PSC 151Laboratory Activity 1

Measurement

The Importance of Measurement and Mathematics in the Physical Sciences

•Also since numbers are much less ambiguous than words, our hypotheses can be stated in a way so that they may be tested by many persons which increases our confidence in the results.

Measurement allows us to express our observations in numerical form.

•This allows others to make the same observation and compare the results. This leads to increased confidence that the observation is valid. Well-confirmed observations are called facts.

The importance of observation in science would be difficult to exaggerate. Every statement or idea in science must be checked and rechecked by observations of nature, and if the idea conflicts with the observation, the idea must yield and be modified or cast aside. No appeal to "common sense", authority, or anything else can save an idea that conflicts with observation.

•Having our observations, hypotheses, and conclusions expressed numerically allows them to be manipulated using the rules of mathematics.

1. We can replace long verbal statements with more concise mathematical expressions.

For example: Newton’s Second Law of Motion can be expressed in words as follows:“The magnitude of the acceleration of an object is directly proportional to the net force applied to the object, and inversely proportional to the object's mass. The direction of the acceleration is the same as the direction of the net force.”

The statement above can be expressed mathematically as:

r a =

r F netm

2. Once an idea is expressed in mathematical form, we can use the rules (axioms, theorems, etc.) of mathematics to change it into other statements. If the original statement is correct, and you follow the rules faithfully, your final statement will also be correct.

For example: the acceleration of an object can be expressed mathematically as:

r a =

r v f −r v it

We can multiply both sides of the equation by “t” to get:

r a t =

r v f −

r v i

r v f =

r v i +

r a t

We can add "r v i" to both sides to get :

What does it mean to measure something?Suppose you wanted to measure the length of a table.

What would be your first step?Choose a standard unit of length that is appropriate for the measurement.

Meter stickYard stickRuler, etc.

What would be your next step?Compare the meter stick with the table, i.e. place one end of the meter stick at one end of the table, mark the location of the other end, move the meter stick to begin at the mark, continue to the other end of the table, counting as you go?

Let’s choose the meter stick.

Let’s say that we placed the meter stick 5 times.

The final step is to record the measurement.

The length of the table is: 5 meters or 5m

All measurements have two parts:

A numerical part that gives the result of the comparison between the chosen unit and the thing being measured.

A unit part which tells what standard unit was used to make the measurement.

Both parts must be recorded if the measurement is to be useful.

In this activity we will examine both the units with which measurements are made and rules related to the numerical part of a measurement.

We will begin with the fundamental units from which all other units are derived.

Five Five Fundamental Properties of Nature Properties of Nature

meter (m)

kilogram (kg)

second (s)

Coulomb (C)

Ampere (A)

degree Celsius (°C) or Kelvin (K)

Property MKS (metric)

length (distance)

mass

time

electric charge

electric current

temperature

Metric PrefixesUsed to adjust the size of a unit to best fit the quantity being measured.

Prefix Symbol Multiplies by Power of 10 notation

Mega M 1,000,000 106

kilo k 1,000 103

centi c 0.01 10-2

milli m 0.001 10-3

micro 0.000001μ 10-6

Prefixes that enlarge a unit

Prefixes that reduce a unit

Derived Properties (Units depend on operational definition)

Property MKS (SI) UnitsOperational Definition

Area L ×W, s2, πr2 m×m=m2

Volume L ×W ×H,πr2h, 43πr3

m×m×m=m3

Density massvolume

kgm3

Speed distancetime

ms

Accelerationchange in speed

timemss

mss =m

s ÷s1 =m

s ×1s =m

s2

RectangleSquareCircle

RectangularSolid

Right Circular Cylinder

Sphere

Force

Property MKS (SI) UnitsOperational Definition

mass×acceleration kg×ms2 =

kg⋅ms2

Newton,N

Work force×distance kg⋅ms2 ×m=

kg⋅m2

s2

Joule,J

Practice

Kinetic energy is operationally defined as one-half

the product of an object's mass and the square of its speed.

KE =12 mass×speed2

What are the units of kinetic energy?

mass→ kg

speed2 → ms( )

2= m2

s2

KE → kg⋅ m2

s2⎛ ⎝ ⎜

⎞ ⎠ ⎟ =

kg⋅m2

s2 =Joule,J

Accuracy, Precision and Uncertainty in Measurements

Working with the Numerical Part of a Measurement

Accuracy, Precision and Uncertainty in MeasurementThere is no such thing as a perfect measurement. Each measurement contains a degree of uncertainty due to the limits of instruments and the people using them. In laboratory exercises, students are expected to follow the same procedure that scientists follow when they make measurements. Each measurement should be reported with some digits that are certain plus one digit with a value that has been estimated.

For example, if a student is reading the level of water in a graduated cylinder that has lines to mark each milliliter of water, then he or she should report the volume of the water to the tenth place (i.e. 18.5 ml.) This would show that the 18 ml are certain and the student estimated the final digit because the water level was about half way between the 18 and 19 mark.

Two concepts that have to do with measurements are accuracy and precision.

The accuracy of the measurement refers to how close the measured value is to the true or accepted value. For example, if you used a balance to find the mass of a known standard 100.00 g mass, and you got a reading of 78.55 g, your measurement would not be very accurate. One important distinction between accuracy and precision is that accuracy can be determined by only one measurement, while precision can only be determined with multiple measurements.

Precision refers to how close together a group of measurements actually are to each other. Precision has nothing to do with the true or accepted value of a measurement, so it is quite possible to be very precise and totally inaccurate. In many cases, when precision is high and accuracy is low, the fault can lie with the instrument. If a balance or a thermometer is not working correctly, they might consistently give inaccurate answers, resulting in high precision and low accuracy.

