Measurement and Problem Solving

1
Measurement and Problem Solving

• Holt Physics, pages 3-45

2
Prerequisite Objectives

• Associate the units used for the measurement of
  mass, length and time with the instruments used
  for the measurements of these quantities.
• Mass kilogram triple-beam balance
• Length meter meterstick
• Time second stopwatch

3
Prerequisite Objectives, continued

• Use correctly the instruments for the measurement
  of mass, length and time in laboratory
  situations.
• See labs

4
Prerequisite Objectives, continued

• Use the SI (MKS) system of measurement.
• SI International System of Units (Système
  International dUnites French name)
• MKS meter, kilogram, second
• Established in 1800 in France
• In widespread use by scientists everywhere

5
Prerequisite Objectives, continued

• Units are defined in terms of a standard.
• Originally, meter was defined as 10-7 times the
  distance from the North pole to the equator at
  the prime meridian.
• In 1982, a more precise length measurement
  defined the meter as the distance light travels
  in 3.335 640 95 x 10-9 second.

6
Prerequisite Objectives, continued

• The kilogram is the only unit not defined in
  terms of properties of atoms. It is the mass of a
  platinum-iridium metal cylinder kept in Sèvres,
  France.
• In 1967, the second was redefined as the duration
  of 9 192 631 770 periods of certain radiation
  emitted by an atom of cesium-133.

7
Prerequisite Objectives, continued

• Express the numbers correctly in terms of
  scientific notation and use these values
  effectively in calculations involving the usual
  arithmetic operations.

8
Prerequisite Objectives, continued

• Scientific notation is based on exponential
  notation. In scientific notation, the numerical
  part of a measurement is expressed as a number
  between 1 and 10 multiplied by a whole-number
  power of 10.
• To write measurements using scientific notation,
  move the decimal point until only one non-zero
  digit remains on the left. Then count the number
  of places the decimal point was moved and use
  that number as the exponent of ten.
• Ex. 1.04 x 103 m 2.70 x 10-4 kg

9

• Which of the following is the number 1274000 in
  proper scientific notation?
• A. 12.74 105
• B. 1.274 103
• C. 1.274 106
• D. 1.274 10-6

10
Prerequisite Objectives, continued

• To add or subtract in this system, all numbers
  must be changed so that the multiplier is the
  same power of ten. The numerical parts of the
  measurements are then added or subtracted, and
  the sum or difference is multiplied by the
  exponent of the original measurements.

11
Prerequisite Objectives, continued

• To multiply in this system, multiply the
  numerical parts and add the exponents
  algebraically.
• To divide in this system, divide the numerical
  parts and subtract the exponents.
• If the sum, difference, product, or quotient of
  the numerical parts is smaller than one or larger
  than ten, the accompanying exponents should be
  adjusted to bring the portion of the answer
  containing the numerical part within the range of
  one to ten.

12
Prerequisite Objectives, continued

• Use significant figures effectively in
  calculations involving the usual arithmetic
  operations.
• Significant figures are those digits in a
  measurement that are known with certainty plus
  the first digit that is uncertain.

13
Rules for Determining Whether Zeros are
Significant Figures

• Zeros between other nonzero digits are
  significant.
• Ex. 120034 6 significant figures
• Zeros in front of nonzero digits are not
  significant.
• Ex. 0.01234 4 significant figures

14
Rules, continued

• Zeros that are at the end of a number and also to
  the right of the decimal are significant.
• Ex. 1234.00 6 significant figures
• Zeros at the beginning of a number but to the
  left of a decimal are significant if they have
  been measured or are the first estimated digit
  otherwise they are not significant.
• Ex.123400 4 significant figures

15

• How many significant digits are in the following
  10203000
• A. 3
• B. 5
• C. 6
• D. 8

16

• How many significant digits are in the following
  102.0300
• A. 3
• B. 4
• C. 5
• D. 7

17

• How many significant digits are in the following
  0.0020340
• A. 3
• B. 4
• C. 5
• D. 7

18
Prerequisite Objectives, continued

• One way of indicating the precision of a
  measurement is by means of significant figures.
• Significant figures should be used throughout
  the course in laboratory investigations.

