LVang+-+L1+-+Measurement+Lab

Lue Vang Mr. Kellogg AP Physics - Pd. 7 11 September 2011 Performed On: 6-7 September 2011 Lab Partners: Arielle Roth, Kevin Trinh

**Measurements Lab** The following lab with let AP Physics students practice using standard measurement tools such a ruler, Vernier caliper, and micrometer. The lab also reviews calculations for volume, density, and errors. In addition, it will get the students to apply the rules of significant figures. Most importantly, the lab will study the relationship between English and Metric units for length.
 * Purpose:**

Measurements of the English and metric system differ because they use different units to measure length, mass, time, temperature, etc. Even so, measurements can be converted back and forth by using conversion units. The lab will study the conversion from inches to centimeters, along with their derived units, and vice versa. The experiment will then compare its data with the accepted value and calculate the experimental error with the following formula:
 * Background:**



% error = |expected – experimental value|/expected value * 100

The lab will also be comparing density and volume cylinders and spheres, which will require use of the respective equations:   

p(density) = m/v

V(cylinder) = πR2L

V(Sphere) = 4/3* πR3

If the materials and shape are measured carefully, the calculated percent error (equation above) will be very low (5- 15%). In the perfect world, the percent error will be 0%, but it is impossible to collect data measured data with 100% accuracy, so this will not be the case: the person reading will have to estimate the measurement’s mark, the object might be slightly misaligned, or the measuring tool might not be able to distinguish smaller fractional units.
 * Hypothesis:**


 * Apparatus:**
 * 1 Ruler (cm/in.)
 * 1 Vernier caliper
 * 1 Micrometer
 * 1 Balance/Scale
 * 1 Card
 * 1 Block #1
 * 1 Block #2
 * 1 Cylinder #1
 * 1 Cylinder #2
 * 1 Sphere #1
 * 1 Sphere #2
 * 1 Calculator


 * Procedure:**
 * Part A:** Take the card and take ten measurements of its length and width using both centimeters and inches. Calculate the mean of the collected data, the deviation, the percent deviation. Next, calculate the area of the card. Then, take the average measurement of the card’s length in centimeters and divide by the average measurement of the card’s length in inches to find the conversion factor of centimeters to inches. Finally, calculate the percent error with the accept value of 2.540 cm/in.


 * Part B:** Take Block #1 and use a ruler to measure its dimensions in inches, then centimeters, for three trials. Find the average of its length, width, and height. Use the average values to calculate the block’s volume in both centimeters and inches. Divide the cubic centimeter volume by the cubic inches volume to find the experimental conversion factor. Then, compare it with the accepted value of 16.39 cm³/in.³ by calculating error and percent error. Repeat the steps with Block #2.


 * Part C:** Take Cylinder #1 and use a Vernier caliper to measure the length and diameter of the cylinder for three trials. Use the average radius and length to calculate and record the volume of the cylinder. Then, weigh the cylinder on a balance or scale and use the mass to along with the calculated volume to find the cylinder’s density. Repeat the steps with Cylinder #2.


 * Part D:** Take Sphere #1 and measure its diameter with a micrometer for three trials. Using the collected data, find the radii and use their average to calculate the sphere’s volume. Weigh the sphere and record its mass. Then, use the data collected to find the density of the sphere. Repeat the steps with Sphere #2.


 * Data:**
 * ** Table A-1: Card Measurements ** ||
 * ** Trial ** || ** Length (L) ** || ** Width (W) ** || ** Length (L) ** || ** Width (W) ** ||
 * ** No. ** || ** cm ** || ** cm ** || ** in. ** || ** in. ** ||
 * ** 1 ** || 18.21 || 6.62 || 7.16 || 2.63 ||
 * ** 2 ** || 18.11 || 6.62 || 7.16 || 2.63 ||
 * ** 3 ** || 18.29 || 6.61 || 7.13 || 2.63 ||
 * ** 4 ** || 17.99 || 6.52 || 7.13 || 2.50 ||
 * ** 5 ** || 18.13 || 6.59 || 7.13 || 2.63 ||
 * ** 6 ** || 18.20 || 6.60 || 7.16 || 2.59 ||
 * ** 7 ** || 18.18 || 6.61 || 7.19 || 2.28 ||
 * ** 8 ** || 18.20 || 6.60 || 7.13 || 2.59 ||
 * ** 9 ** || 18.08 || 6.51 || 7.13 || 2.56 ||
 * ** 10 ** || 18.12 || 6.69 || 7.13 || 2.63 ||
 * ** Mean ** || 18.15 || 6.60 || 7.14 || 2.57 ||


