far away from each other leading it to move easil y. On the other hand, solid is the least affected. Most solids, like a metal rod expand when heated and this expansion always occurs in three dimensions. However, the thermal expansion of solids is usually too small to be seen, but since the expansion is proportional to the length of the object, the expansion will be considerable if the object is large. This can be attain by using the formula
Where: is change of length is change in temperature is the initial length The formula is obtained from the principle that is the change in length of solids is proportional to the original length times the change temperature. As we had learned, the expansion is a function of temperature. The proportionality constant is represented by the Greek letter, alpha. This constant represents an individual
material which makes it as an identifier when dealing with this type of matter. The formula originally came from two observations. It is found experimentally that the increase in length of rod is directly proportional to: (a) Original length i.e., ∆L is prop. to L (b) Temperature is raise, i.e., ∆L is prop. to ∆T
Steam Micrometer
Generator
Hose Rod
Thermistor Expansion Base and Meter Sick Multitester
Combining (a) and (b):
Figure 1. Equipments Used in Experiment 301
So,
For simplification purposes, we can rearrange the formula in such a way that the constant is to be determined given the other variables known.
The expansion of metal rod is proportional to its length and the change in temperature. It is increase in length per unit of length at initial temperature per degree change in temperature. For this particular experiment, the data we had obtained allows us to determine the coefficient of linear expansion of a metal rod. Also, it gave us the chance to determine the factors affecting the change in length in thermal expansion. METHODOLOGY The type of apparatus to be used in this experiment may be one of the forms illustrated in the figure found on Figure 1. These are the metal rods, digital multitester, expansion base with built in gauge and thermistor, foam insulator, rubber tubing and steam generator. In either form, the metal rod to be studied is placed in the expansion base with built-in gauge and thermistor. The two metal rods used are made from aluminum and copper for the other. The rods are locked in the expansion base using an adjustable stop at one end. While on the other
side, makes contact with a device design to indicate the change in length as the rod expands (with the gauge). The steam for heating the rod is supplied by a boiler which is connected by a rubber hose to the inlet. The thermistor is get in touch with the metal rod which tells resistance using the temperature of the object. The multitester is then be used to read the resistance told by the thermistor. For this experiment, we are to use a digital multitester for a more convenient measuring. The steam generator which produces the steam is covered by a foam insulator. First thing we should do in the experiment is to record what type of rod is to be use, the initial length of the rod and the temperature of the system using its resistance. The resistance is showed in the multitester. To use the multitester, the two pointers is to be placed in the holes found in the expansion base. The two pointers (red and black) can be interchanged. The resistance n the multitester has a counterpart temperature found in the table (still, can be found in the expansion base). The initial length of the rod is measured using a meter stick. Instead of using highly accurate measuring equipment, we just use it since it doesn’t affect much the result. The gathering of the data must be done before heating the rod.
The adjusting end where the gauge is located must be pushed until the pointer is not moving to make sure that there is no space. The pointer is adjusted in the zero mark by rotating the gauge. By that we can easily read the gauge reading later.
percentage error.
For heating, we will use a steam generator. We must remember that carefulness is to be observed in using this equipment. It is filled with water of about half of its full capacity. It is covered by the foam insulator. There is a mounted hole there where the rubber hose is to be connected. Once generating steam, the other end of the hose is then connected to the rod and let steam pass through the rod. At this time, we will wait for about half of a minute until the steam come out. We will start recording the temperature or resistance when the gauge pointer stops to rotate. We will also record the gauge reading or the change in length of the rod. One complete revolution of the pointer means one millimeter. It is divided into 100 parts so a half revolution means 50 parts or 0.50 mm. The obtained data will be used in getting the experiment al value of the alpha, linear coefficient of expansion. It is compared with the theoretical value by computing the
The set up shown above is composed of steam generator, expansion base, a dial gauge, thermistor and the rods being studied. As temperature of the rod increases due to steam, the rod expands which is measured using the dial gauge. It is seen that the right part is higher than the left part for the water to let out. A thermistor will be used in obtaining the temperature by measuring the resistance in the rod.