19
Operations With Significant Figures

• To add or subtract measurements, first perform
  the operation, then round off the result to the
  correspond to the least precise value involved.
• Ex. 123.567 thousandths
• 78.9 tenths
• 63.25 hundredths
• 372.644 thousandths
• 638.361

Answer 638.4 tenths
20
Operations, continued

• When multiplying or dividing, first perform the
  calculation. Then note the factor with the least
  number of significant figures and round the
  product or quotient to this number of figures.
• Ex. 10.6 3 significant digits
  12.34 4 significant digits
  130.804

Answer 131 3 significant digits
21

• Using significant digits, what is total mass of
  system which contains a 145.02 g toy car, 0.3 g
  piece of string, and paperclips with a total mass
  of 4.622g?
• A. 149.942 g
• B. 149.94 g
• C. 149.9 g
• D. 150 g
• E. 200 g

22

• Using significant digits, what is the area of a
  room with dimensions of 6.23 m by 10.025 m?
• A. 62. 45575 m2
• B. 62.457 m2
• C. 62.46 m2
• D. 62.5 m2

23
Operations, continued

• Note Textbook answers will always show only the
  number of significant figures that the
  measurements justify.
• The rules for rounding are in Table 1-6, page 19,
  in Holt Physics.

24
Prerequisite Objectives, continued

• Transpose a simple equation quickly and
  efficiently.

Multiply by s ays cbx
25
Objective 1

• Distinguish physics from other areas of science.
• Physics is a science which describes and explains
  the interaction of matter and energy.
• Physicists gather information and organize it in
  words or in mathematical symbols.
• Some of the major areas of physics are
  mechanics, thermodynamics, vibrations and wave
  phenomena, optics, electromagnetism, relativity
  and quantum mechanics.

26
Questions

• How many significant figures?
• 2804 m
• 0.0029 m
• 0.007060 kg
• Solve, giving the answer with the correct number
  of significant figures.
• 6.201 cm 7.4 cm 0.68 cm 12.0 cm
• 3.2145 km X 4.23km

(4)
(2)
(4)
26.3 cm
13.6 km2
27
Objective 2

• Distinguish between base and derived units.
• A base unit is one that is not defined in terms
  of other units.
• The seven base physical quantities and their
  units are length (meter), mass (kilogram), time
  (second), temperature (Kelvin), luminous
  intensity (candela), electric current (Ampere)
  and molecular quantity (mole).

28
Objective 2, continued

• A derived unit is one that is defined in terms of
  base units.
• For example, the SI unit for area is the square
  meter, and
• The SI unit for velocity is the meter per second.
  
• These are derived units.

29
Objective 3

• Recognize that all measured quantities have
  uncertainties.
• All measurements are in error to some degree.

30
Errors in Measurements

• Human error mistakes made in reading an
  instrument, or recording the results.
• To avoid, take repeated measurements to be
  certain they are consistent.
• Measurements must be made by looking at the
  device straight on.
• If they are not read straight on, an error due to
  parallax is possible.
• Parallax is the apparent shift in position of an
  object as its viewed from different angles.

31
Errors, continued

• Method error measurements taken by different
  methods.
• To avoid this error, you should standardize the
  method of taking measurements.
• Instrument error equipment not in good working
  order.
• It is important to be careful with equipment.

32
Errors, continued

• External error some equipment changes due to
  external causes.
• Ex. length of a ruler changes with changes in
  temperature electric measuring devices are
  affected by magnetic fields near them.

33
Objective 4

• Distinguish between accuracy and precision.
• The uncertainty of a measurement can be expressed
  in terms of accuracy or precision.

34
Accuracy

• Accuracy of a measuring device depends upon how
  well the value obtained by using the instrument
  agrees with the accepted value.
• When a measurement to be made, the measuring
  device should first be checked for accuracy.
• This can be done by using the instrument to
  measure quantities whose values are known.
• The measured values are then compared to the
  known values.
• This is known as calibrating the instrument.

35
Precision

• Precision is the degree of exactness with which
  the measurement is made or stated.
• The precision of a measuring instrument is
  limited by the smallest division on its scale.
• Errors in measurements affect the accuracy of a
  measurement.
• But the precision is not affected since values
  are still stated in terms of the smallest
  division on the instrument.

36

• What was the precision of the instrument which
  made the following measurement 3.024 g?
• A. 10 g
• B. 1 g
• C. 0.1 g
• D. 0.01 g
• E. 0.001 g

37

• If your lab partner is using a cylinder that is
  marked to the nearest ten ml and tells you that
  there is 100 ml of water in the graduated
  cylinder. How should this amount be recorded on
  your lab sheet?
• A. 100.00
• B. 100.0
• C. 100
• D. 10.0 101
• E. 1.00 102

38
Relative Error

• The accuracy of measurements can be determined by
  comparing your results with the accepted value.
• The percentage error, or relative error, of a
  measured value can be found with the following
  equation

39
Objective 5

• Demonstrate an understanding of the use of
  significant figures as a means of stating the
  precision of measured quantities.
• One way of indicating the precision of a
  measurement is by means of significant figures.
• Significant figures should be used throughout
  the course in laboratory investigations.

40
Objective 6

• Use significant figures in measurements and
  calculations.
• See prerequisite objective 5.

41
Investigation Precision and Significant Figures

• Purpose
• The purpose of this investigation is to learn how
  the number of significant figures in a
  measurement is related to the precision of the
  measurement.
• Equipment
• four-scale meterstick
• magnifying lens
• graph paper

42
Uncertainty in Measurement

• Suppose the measurement, using the decimeter
  scale of the meterstick, is 0.23 m. To construct
  a bar graph of the uncertainty in the measurement
  extend a bar graph to the 0.2 m mark as in the
  figure to the right. Darkly shade this region and
  label it certain since we are certain that the
  length is at least 0.2 m.

Certain
43
Uncertainty in Measurement, continued
Now extend the bar graph up to the 0.3 m mark.
Lightly shade this region and label it doubtful
since we only know that the length is at least
0.2 m and not more than 0.3 m.
0.3
Doubtful
0.2
Length in meters
Certain
0.1
Now place a line across at 0.23 m.
Decimeter
44
Questions

• What determines the precision of a measurement?
• Explain how a measurement can be precise but not
  accurate.
• How does the last digit differ from other digits
  in a measurement?
• Your lab partner recorded a measurement as 100
  ml.
• What precision has he / she indicated is marked
  on this cylinder?

45
Investigation Precision and Accuracy in
Measurement

• Purposes
• to make precise linear measurements using the
  meterstick and to use these measurements to
  compute quantities derived from length and
• to distinguish between precision and accuracy by
  determining the mass densities of some liquids
  and solids and comparing these values with
  accepted values.

46
Densities of Metals
To the right are the accepted values for the
densities of several metals.
47
Objective 7

• Distinguish between dependent and independent
  variables determine the effect of changing the
  independent variable.

48
Independent Variable

• The independent or manipulated variable is any
  quantity that an experimenter can change at will.
  
• It is carefully varied during the experiment.
• It is placed in the first column of the data
  table.
• It is plotted horizontally on the x-axis, or
  abscissa. Remember This is the MIX of DRY
  MIX Manipulated, Independent on the X-axis.

49
Dependent Variable

• The dependent or responding variable is any
  quantity that changes in response to the
  experimenters manipulation of one or more
  independent variables.
• It is measured for each variation of the
  independent variable.
• It is placed in the second column of the data
  table.
• It is plotted vertically on the y-axis, or
  ordinate. Remember This is the DRY of DRY MIX
  Responding, Dependent on the Y-axis.

50
Changing the Independent Variable
51

• In the equation I (V/R), where I is the
  dependent variable, but V and R are independent
  variables. How are I and V related?
• A. Directly
• B. Inversely
• C. They are not related to each other.

52

• In the equation I (V/R), where I is the
  dependent variable, but V and R are independent
  variables. How are I and R related?
• A. Directly
• B. Inversely
• C. They are not related to each other.

53

• In the equation I (V/R), where I is the
  dependent variable, but V and R are independent
  variables. How are V and R related?
• A. Directly
• B. Inversely
• C. They are not related to each other.

54

• In the equation I (V/R), where I is the
  dependent variable, but V and R are independent
  variables. If V halves, what happens to I?
• A. Doubles
• B. Halves
• C. Does not change.

55

• In the equation I (V/R), where I is the
  dependent variable, but V and R are independent
  variables. If R halves, what happens to I?
• A. Doubles
• B. Halves
• C. Does not change.

56

• In the equation I (V/R), where I is the
  dependent variable, but V and R are independent
  variables. If V halves, what happens to R?
• A. Doubles
• B. Halves
• C. Does not change.

57
Objective 8

• Interpret data in tables and graphs, and
  recognize equations that summarize data.

58
Data in Tables

• The purpose of a scientific investigation is to
  discover relationships that may exist among the
  quantities measured.
• Physicists make their work easier by summarizing
  data in tables and graphs.
• Tables organize data and, usually, a clear trend
  can be seen in the data.

59
Graphs

• A graph is a pictorial display of data.
• The shape of a graph may frequently reveal a
  relationship that is not apparent from a quick
  look at the data.
• Because a graph usually shows an obvious pattern,
  a smooth curve is drawn through the data points
  to make estimations for points without data.

60
Direct Proportion
61
Quadratic Relationships
62
Inverse Proportion
Note that as i increases, d decreases much more
rapidly in an inverse square relationship than it
does in the simple inverse relationship.
63
Sample Problem

• The data to the right was gathered in a
  laboratory using various batteries connected to a
  resistor and measuring the current flowing
  through the resistor.

Current vs. Voltage
64
Sample Problem, continued

• Which quantity should be plotted on the x-axis?
• Which quantity should be plotted on the y-axis?
• Sketch the graph of this data. Be sure to label
  each axis and include units.
• What type of function is shown by the graph?
• What is the slope of this graph?
• What are the units of the slope of this function?

