Physics

Density

Experiment objectives:

1. Learn to correct read a digital caliper, micrometer and meter stick

2. Achieve an understanding of significant digits in measurement

3. Achieve a basic understanding of instrument precision

4. Cultivate the habit of keeping all experimental data

Experiment introduction:

Density

The mass density, 𝜌, of a material is defined as

𝑚

𝜌 = , (1)

𝑉

where 𝑚 is the mass of an object made of the material, 𝑉the volume of the object. The SI unit of density is kg/m3, another commonly used units is g/cm3.

Density is a characteristic property of materials; generally speaking, each different material will have its own density value, which makes it a valuable tool to identify unknown materials.

Instrument precision and significant figures in measurement

The following paragraphs assume that the reader has a clear understanding of how to use a meter stick, a digital caliper and a micrometer for measurement. The instruction can be found in Addendum A: how to use a digital caliper and a micrometer. at the end of this manual.

In any discipline where experiment is involved, people need to deal with the uncertainty of experiment. Please note that experiment uncertainty, which depends on both the skill of the experimenter and the apparatus used, is inevitable in any experimental work; we can work to minimize it, but we cannot eliminate it.

One major source of experimental uncertainty is the precision of the measurement instruments.

For an analogue instrument, such as the meter stick or the micrometer used in this experiment, the general rule is that the precision of it is one-tenth of the smallest increment in the scale of the instrument. The reason is that when making measurement as seen in Figure 1 (a), since the edge of the object falls between the smallest increment, one has to make an estimate. In the case shown in Figure 1 (a), one may estimate 0.5mm, it is also perfectly fine to estimate 0.6mm. Therefore, the precision is limited to the level of 0.1mm.

For a digital instrument, we usually need to refer to the instrument manual for its precision. Table 1 summarizes the 4 instruments used in this experiment along with their precisions.

Millimeter mark Centimeter reading ( a ) ( b )

Figure 1: Meter stick and its measurement scales. The numbers on the meter stick represent centimeter scale. The smallest scale increment is the millimeter mark.

Since different instruments have different precision, when recording measured data, we need to properly document the precision level using significant figures. In other words, the measured number obtained using an instrument should have a decimal place matching the decimal place of the instrument’s precision. For example, since the precision of a meter stick is 0.1mm or 0.01cm, the reading in Figure 1 (a) should be recorded as 428.6mm or 42.86cm. For the case when the edge of an object seems to fall exactly on a mark, as seen in Figure 1 (b), the reading in that case should be 420.0mm or 42.00cm. Please note that the trailing zero in the reading should NOT be dropped, because it is non-trivial, it carries the meaning of uncertainty for the reading. Of the number

420.0mm, the last digit, 0, is uncertain since it is limited by the precision of meter stick; the preceding 3 digits, 420, are certain, i.e. the number 420.0mm is certain to millimeter level, uncertain at tenth millimeter level. If the number is mistakenly recorded as 420mm, it would mean that the digits 42 are certain, the last digit 0 is uncertain, i.e. the number 420mm is certain to ten millimeter level, uncertain at millimeter level.

Table 1: Instruments used in this experiment

Instrument Type Smallest increment in the scale precision
Meter stick Analogue 1mm 0.1mm
Micrometer Analogue 10𝜇m 1𝜇m
Digital caliper Digital N/A 10𝜇m
Mass scale Digital N/A 0.5gram

Exploration:

During the Exploration, conduct the following investigation for two metal rods, aluminum and brass:

1. Determine the diameter of the rods using both a digital caliper and micrometer

2. Determine the length of the rods using a meter stick

3. Determine the mass of the rods using a mass scale

4. Determine the densities of the materials, and compare them to the reference values, 𝜌aluminum = 2.84g/cm3, 𝜌brass = 8.56g/cm3.

Exploration grade: 20 points

Please answer the following questions with one or two sentences. Some of the questions may appear in the post-lab quiz. Your instructor will randomly check your answers.

1. What is the zero offset of your micrometer (refer to the Addendum A: how to use a digital caliper and a micrometer. at the end of this manual)?

2. When measuring the diameter of the rods, do you make one single reading or the average of a few readings along different spots on the rods?

3. When measuring the diameter of the rods, are the readings from the digital caliper and micrometer close to each other?

4. Using the following formula, calculate the percent uncertainties of the 3 measurements (length by meter stick, diameter by the digital caliper and micrometer), and rank from lowest to highest the percent uncertainties.

instrument precision

percent uncertainty = ×100%

measured value Please also present the following to your instructor for a grade:

1. All measured data;

2. Relevant calculation;

3. Metal rods density values with percent errors determined using the following formula; experiment value − reference value

percent error = ×100%

reference value

Please note that points will NOT be marked down if any of the above is wrong; however, points will be deducted based on the following guideline.

-10 -5 No deduction
More than half of the materials are missing, illegible and/or poorly organized; results cannot be understood. Some but less than half of the materials are missing, illegible and/or poorly organized; efforts have to be made to understand the results. Everything is legible and well organized; instructors can easily understand the results.

Exploration notes:

The figure below show s the major equipment needed in this lab . The mass scale is not shown. micrometer Digital caliper Metal rods Meter stick

Application:

Ask your lab instructor for an unknown metal rod, and determine the density of the rod. The rod may have a shiny or tarnished appearance.

Inform your instructor once you confidently determine the rod density. Your instructor will compare it to the reference value and determine your Application grade.