The method is to be repeated from the start for the second rod. But before doing that, make sure that the thermistor is at room temperature. Also, we should avoid tapping the table to obtain an accurate gauge reading. If desired, we could perform another trial but make sure that the rod to be used is at room temperature. Set up of Experiment 1:
Similarly to the diagram above we proved the theory of the coefficient of thermal linear expansion. The micrometer form was referred to as the “dial gauge.” As well as the thermometer was used along with the “thermistor” as shown in the diagram. The metal used were aluminum and copper with a standard expansion coefficient of 2.38 x 10 -5 and 1.68 x 10 -5, respectively.
RESULTS AND DISCUSSION During heat transfer, the energy that is stored in the intermolecular bonds between atoms changes. When the stored energy increases, so does the length of the molecular bond. As a result, solids expand in response to heating and contract on cooling. This response to temperature change is expressed as its coefficient of thermal expansion. The linear thermal expansion coefficient relates the change in temperature to the change in a material’s linear dimensions. It is the fractional change in the length of a bar per degree of temperature change. From our preliminary observations, we had obtained that aluminum and copper rods have the following: Type of Tube
Aluminum Tube of 705.1 mm
Copper Tube 705.2 mm
114200 Ω
111700 Ω
Initial Length Tube, Lo Initial Resistance of Thermistor at Room Temperature, Rrm Initial Temperature, trm
20 °C
22 °C
In measuring the initial length of the tubes, we only used a meter stick because it does not affect much the percentage error. It is reasonable because based on the formula of percentage error; the relativity of the length is used. An error of 1 mm is too small for the whole value or let me say: 705.1 mm >>> 1 mm We observed that the room temperature is below normal because of the air conditioner. Since the metals are strong conductor, the temperature of the room is the temperature of the rod. Instead of using a thermometer (and measure the temperature of the room), we use the resistance of the rod which can be used in determining temperature. This is only applicable for temperature lower or equal to 100 0C. By that
usage of resistance, we could say that the temperature determination is nearly accurate because it directly measures the temperature of the rod. Since the resistance range have a temperature counterpart (increment of 1 0C), we could use interpolation of the data for the corresponding temperature of a specific resistance. Since the temperature is based on resistance, the thermistor, which measure the resistance of the rod must be properly contacted to it. For an error of 10,000 Ω, the result is still the same. So, it doesn’t produce much deviation. In heating the rod, the steam produced passed through the rod. As steam passes, heat transfer occurs from the steam to the rod. Because of that, sudden expansion of the rod occurs. Since enough heat is transferred, the steam returns back into water and stuck inside the tube. Because of that, heat from the rod may be transferred to water. To avoid that, the rod should be oriented in an inclined position for water not to stock. The heat transfer between the rod and steam is limited since the experiment is designed such that steam can only supply a certain amount of heat. Once the rod reached its maximum expansion (maximum temperature), we could start measuring the resistance of the rod. If we start measuring the resistance not at maximum expansion, we might be getting the correct measure (much smaller than the actual). The fractional change in length which is a natural quantity to use is the difference in length of the micrometer measurements (initial minus the final) divided by the length of the metal rod used. This is equivalent to the expansion coefficient multiplied by the change in temperature of the metal rod used. For our result, it is shown below: Type of Tube
Aluminum Tube Change in Length of 1.27 mm Tube, ∆L
Copper Tube 0.92 mm
Resistance of Thermistor at Final Temperature, Rhot Final Temperature of the Tube, thot Change in temperature of the Tube, (thot - trm) Experimental Coefficient of Linear Expansion, αexperiment
13100 Ω
11500 Ω
75 °C
78 °C
55 C°
56 C°
3.27484E05 /C°
2.32963E05 /C°
The values of coefficient of linear expansion obtained using the formula obtained are 2.27x10 5 /C0 and 2.33x10 -5 /C0 for aluminum and copper respectively. It is observed that aluminum has higher value of coefficient copper. It is due to copper’s molecular weight is greater than aluminum. The aluminum can vibrate greater than copper because it is lighter. So, as temperature changes, the molecules of aluminum move more rapidly than heavier molecules of copper. As a result, expansion of aluminum became significant even at smaller temperature change than copper. Among all parameters, the change in temperature greatly affects the experimental value of coefficient of linear expansion. Taking aluminum as our example, based on our result (assume that other numerical values are accurate and doesn’t produce any error): For every mistake of 1 0C:
The x values are the ∆t, while the y values are the percent error. The graph shown above is logarithmic. For an actual value of 75 C 0, produces an error of 0%. While in our experiment, we produce an error of 37% that has 20 Celsius degree difference from the assumed actual value. Based on the graph, for the change in temperature of 38 Celsius degree produces an error of 100%. Aside from that great source of error, I couldn’t see any significant source from the values. Despite of that, we still got a great error of 37% because we are dealing with microscopic measurements. So, it is possible to say that deviation in the numerical values leads to error. The errors for both rods used are: Type of Tube Percentage error
Aluminum Tube of 37.5983923 3%
Copper Tube 38.6686268 4%
Aside from linear expansion, there is also area and volume expansion for two and three dimensions. For this experiment, we focused on one dimension, just to prove the law of thermal expansion. CONCLUSION In the experiment, we are aimed to determine the coefficient of linear expansion of a metal rod and to determine the factors affecting the change in length in thermal expansion. And upon by carefully doing the necessary procedures of the experiment and by interpreting the obtained data and results, we found out that dimension of solid objects are affected by temperature. The change in length of an object is proportional to its original length and to the change in temperature. As we increase temperature, the length also increases (expansion) while when we decrease the temperature, the length of the object decreases (contraction).