 * ** Table A-2: Deviation of Card Measurements ** ||
 * ** Trial ** || ** Length (L) ** || ** Width (W) ** || ** Length (L) ** || ** Width (W) ** ||
 * ** No. ** || ** cm ** || ** cm ** || ** in. ** || ** in. ** ||
 * ** 1 ** || 0.06 || 0.02 || 0.02 || 0.06 ||
 * ** 2 ** || 0.04 || 0.02 || 0.02 || 0.06 ||
 * ** 3 ** || 0.14 || 0.01 || 0.02 || 0.06 ||
 * ** 4 ** || 0.16 || 0.08 || 0.02 || 0.07 ||
 * ** 5 ** || 0.02 || 0.01 || 0.02 || 0.06 ||
 * ** 6 ** || 0.03 || 0.01 || 0.05 || 0.28 ||
 * ** 7 ** || 0.03 || 0.01 || 0.05 || 0.28 ||
 * ** 8 ** || 0.05 || 0.00 || 0.02 || 0.03 ||
 * ** 9 ** || 0.07 || 0.09 || 0.02 || 0.00 ||
 * ** 10 ** || 0.03 || 0.09 || 0.02 || 0.06 ||
 * ** Mean ** || 0.06 || 0.04 || 0.02 || 0.10 ||
 * ** % Deviation ** || 0.35% || 0.53% || 0.31% || 3.75% ||


 * ** Table A-3: Area of Card ** ||
 * ** Area (cm²) ** || ** Area (in.²) ** ||
 * 119.74 || 18.32 ||


 * ** Table A-4: Percent Error of cm-in. Conversion Factors ** ||
 * ** Experimental Conversion Factor (cm/in.) ** || ** Accepted Value (cm/in.) ** || ** Error (cm/in.) ** || ** % Error ** ||
 * 2.54 || 2.54 || 0.00 || 0.00 ||


 * ** Table B-1: Block #1 Measurements ** ||
 * ** Trial ** || ** Length (L) ** || ** Width (W) ** || ** Height (H) ** || ** Length (L) ** || ** Width (W) ** || ** Height (H) ** ||
 * ** No. ** || ** cm ** || ** cm ** || ** cm ** || ** in. ** || ** in. ** || ** in. ** ||
 * ** 1.00 ** || 8.45 || 3.30 || 2.50 || 3.34 || 1.31 || 1.00 ||
 * ** 2.00 ** || 8.49 || 3.32 || 4.49 || 3.31 || 1.31 || 1.00 ||
 * ** 3.00 ** || 8.54 || 3.31 || 2.51 || 3.31 || 1.25 || 1.00 ||


 * ** Table B-2: Volume of Block #1 ** ||
 * || ** Average of Volumes ** || ** Volume Derived from Average of Dimensions (cm³) ** ||
 * ** cm³ ** || 89.07 || 89.02 ||
 * ** in.³ ** || 4.29 || 4.29 ||


 * ** Table B-3: Percent Error of cm³-in.³ Conversion Factors for Block #1 ** ||
 * ** Experimental Conversion Factor (cm³/in.³) ** || ** Accepted Value (cm³/in.³) ** || ** Error (cm³/in.³) ** || ** % Error ** ||
 * 20.74 || 16.39 || 4.35 || 26.55 ||


 * ** Table B-4: Block #2 Measurements ** ||  ||   ||
 * ** Trial ** || ** Length (L) ** || ** Width (W) ** || ** Height (H) ** || ** Length (L) ** || ** Width (W) ** || ** Height (H) ** ||
 * ** No. ** || ** cm ** || ** cm ** || ** cm ** || ** in. ** || ** in. ** || ** in. ** ||
 * ** 1.00 ** || 8.80 || 4.21 || 1.89 || 3.44 || 1.69 || 0.67 ||
 * ** 2.00 ** || 8.75 || 4.21 || 1.89 || 3.44 || 1.66 || 0.75 ||
 * ** 3.00 ** || 8.71 || 4.20 || 1.88 || 3.44 || 1.66 || 0.75 ||


 * ** Table B-5: Volume of Block #2 ** ||
 * || ** Average of Volumes ** || ** Volume Derived from Average of Dimensions (cm³) ** ||
 * ** cm³ ** || 69.47 || 69.47 ||
 * ** in.³ ** || 4.14 || 4.14 ||