65

• What is the relationship between X and Y?
• Directly related
• Inversely related
• They are not related

66

• What is the relationship between X and Y?
• Linear
• Parabolic
• Hyperbolic

67
Objective 9

• Practice appropriate methods (graphical and
  algebraic) for the solution of problems growing
  out of laboratory exercises or problem
  assignments.

68
Problem Solving

• Read the problem carefully and make sure that you
  know what is being asked and understand all the
  terms and symbols that are used in the problem.
  Write down all given data.
• Write down the symbol for the physical quantity
  or quantities called for in the problem, together
  with the appropriate unit.

69
Problem Solving, continued

• Write down the equation relating the known and
  unknown quantities of the problem. This is called
  the basic equation. In this step you will have to
  draw upon your understanding of the physical
  principle involved in the problem. It is helpful
  to draw a sketch of the problem and to label it
  with the given data.

70
Problem Solving, continued

• Solve the basic equation for the unknown quantity
  in the problem, expressing this quantity in terms
  of those given in the problem. This is called the
  working equation.
• Substitute the given data into the working
  equation. Be sure to use the proper units.

71
Problem Solving, continued

• Perform the indicated mathematical operations
  with the units alone to make sure that the answer
  will be in the units called for in the problem.
  This process is called dimensional analysis.
• Estimate the order of magnitude of the answer.
• Perform the indicated mathematical operations
  with the numbers.
• Review the entire problem and compare the answer
  with your estimate.

72
Grading Word Problems

• Write down given data. (2 points)
• Write down basic equation. (1 point)
• Substitute the given data into the equation. (1
  point)
• Perform dimensional analysis. (2 points)
• Write the correct answer with units. (2 points)

73
Sample Problem

• According to the laws of planetary motion,
  planetary orbits can be described by the
• following equation , where k is a
• constant that is the same for all the suns
  planets, p is the period of revolution (in earth
  years) for a specific planet, and d is its
  average distance (in km) from the sun. The
  average distance between the sun and the earth is
  1.496 x 108 km. Find the value of k for the solar
  system.

• 2.987 x 10-25 earth year2/km3

74
Sample Problem

• Using the value of k found in the previous
  problem, find the average distance between the
  sun and Jupiter, if the period of revolution is
  equal to 11.86 earth years.

• 7.780 x 108 km

75
Sample Problem

• Jupiter has a period (time for one revolution
  around the sun) of 11.86 years. If its mean
  radius of orbit is 7.78 x 1011 m, calculate the
  average speed, in km/h, of Jupiter around the
  sun.

• 4.702 x 104 km/h

76
Sample Problem

• A farmer must move six bales of hay, each of
  which has a mass of 50 kg, from the floor of the
  barn to the hayloft, which is 5 m above the
  floor. How much work must the farmer do to move
  all six bales?
• Formulas F mg W Fd
• where F is the force, in Newtons (1 kg-m/s2)
• g is the acceleration due to gravity (9.8 m/s2)
• m is the mass in kilograms
• d is the distance in meters
• W is the work in Jules (1 Nm 1 J)

• 1.47 x 104 J

77
Sample Problem

• Each cubic centimeter of aluminum has a mass of
  2.6 grams. What is the mass of a cube that
  measures 2 cm along the edge? If the cube is then
  measured to have a mass of 13 g, what volume of
  the cube is hollow?

• 7.8 g

78
Objective 10

• Use dimensional analysis to check the validity of
  an equation.
• Units must be uniform when solving a problem.
• When writing an equation in physics, you must
  state the units as well as the numerical values.
• This helps you to keep consistent units in the
  equation.
• This also tells you if the equation is
  dimensionally correct (dimensional analysis).

79
Dimensional Analysis

• mass density ? volume

• velocity distance ? time

• kg kg
• Correct!

Incorrect!
80
Sample Problem

• In an alternative reality, scientists have found
  that
• where F is measured in chirps, q is measured in
  wogs, and r is measured in nerds. Using
  dimensional analysis, determine the units for k.

• (chirpsnerds2)/wogs2

81
Question
The mathematical relationship between
gravitational force (F) and distance (d) can be
written as

• Which graphing procedure would indicate this
  relationship by yielding a straight line graph?
• plotting F against d2 b.. plotting F against 1/d2
• plotting F against d d. plotting F
  against 1/d
• plotting d against F

Answer b
82
Question

• Which of the following equations cannot be
  correct because of dimensional inconsistencies?
  In each case d is distance (in m), v is velocity
  (in m/s), a is acceleration (in m/s2), and t is
  time (in s)?

?
A
?
B
C
?
?
D