120 100 80 60 40 20 0 38
48
58
68
The objects expand and contract based on temperature due to molecules of an object vibrates greater than at lower temperature. The
amplitude of vibration becomes greater resulting to expansion.
some of his expansion.
It is also found out that copper has less coefficient of linear expansion than aluminum. It is due to copper has greater molecular weight than aluminum. So, aluminum expands greater than aluminum.
Finally, I would like to thank my brother for letting me first to use the computer to make this work report.
It is obtained that the greatest factor in error accumulation is the error in measuring temperature. Also, deviation can be considered because we are dealing with microscopic measuring. We also account for not tilting the rod while steam is not flowing. It is because the water may stock inside the tube and causes error in the actual temperature. Despite of this, we can say that our experiment is quite well done. The application of this experiment has a great significant to us. We may strengthen bridges and other constructions by considering expansion of the body. Because of this principle, we may be able to predict how much space must be allocated to in constructions to avoid harmful accidents. ACKNOWLEDGEMENT First and foremost, I want to acknowledge my fellow group companion for giving me her full hands in making up the experiment. My warmest gratitude to Mr. Deduyo, Mami Red and Ms. Gimena for the help on performing the experiment, to Ms. Ang for the recording of the data and Mr. Borres, even though he is not one of our group mate, for helping us in making up the experiment. If you weren’t there, I can’t make a good result so thus a good report.
ideas
there
regarding
thermal
REFERENCES
[1] Young, H., Freedman, R., University Physics with Modern Physics, 11th Edition, 2004 [2] Bernard, C.H., Laboratory Experiment in College Physics, 7th Edition, 1995 [3] en.wikipedia.org/wiki/Thermal_expansion [4] http://www.physics.info/expansion/ [5] http://www.britannica.com/EBchecked/topic/ 591408/thermal-expansion FREE SPACE Burj Khalifa known as Burj Dubai prior to its inauguration, is a skyscraper in Dubai, United Arab of Emirates, and the tallest man-made structure ever built, at 828 m (2,717 ft). Construction began on 21 September 2004, with the exterior of the structure completed on 1 October 2009. The building officially opened on 4 January 2010, and is part of the new 2 km2 (490-acre) flagship development called Downtown Dubai at the 'First Interchange' along Sheikh Zayed Road, near Dubai's main business district. The total cost for the project was about US$1.5 billion; and for the entire "Downtown Dubai" development, US$20 billion. In
March
2009, Mohamed
Ali developer, Emaar
Alabbar, chairman of the project's Properties, said office space pricing at Burj Khalifa reached US$4,000 per sq ft (over US$43,000 per m2) and the Armani Residences, also in Burj Khalifa, sold for US$3,500 per sq ft (over US$37,500 per m 2).
I also like to express our earnest gratitude to Prof. De Leon not only for imparting to us some of her insights regarding the experiment, but also for some of her ideas of what would be the theoretical result. I also thank again the Library for lending to us their computer services. I want to acknowledge the author of our text book, Mr. Young of University Physics, for writing
Imagine if it’s giant steel, a one degree oC rise in temperature means 0.032 ft taller than its original height. Not bad at all! Note:
