UOIT PHY1010U Lab # PhyI-01: Measurement and Evaluation of Physical Parameters

Physics I Laboratory

Faculty of Science, UOIT

Lab # PhyI-01: Measurement and Evaluation of Physical Parameters Introduction (Knight, Ch.1: 1.8) This laboratory is an introduction to physical measurement and error analysis procedures. There are many different techniques for determining physical parameters. Some of these methods are called direct measurements as one reads the desired physical quantity directly from a measuring device (for example, a time interval t is is measured with a stopwatch and a distance L is measured with a ruler); others called indirect measurements are based on obtaining the results from calculations involving the directly measured quantities (as in the case of velocity ϖ = L/t ). ). Any measured value of a physical quantity is not an exact number. We can determine their value only within a certain accuracy that depends on the method and equipment used. Thus, each determined quantity X is

[

]

valid within a certain interval of values, namely X −σ X , X + σ X , that is recorded as X = X ± σ X , where

X is a mean value of X, and

σ X – uncertainty (see Fig.1). Therefore, the higher the precision of the method,

the smaller is the interval or uncertainty σ X .

Figure 1: Confidence/Valid Interval Illustration In this experiment, we will be using calipers (Appendix 1) to measure an object’s dimensions, a triple beam balance (Appendix 2) to determine its mass, and Archimedes’ Principle of Displacement to indirectly determine the volume and density of irregularly shaped objects.

Archimedes’ Principle Archimedes’ principle states that a body wholly or partially submerged in a fluid is buoyed up by a force equal in magnitude to the weight of the fluid displaced by the body. It is important to remember that fluids include liquids and gases, and that the buoyant force exerted by a fluid is not determined by the properties of the object but only by those of the fluid. This force f orce is given by:

F b

= ρ f V f g

(1)

where ρf is the density of the fluid, V f f is is the volume of fluid displaced and, g is acceleration due to gravity. It is the buoyant force that keeps ships afloat and hot air balloons floating in air. Archimedes’ principle is useful for determining the volume, and therefore the density of an irregularly shaped object by measuring its mass in air, m , and its effective mass when submerged in water, me. This effective mass under water will be its actual mass minus the mass of the fluid displaced. The difference between the real and effective mass ∆m = m - me therefore gives the mass of water displaced and allows V = ∆m/ ρwater . The real mass divided by the the calculation of the volume of the irregularly shaped object V = volume thus determined gives a measure of the average density of the object ρ = m/V .

Lab # PhyI-01: Measurement and Evaluation of Physical Parameters



Equipment Density set (one aluminum/brass cylinder or block and one irregular-shaped aluminum block), calipers, string, overflow can, beaker for c atching water, graduated cylinder (50 ml), and triple-beam balance.

Figure 2: Experiment Equipment

Purpose In this lab, you will examine both regular- and irregular-shaped objects and apply different experimental methods to determine the physical parameters of length, mass, volume, density and buoyant force. Both the volume and the density will be determined by two different methods, and the calculated values of density will be compared to reference data. This will allow you to put side by side the precision of different methods and discuss reasons and sources for their inaccuracy.

Experiment #1: Length, Diameter and Volume Purpose The objective of this experiment is to measure the dimensions and volume of objects and to perform an error analysis for both direct and indirect methods of measurements.

Procedure (Experimental Method) In this experiment, you will be determining the volume using two different methods. The f irst method is an indirect and is based on calculating the volume using measured linear dimensions. It can only be used for regular-shaped objects and is called the Volume Equation Method . The second method is a direct volume measurement that can be used for both regular- and irregular-shaped objects and is sometimes called the Displaced Volume Method .

Volume Equation Method This method is applied effectively only for regular-shaped objects. Using the calipers, measure the geometrical parameters given below at least 5 times and at different positions (even regularly shaped objects have surfaces that are not perfectly ideal): for the cylinder : measure the height h and the diameter of the cylinder d ; for the regular-shaped block : measure the length l, the width w and the height h of the block. You have to take so many decimal places in calipers’ readings as you can (See Appendix 2 of this manual). Record the results in Table 1.1.


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Table 1.1. Measurements Object

Height h, cm

Diameter d or width w, cm

Length l, cm

N/A

N/A

N/A

Displaced volume V D , ml (cm3)

regular-shaped object irregularshaped object

Displaced Volume Method For both the regular- and irregular-shaped objects, find the object’s volume by finding the volume of water that each one displaces: 1. Put the beaker under the overflow can spout as shown in Figure 3.