Application Grade: 20 points

Please present the metal rod assigned to you, relevant calculation and the determined density to your instructor for a grade, which will be determined based on the following guideline:

Tarnished rod
Calculated current is within ____ of the measured value. ≤ ±2.5% ≤ ±3% ≤ ±3.5% ≤ ±4% ≤ ±4.5% > ±5%
Points 20 17 14 11 8 5
Shiny Rod
Calculated current is within ____ of the measured value. ≤ ±3.5% ≤ ±4% ≤ ±4.5% > ±5% ≤ ±5.6% > ±6%
Points 20 17 14 11 8 5

Additionally, points will be deducted based on the guideline below.

-10 -5 No deduction
More than half of the materials are missing, illegible and/or poorly organized; results cannot be understood. Some but less than half of the materials are missing, illegible and/or poorly organized; efforts have to be made to understand the results. Everything is legible and well organized; instructors can easily understand the results.

Addendum A: how to use a digital caliper and a micrometer.

How to operate a micrometer:

Figure 1 shows a typical micrometer, which is a measurement instrument capable of precision to the micrometer (1𝜇m = 0.001mm) level. It measures the spacing between its anvil and spindle. The precise measurement of a micrometer depends on delicate screw threads inside, which are vulnerable to damage if overtightened. To prevent damage, please use the ratchet on a micrometer to tighten the spindle, it will stop tightening and make a clicking sound when it is adequately tightened. Another advantage of using the ratchet is that it ensures, for different measurements, all readings are obtained with consistent tightenin;, there is no error caused by over- or under-tightening.

The reading of a micrometer is obtained by combining the linear scale and thimble scale as shown in Figure 2. The numbers on the linear scale are millimeter marks; the numbers on the thimble scale are 10 micrometer marks. For example, in the zoomed-in view of Figure 2, the number 5 on the thimble scale means 50 micrometers. The reading of Figure 2 is obtained with the following steps:

1. Locate the thimble edge: the edge of the thimble falls slightly beyond the 7mm mark on the linear scale;

2. Locate the horizontal line on the linear scale: the horizontal line lines up at about 2.4 marks on the thimble scale (keep in mind, the last digit is a personal estimate, one may have a slightly different estimate), which translates to 24𝜇m,

3. Combine the two readings: so the total reading is 7mm + 24𝜇m = 7.024mm.

Figure 2 : Detailed view of a micrometer . mm mark with 10 𝜇 m mark anvil spindle thimbleLinear scale with Thimble scale ratchet

You may notice that on the linear scale, there are two sets of marks. The set below the horizontal line is the half millimeter mark, which, when fully exposed, indicates that 0.5mm must be added to the reading. For example, in Figure 3, the thimble edge falls beyond 12mm, and fully exposes the half millimeter mark after it; meanwhile, the thimble scale lines up at the 25.3 mark, which is 253𝜇m, so the final reading is 12mm + 0.5mm + 253𝜇m = 12.753mm.

Half millimeter mark

Figure 3: Detailed view of the half millimeter mark on a micrometer.

Zero offset

When the anvil and spindle of a micrometer is closed in on each other, the reading on it should be zero. But more often than not, it is not zero. The zero offset has to be subtracted for correct measurement.

To determine the zero offset of a micrometer, one needs to close the anvil and the spindle, and record the reading on the micrometer. Please note that the zero offset could be either positive or negative. If the zero mark on the thimble scale is below the horizontal line of the linear scale, as seen in Figure 4 (a), the zero offset is positive; if it is above the horizontal line, as seen in Figure 4 (b), the zero offset is negative.

(a)
(b)

Figure 4: Zero offset of micrometer. A micrometer’s zero offset could be positive as in (a) or negative as in (b). In situation (a), the zero offset is +28𝜇m; in situation (b), the zero offset is −18𝜇mbecause the horizontal line on the linear scale lines at about 1.8 mark below the thimble zero.

How to operate a micrometer:

Figure 5 shows a typical digital caliper, which can be used to measure external diameter, internal diameter and depth of objects. In this experiment, we will use it to measure the external diameter of metal rods.

Zero offset is less of an issue for digital calipers because it can be easily corrected. Before any measurements, close the caliper jaws (a) indicated in Figure 5. The reading should be 0.00; if it is not, one simply presses the ZERO button on the caliper to reset it to zero. Then, the caliper is ready for measurement.

( a ) ( b ) ( c )

Figure 5: Detailed view of a digital caliper. This digital caliper can be used to measure (a) external diameter, (b) internal diameter and (c) depth.

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Lab 1 Report – Table

Using table is a convenient technique to organize, present and analyze information. When used correctly, tables can help the reader quickly understand abstract concepts, trends, etc. hidden with the information.

Please refer to Table 1 in the lab manual as example.

Requirements:

In this report, please make one data table to present your experiment data for the two metal rods you worked on in the Exploration part.

Please think about how to organize the data in the table so that it is easy for readers to follow the data and easy for the important information to stand out (the percent error is the important information in this case). Format

Item 0 points 2 points
Requirements The requirements above are neglected. The requirements above are met.
Table title and numbering Table title and numbering are missing Table title and correct numbering are presented.
Font format Non-standard font Times New Roman font in size 11 and bold type face is used for title numbering;

Regular Times New Roman font in size 11 for everything else.

Content

Item 0 points 2 points
Table title content Table title is missing, or does not describe what the data are. Table title briefly and clearly summarizes what data are presented.
Table Header Table header missing Table header for each row or column are presented to show the meaning of the numbers.
Units in table header Units are missing even when applicable. Appropriate units, when applicable, are presented.
Data to be presented: mass, length, average radius, volume, calculated density, reference density and percent error for all metal rods (8 points) All data are missing. All data are presented in a well-organized way.

Don’t forget to attach a copy of this rubric to your lab report, otherwise 5 points will be marked down.

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DATA:

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