 * ** Table B-6: Percent Error of cm³-in.³ Conversion Factors for Block #2 ** ||
 * ** Experimental Conversion Factor (cm³/in.³) ** || ** Accepted Value (cm³/in.³) ** || ** Error (cm³/in.³) ** || ** % Error ** ||
 * 16.79 || 16.39 || 0.40 || 2.44 ||


 * ** Table C-1: Cylinder #1 Measurements ** ||
 * ** Trial ** || ** Length (L) ** || ** Diameter (D) ** || ** Radius (R) ** ||
 * ** No. ** || ** cm ** || ** cm ** || ** cm ** ||
 * ** 1 ** || 9.68 || 1.58 || 0.79 ||
 * ** 2 ** || 9.69 || 1.58 || 0.79 ||
 * ** 3 ** || 9.68 || 1.59 || 0.80 ||


 * ** Table C-2: Volume and Mass of Cylinder #1 ** ||
 * ** Volume (cm³) ** || ** Volume Derived from Average of Dimensions (cm³) ** || ** Mass (g) ** ||
 * 19.06 || 19.06 || 22.28 ||


 * ** Table C-3: Percent Error of Experimental Mass Density of Cylinder #1 ** ||
 * ** Experimental Mass Density (g/cm³) ** || ** Accepted Mass Density (g/cm³) ** || ** Error (g/cm³) ** || ** % Error ** ||
 * 1.17 || 1.24 || 0.07 || 5.71 ||


 * ** Table C-4: Cylinder #2 Measurements ** ||
 * ** Trial ** || ** Length (L) ** || ** Diameter (D) ** || ** Radius (R) ** ||
 * ** No. ** || ** cm ** || ** cm ** || ** cm ** ||
 * ** 1 ** || 6.10 || 1.62 || 0.81 ||
 * ** 2 ** || 6.10 || 1.63 || 0.82 ||
 * ** 3 ** || 6.10 || 1.63 || 0.82 ||


 * ** Table C-5: Volume and Mass of Cylinder #2 ** ||
 * ** Volume (cm³) ** || ** Volume Derived from Average of Dimensions (cm³) ** || ** Mass (g) ** ||
 * 12.67 || 12.67 || 17.05 ||


 * ** Table C-6: Percent Error of Experimental Mass Density of Cylinder #2 ** ||
 * ** Experimental Mass Density (g/cm³) ** || ** Accepted Mass Density (g/cm³) ** || ** Error (g/cm³) ** || ** % Error ** ||
 * 1.35 || 1.36 || 0.01 || 1.09 ||


 * ** Table D-1: Sphere #1 Measurements ** ||
 * ** Trial ** || ** Length (L) ** || ** Diameter (D) ** || ** Radius (R) ** ||
 * ** No. ** || ** cm ** || ** cm ** || ** cm ** ||
 * ** 1 ** || 2.54 || 2.54 || 1.27 ||
 * ** 2 ** || 2.53 || 2.53 || 1.27 ||
 * ** 3 ** || 2.52 || 2.52 || 1.27 ||


 * ** Table D-2: Volume and Mass of Sphere #1 ** ||
 * ** Volume (cm³) ** || ** Volume Derived from Average of Dimensions (cm³) ** || ** Mass (g) ** ||
 * 8.52 || 8.52 || 9.68 ||


 * ** Table D-4: Sphere #2 Measurements ** ||
 * ** Trial ** || ** Length (L) ** || ** Diameter (D) ** || ** Radius (R) ** ||
 * ** No. ** || ** cm ** || ** cm ** || ** cm ** ||
 * ** 1 ** || 2.53 || 2.53 || 1.27 ||
 * ** 2 ** || 2.52 || 2.52 || 1.26 ||
 * ** 3 ** || 2.52 || 2.52 || 1.26 ||


 * ** Table D-5: Volume and Mass of Sphere #2 ** ||
 * ** Volume (cm³) ** || ** Volume Derived from Average of Dimensions (cm³) ** || ** Mass (g) ** ||
 * 8.42 || 8.42 || 65.78 ||


 * ** Table D-6: Percent Error of Experimental Mass Density of Sphere #2 ** ||
 * ** Experimental Mass Density (g/cm³) ** || ** Accepted Mass Density (g/cm³) ** || ** Error (g/cm³) ** || ** % Error ** ||
 * 7.81 || 7.80 || 0.01 || 0.17 ||

To get the average of any measurement, take all the collected data for that measurement and divide by the total number of data.
 * Analysis:**



Average = (a1 + a2 … + an)/n Example of Averaging in Part B:



Average Length (cm) = (8.80cm + 8.75cm + 8.71cm)/3 = 8.75cm

To get the average deviation from part A, first get the deviation by taking the absolute value of the average measurement of the wanted dimension subtracted by the specific trial value for the same dimension. Then, average all the deviations of that dimension to get the average deviation.