Figure 3: Displacement of Water Experiment 2. Pour water into the overflow can until it overflows into the beaker. Allow the water to stop overflowing on its own and empty the beaker into the sink. Now, return it to its position under the overflow can spout without jarring the overflow can. 3. Tie a string to each of the objects (including the irregularly-shaped object). 4. Gently lower the first object into the overflow can until it is completely submerged. Allow the water to stop overflowing and then pour the water from the beaker into the graduated cylinder. Measure the volume of water that was displaced by reading the water level in the graduated cylinder in milliliters (1ml = 1 cm3). Record this volume V D in Table 1.1. Repeat this procedure at least 5 times. 5. Repeat this procedure for the other object.

Analysis Since the shape of an object is not perfectly-ideal, when you try to determine the value of one of its dimensions, measurements in even slightly different places can give you slightly different results. In this case, one can only state a mean value of the measurements, not an exact value. The measured spread in data gives you a statistical uncertainty. One more uncertainty – instrumental – comes from the use of non-ideal measurement tools; in this case, from the calipers. When we want to measure an object’s volume by the Displaced Volume Method, we also can obtain slightly different results for each different measurement. The


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reasons for this can be numerous: water left in the beaker, the water’s surface shape in the overflow can due to its surface tension, and others. To calculate the measurement’s uncertainties, follow the recommendations found in the Significant Figures and Error Analysis file. This file is posted on WebCT. 1. First, calculate the mean values of all measured quantities in Table 1.1 ( h, w or d, l, V D) using the following formula: N

∑ x x

i

i =1

=

,

N

where xi is the ith result of measurement of quantity x, and N is the number of such measurements. 2. Calculate the absolute statistical uncertainties of these ( h, w or d, l, V D) measured quantities: N

∑ ( x

i

− x

)2

i =1

σ x , stat =

N ( N − 1)

3. Calculate the absolute uncertainties of all measured quantities taking into account the instrumental uncertainty: σ x =

(σ , )2 + (σ , )2 , x stat

x inst

where σ x,inst = 0.05 mm is the instrumental uncertainty of calipers and σ x,inst = 0.5 ml (cm3) is the graduated cylinder instrumental uncertainty. 4. Calculate the mean volume for the regular-shaped object: 2

 d  V C = h ⋅ π ⋅   for the cylinder or  2  V B

=

l ⋅h ⋅w

for the block.

5. Calculate the absolute uncertainty of the volume V C or V B measurements:

 σ d  σ C = V C  2  d   σ B

 σ l  = V   B  l 

2

2

2

 σ h  +  , h  

 σ h  +  h  

2

2

 σ w  +  . w  

6. Record the data obtained in Table 1.2. Compare the results for the two different volume methods. 7. Make a conclusion stating which method is more accurate and why. 8. Example of calculations can be found on Blackboard.


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Table 1.2. Analysis

Regular-shaped object: Mean (average) value, x

Instrumental uncertainty,

Statistical uncertainty,

Absolute uncertainty,

σ x,inst

σ x,stat

σ x

N/A

N/A

Instrumental uncertainty,

Statistical uncertainty,

Absolute uncertainty,

σ x,inst

σ x,stat

σ x

Height, h Diameter of cylinder or width of block, d/w Length of block only, l Volume calculated, V B/C Volume displaced, V D

Irregular-shaped object:

Mean value,

x Volume displaced, V D

Experiment #2: Mass and Density Purpose The objective of this experiment is to measure mass and density parameters, and to carry out an error analysis for both direct and indirect methods of measurements.

Procedure (Experimental Method) In this experiment, you will find the densities of regular and irregular shaped objects made from two different materials (brass or aluminum) and will compare the obtained results with the known density values of these materials (for reference you can use CRC Handbook of Chemistry and Physics , CRC Press, 2001, available at the UOIT library or go to: http://www.matweb.com – property of materials web site). Using the triple-beam balance, find the mass of all the objects used in Experiment 1. Record the results in Table 2.1. Since the result of the measurement of the object’s mass depends only on the property of used balance, it is not necessary to repeat measurements several times, do just one measurement and for uncertainty use the instrumental uncertainty of the balance. Note: do not forget to carefully balance the balance before weighing the object.


