Physics Lab 1: Measurement and Error Analysis Section: Name:

1

Objectives Measurement is a fundamental skill for all the physics experiments. How to acquire data through measurement, and how to represent them in a meaningful way are crucial to the success and validity of physics experiments. In this lab, we are going to measure the distance of an inaccessible point, and analyze results and errors. The following are the learning objectives:

• SI Unit: Be familiar with the SI unit of length, its prefix and conversion.

• Length Measurement: Use ruler to measure distance.

• Angle Measurement: Use protractor to measure angles.

• Significant Figures: Determine the significant figures, and their meaning in measurement.

• Percent Error: Learn the formula of percent error, and its meaning in measurement.

• Scientific Notation: Represent all the measured values in scientific notion, and be fluent in their operations.

• Geometry/Trigonometry: Use geometric and/or trigonometric formulas/rules to solve real-world problems.

• Scale of Map: Learn to find and use the scale of a map to calculate the real distance.

• Error Analysis: Identify possible sources of errors in an experiment.

Problem: Measuring the distance of an inaccessible point Measuring the distance of an inaccessible point has been a classical problem since ancient time. For example, the distance to the top of a mountain and the distance of a remote star are both hard to be measured directly. So, being able to perform the measurement indirectly, and calculate the estimated distance through geometry and trigonometry is the key to solve these practical problems in engineering and science.

In this lab, we will use a “shrunk” model (i.e., map) of the real-world environment, and use a simple self-made survey equipment to solve a real-world measurement problem. Figure 1 shows a scaled-down picture of the map you will be given to work on during the lab. Point A is your current location on the South Rim of the Grand Canyon. Point B is a nearby tower on the same side of the Canyon. The distance between Point A and Point B is known (2,500.00 m). Point X is an inaccessible landmark across the Canyon. All three points have the same altitude. Since you cannot cross the Canyon to conduct direct measurement in real world, you are NOT ALLOWED to measure the distance between Point A and point X directly using a ruler. However, you are free to use any locations on the South Rim to build a simple structure of the same altitude for measurement. The goal of this lab is to find the distance between Point A and Point X.

Work with your group partners to brainstorm, research the possible measurement methods, and design a lab procedure that can help you find the distance. The available materials, equipment, and resource are listed in the following sessions. Have your solution(s) ready before coming to the lab such that you will have enough time to conduct the measurement, perform the calculation, and analyze the results/errors during the lab time. Materials

A computer A ruler Color pencils Cardboards Map worksheets A protractor Pushpins

Geometry/Trigonometry Similar Triangles:

∆ABC ~ ∆DEF, then !"!"= !"

!"= !"

!"

Trigonometric Ratios: ∆ABC

sin 𝜗 = !"!"

, cos 𝜗 = !"!", tan 𝜗 = !"

!"

A

B C

D

E F

A

B C θ

Figure 1: A scaled-down map

Physics Lab 1: Measurement and Error Analysis Section: Name:

2

Law od Sine & Law of Cosin:

𝒄𝒔𝒊𝒏  𝑪

= 𝒃𝒔𝒊𝒏  𝑩

= 𝒂𝒔𝒊𝒏  𝑨

𝒂𝟐 =  𝒃𝟐 +  𝒄𝟐 − 𝟐𝒃𝒄  𝒄𝒐𝒔  𝑨

𝒃𝟐 =  𝒂𝟐 +  𝒄𝟐 − 𝟐𝒂𝒄 𝐜𝐨𝐬𝑩

𝒄𝟐 =  𝒂𝟐 +  𝒃𝟐 − 𝟐𝒂𝒃  𝒄𝒐𝒔  𝑪 Measurement Tools Ruler: An online video tutorial for using a metric ruler to read out measurement with significant figures can be found in the following link: https://www.youtube.com/watch?v=7MzuinoHJg8.

Protractor: An online video tutorial for using a protractor can be found in the following link: https://www.youtube.com/watch?v=t4xCOUNEInI

Pushpins: A simple measurement tool made of pushpins is shown in Figure 2. When you arrange the locations of three pushpins in a way that you can see them overlapping each other using only one eye, these three points are collinear. Accuracy and Precision Accuracy: The accuracy is a measure of the degree of closeness of a measured or calculated value to its actual or accepted value. Accuracy is often reported quantitatively by using percent error. 𝑝𝑒𝑟𝑐𝑒𝑛𝑡  𝑒𝑟𝑟𝑜𝑟 =   |!"#$%&"'  !"#$%!!""#$%#&  !"#$%|

!""#$%#&  !"#$%  ×  100%

Precision: Precision is a measure of how well a result can be determined (without reference to a theoretical or actual value). It is the degree of consistency and agreement among independent measurements of the same quantity. In many cases engineers and scientists choose to use an implied precision via significant digits. Significant digits carry with them an implied precision of ± ½ unit in the rightmost significant digit. For example,

3280 ± 5 220. ± 0.5 6.47 ± 0.005 0.190 ± 0.0005

Unit Conversion The SI unit for length is meter (m). However, it is probably that all your measurements in this lab will be in centimeters (cm). 1 cm = 1 x 10 – 2 m. You should convert all your measurements into meters (m). For example,

2.6 cm = 2.6 x 10 – 2 m

16.34 cm = 16.34 x 10 – 2 m = 1.634 x 10 – 1 m

Errors It is useful to think of measurement errors in two categories: system errors and random errors.

Systematic Errors: Systematic errors are those differences between an observation and the true value that are consistent from one observation to the next. They are most closely associated with inaccuracy.

Random Errors: Random errors are more difficult to characterize. They are unpredictable and change from one observation to another. Common sources of random errors include:

• measurement read by different people,

• inherent randomness in the instrumentation (usually electronic components),

• uncontrolled and unobserved external influences on the measurement, and

• random differences in the quantity being measured.

A

B C a

b c

Figure 2: Pushpin measurement tool

Physics Lab 1: Measurement and Error Analysis Section: Name:

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Data & Analysis Collect data from at least six different measurements performed by all the members of the group, calculate the distance between Point A and Point X, calculate the percent errors, and summarize them in the following table. The actual distance between point A and Point X (accepted value) is 9,810.51 m. The number of columns and the contents of the table depend on the measurement method you have chosen in the lab. All the measured values should be recorded in proper significant figures, represented in scientific notation, and with correct SI unit for length.

No Map Scale Distance between A & X (m) Percent error (%) Conducted by (initial)

1

2

3

4

5

6

Average

Table 1: Data table of the measurement of an inaccessible point

Discussion Questions 1. Which method do you choose to solve this measurement problem? Explain how it works.

2. Describe the step-by-step procedure you use to conduct this experiment.

3. Show step-by-step, complete calculations of all the data in one row of your data table including unit conversions, map scale, geometric/trigonometric formula(s), and percent error. You should attach proper unit to each value in your calculation.

4. How do you determine the number of significant figures in your data? What is the meaning of “significant figure” in measurement?

5. How can you increase the number of significant figures in your measurement?

6. What is the meaning of “percent error” in measurement?

7. What are the possible sources of errors in this experiment? List and explain them in details.

8. How can you reduce the percent error in your measurement?