Deviation = |E1 – a1| = b1, E1 = average of dimension 1, a1 = value of trial 1

Example of Deviation for Part A:



Deviation of Length (cm) Trial 1 = |18.15cm – 18.21cm| = 0.06cm



Average Deviation = (b1+b2 … +bn)/n = D Example of Average Deviation in Part A

 Average Deviation of Length (cm)= (0.06cm + 0.04cm + 0.14cm + 0.16cm + 0.02cm + 0.03cm + 0.03cm + 0.05cm + 0.07cm + 0.03cm)/10 = 0.06cm

To get the percent deviation, take the average deviation and divide it by the mean value of the dimension. Then, multiply by 100.



Percent Deviation = (D1/E1)*100

Example of Percent Deviation in Part A:



Percent Deviation for Length (cm) = (0.06cm/18.15cm)*100 = 0.35%

To find the experimental conversion factors, divide the average metric value by the average english value.



Experimental Conversion Factor= m/e = Y

Example of Experimental Conversion Factor in Part B:



Experimental Conversion Factor of cm3 to i n3 = (69.47cm3)/(4.14in3 )=16.79(cm3/in.3)

To find the error when comparing experimental value with accepted values, just subtract the experimental value from the expected value, and take the absolute value of the answer.



Error = |X-Y| = Z, X=accepted value, Y=experimental value

Example of Error in Part B:

<span style="font-family: 'Calibri','sans-serif'; font-size: 14.6667px;">

Error of Conversion Factors in Part B = |16.79 (cm3/in.3) - 16.39c (cm3/in.3)| = 0.40(cm3/in.3)

The error can then be used to calculate % error by dividing error by the accepted value and multiplying by 100.

<span style="font-family: 'Calibri','sans-serif'; font-size: 14.6667px;">

Percent Error= (Z/X)*100 = P

Example of Percent Error in Part B:

<span style="font-family: 'Calibri','sans-serif'; font-size: 14.6667px;">

Percent Error of Conversion Factors in Part B = ((0.40 cm3/in.3)/(16.79 cm3/in.3 ))*100 = 0.17%

During the experiment, it is also necessary to find volume and density with the formulas from the “Background:” section.

Example of Density for Part C:

<span style="font-family: 'Calibri','sans-serif'; font-size: 14.6667px;">

Density of Cylinder #1 = 22.28g/19.06cm3 = 1.17g/cm3

Example Cylinder Volume for Part C:

<span style="font-family: 'Calibri','sans-serif'; font-size: 14.6667px;">

Volume of Cylinder #1 = (π0.79cm2)9.68cm = 19.06cm3

Example of Sphere Volume for Part D:

<span style="font-family: 'Calibri','sans-serif'; font-size: 14.6667px;">

Volume of Sphere #1 = 3/4 π1.27cm3 = 8.52cm3

According to the experiment’s set of percent error values, the data is within an acceptable range because the percentages of error are low. Most of them are about 15% or lower, and the highest is only about 27%. The slight percentages of error may be due to human error such as estimating when recording measurements or misplacing/misaligning the measured object in correlation to the measuring tool. Also, the objects themselves may not have been created proportionally which may affect the reading of its dimensions.


 * Conclusion:**

According to the experiment, the hypothesis is not supported. It predicted that the percent error will fall between 5% and 10% when it actually ranged from 0% to about 27%. For example, Table A-4’s percent error is 0% while Table B-3 is 26.55%. Likewise, tables B-6, C-6, D-3, and D-6 are 2.44%, 1.09%, 16.48%, and 0.17% respectively. In fact, only data that falls into the range of the hypothesis is Table C-3 with a value of 5.71%. Also, the hypothesis doubts the appearance of a 0% error, but Table A-4 disproves it. Even so, the 0% is only due to the effects of significant figures during calculation. The percent error also raises a new question: at what point does the percent error indicate faulty data? 40%? 50%? 60%?

In redoing the experiment, it is best to have something to hold the object in place, while measuring. If not, some objects like the sphere may move around as the reader tries to measure its dimension, thus leading to inaccurate data. Also, the objects to should be more clearly labeled to distinguish the object’s material, which will affect the parts of the experiment that require the use of mass.