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Object


Table 2.1. Measurements Instrumental Absolute uncertainty, uncertainty, σ m,inst , g σ m , g

Mass m , g

regular-shaped object irregular-shaped object

Analysis 1. Calculate the density, ρ , of each object using the mean volume values from Table 1.2: ρ =

m V

2. Calculate the density’s absolute uncertainty:

σ ρ

 σ m  =ρ    m 

2

2

 σ V  +  , V  

where σ m = 0.05 g is the balance uncertainty, and σV is the volume uncertainty σ C ,, σ B, or σ D in Table 1.2. 3. Record all of the calculated density and uncertainty values in Table 2.2. 4. Obtain the accepted value for the density of brass and aluminum from a reference book CRC Handbook of Chemistry and Physics, CRC Press, 2001 and record the material name. 5. Compare your results and make a conclusion.

Object

Volume, V, cm3

Absolute uncert., σ V, cm3

Mass, m , g

Absolute uncert., σ m , g

Table 2.2. Analysis Density, Absolute uncert., ρexp , 3 σ ρ , g/cm3 g/cm

regular-shaped object, indirect volume equation method regular-shaped object, direct displaced volume method irregular-shaped object, direct displaced volume method


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Appendixes Appendix 1. Vernier Calipers: Instructions on Use

Figure 4: Vernier Calipers Reading • • •

• • • • •

•

Vernier calipers is a very precise measuring instrument; the reading error is 1/20 mm = 0.05 mm. Close the jaws lightly on the object to be measured. If you are measuring something with a round cross-section, make sure that the axis of the object is perpendicular to the calipers. This is necessary to ensure that you are measuring the full diameter and not merely a chord. Ignore the top scale, which is calibrated in inches. Use the bottom scale, which is in metric units. Notice that there is a fixed scale and a sliding scale. The numbers on the fixed scale are in centimeters. The tick marks on the fixed scale between the numbers are in millimeters. There are twenty tick marks on the sliding scale. The left-most tick mark on the sliding scale will let you read, from the fixed scale, the number of whole millimeters that the jaws are opened. In the example above, the leftmost tick mark on the sliding scale is between 21 mm and 22 mm, so the number of whole millimeters is 21.

Next we find the hundredth of millimeters. Notice that the twenty tick marks on the sliding scale are the same width as nineteen tick marks on the fixed scale. This means that at most one of the tick marks on the sliding scale will align with a tick mark on the fixed scale; the others will miss.


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•

•

•


The number of the aligned tick mark on the sliding scale tells you the number of tenth of millimeters. In the example above, the 7th tick mark on the sliding scale is in coincidence with the one above it, so the caliper reading is (21.35 ± 0.05) mm. If two adjacent tick marks on the sliding scale look equally aligned with their counterparts on the fixed scale, then the reading is half way between the two marks. In the example above, if the 7 th and 8th tick marks on the sliding scale looked to be equally aligned, then the reading would be ( 21.38 ± 0.05) mm. On those rare occasions when the reading just happens to be a "nice" number like 20 mm, don't forget to include the zero decimal places showing the precision of the measurement and the reading error. So not 20 mm, but rather (20.00 ± 0.05) mm.

Take a look at the Java applet found at: http://www.phy.ntnu.edu.tw/java/ruler/vernier.html

Appendix 2. Triple Beam Balance: Instructions on use

Figure 5: Triple Beam Balance • •

• • • • • •

The triple beam balance is used to measure masses very precisely; the reading error is 0.05 gram. With the pan empty, move the three sliders on the three beams to their leftmost positions, so that the balance reads zero. If the indicator on the far right is not aligned with the fixed mark, then calibrate the balance by turning the set screw on the left under the pan. Once the balance has been calibrated, place the object to be measured on the pan. Move the 100 g slider along the beam to the right until the indicator drops below the fixed mark. The notched position immediately to the left of this point indicates the number of hundreds of grams. Now move the 10 g slider along the beam to the right until the indicator drops below the fixed mark. The notched position immediately to the left of this point indicates the number of tens of grams. The beam in front is not notched; the slider can move anywhere along the beam. The boldface numbers on this beam are in grams and the tick marks between the boldface numbers indicate tenths of grams. To find the mass of the object on the pan, simple add the numbers from the three beams. As with a ruler, it is possible to read the front scale to the nearest tick mark and estimate one more decimal place. Take a look at the Java applet found at: http://www.touchspin.com/chem/DisplayTBB.html


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UOIT PHY1010U Lab # PhyI-01: Measurement and Evaluation of Physical Parameters

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