A dartboard analogy can demonstrate the difference between accuracy and precision. Imagine a person throwing darts, trying to hit the bullseye. The closer the dart hits to the bullseye, the more accurate his or her tosses are. If the person misses the dartboard with every throw, but all of their shots land close together, they can still be very precise.

Students must strive for both accuracy and precision in all of their laboratory activities. Make sure that you understand the workings of each instrument, take each measurement carefully, and recheck to make sure that you have precision. Without accurate and precise measurement your calculations, even if done correctly, are quite useless.

x

xx

x

x

Low AccuracyLow Precision

xx x

xx

Low AccuracyHigh Precision

x

x

xx x

High AccuracyLow Precision

xxxxx

High AccuracyHigh Precision

Significant Figures

Values read from the measuring instrument are expressed with numbers known as significant figures.

For each measurement made it is important to consider significant figures and to keep in mind the uncertainties involved in measurement. When scientists report the results of their measurements it is important that they also communicate how 'close' those measurements are likely to be. This helps others to duplicate the experiment, but also shows how much 'room for error' there was.

Significant figures are digits read from the measuring instrument plus one doubtful digit estimated by the observer.

Example: a measurement of 1.2374 meters was made with a meter stick having millimeter marks. The figures 1, 2, 3, and 7 are certain while the 4 is estimated.

10cm20cm30cm40cm50cm60cm70cm80cm90cm 10cm20cm30cm

23cm22cm 24cm 25cm

23.74cm

10cm20cm30cm40cm50cm60cm70cm80cm90cm 10cm20cm30cm

123.74cm=1.2374m

When you put a sample of a liquid into a graduated cylinder you will notice a curve at the surface of the liquid. This curve, which is called a meniscus, may be concave (curved downward) or convex (curved upward). Hold the graduated cylinder so that the meniscus is at eye level. Read the volume of the liquid at the bottom of a concave meniscus or at the top of a convex meniscus.

Measuring Liquids

graduated cylinder

722mL700

600

500

800mL

Rules For Significant Digits

Digits from 1-9 are always significant.

Zeros between two other significant digits are always significant.

One or more additional zeros to the right of both the decimal place and another significant digit are significant.

Zeros used solely for spacing the decimal point (placeholders) are not significant.

Rules For Rounding

Terminal zeros in a whole number may or may not be significant.

Decide how many significant figures will be kept; look at the first digit to be rejected.If it is less than five (5) drop all rejected digits

If it is greater than five (5) drop all rejected digits and increase the last retained significant digit by one (1).

If it is equal to five (5) drop all rejected digits and increase the last retained significant digit by one (1) only to make it even.

Examples

How many significant digits are in each of the following measurements?

1.75m 302.00g 5.0003s

2000kg 0.00053cm

3 5 5

1 2 1.00053cm 6

Round each of the following measurements to the indicated number of significant digits:

3.08546cm (3) 3.09cm

20304kg (2) 20000kg

0.00000515s (2) 0.0000052s

10.0000m (3) 10.0m

The first zero (0) after the 2 is significant, the other zeros are not significant but only used to place the decimal point.

Measuring Uncertainty in Measurements

When an accepted or standard value of the physical quantity is known, the percent error is calculated to compare an experimental measurement with the standard.

%error = experimental value− accepted value accepted value

×100%

When no standard exists, or when it is desired to measure the precision of an experiment, percent difference is calculated. Percent difference measures how much two or more measurements of the same quantity differ from each other.

%difference = largest value−smallest value average value ×100%

Percent error is a measure of accuracy.

Examples:

An experiment designed to measure the density of copper obtained a value of 8.37g/cm3. If the value listed in a reference table was 8.92g/cm3, what was the percent error in the experimental value?

%error = experimental value− accepted value accepted value

×100%

%error = 8.37g/cm3 −8.92g/cm3

8.92g/cm3 ×100%

%error =0.062 ×100%

%error =6.2%

Note: % error is usually rounded to one or two significant digits.

Five student in a lab group each measured the length of a wooden block. The results are listed below.

Student #1- 25.32cmStudent #2- 25.15cmStudent #3- 25.44cmStudent #4- 24.89cmStudent #5- 25.71cm

What is the % difference of their measurements?

%difference = largest value− smallest value average value ×100%

%difference = 25.71cm−24.89cm25.30cm ×100%

%difference =0.032 ×100%%difference =3.2%

The group should use the average value of 25.30cm in future calculations.

Performing Arithmetic Operations with Measurements

When multiplying or dividing, your answer may only show as many significant digits as the multiplied or divided measurement showing the least number of significant digits.

Example: 22.37 cm x 3.10 cm x 85.75 cm = 5946.50525 cm3

22.37cm has 4 significant digits. 3.10cm has 3 significant digits.85.75cm has 4 significant digits.

The product answer can only show 3 significant digits because that is the least number of significant digits in the original problem.

22.37 cm x 3.10 cm x 85.75 cm = 5950 cm3

When adding or subtracting your answer can only show as many decimal places as the measurement having the fewest number of decimal places.

Example: 3.76 g + 14.83 g + 2.1 g = 20.69 g

3.76g and 14.83g each have two decimal places2.1g has only one decimal place

The sum can only have one (1) decimal place.

3.76 g + 14.83 g + 2.1 g = 20.7 g

Example:

The length and width of a rectangle are measured to be 12.7cm and 8.48cm respectively.

What is the perimeter of the rectangle?

P =2×L +2×W

P =2(12.7cm) + 2(8.48cm)

P =42.36cm

P =42.4cm

Rounding to one decimal place:

What is the area of the rectangle?

A =L ×W

A =(12.7cm)×(8.48cm)

A =107.696cm2

A =108cm2

Rounding to three (3) significant digits: