Assignment - Measurement

Reading: 

 

Chapter  1

 

Objectives/HW:

 

 

The student will be able to:

HW:

1

Define and identify significant digits.

1, 2

2

Use the rules of significant digits to determine the correct number of significant digits in mathematical operations.

3 - 6

3

Define accuracy and precision, and distinguish between the two.

7, 8

4

Distinguish between random and systematic error and demonstrate use of averaging to compensate for random error.

Calculate average, absolute, and relative deviation and error for a given set of measures.

9 - 11

5

Convert numbers from decimal to scientific notation and back.

Cite power of ten and abbreviation associated with metric prefixes:  nano-, micro-, milli-, centi-, kilo-, mega-, and giga-.

Use metric prefixes to express measures so that the numerical value is between 1 and 1000.

Express all results with proper scientific notation or with proper use of metric prefixes.

12, 13

6

Convert measures within the metric system, and without, given appropriate conversion factors.

14

7

Use units as a means to check calculations and solve problems.

15 - 19

8

Construct a proper graph with adequate labels and appropriate scales.  Graph a set of data and draw the best fit line or curve.

Determine the equation of the best fit for cases of linear, hyperbolic, and parabolic relations.

State and discuss the significance and/or meaning of the slope (or other coefficients) on any particular graph.

20, 21

9

Use the graph or regression equation from the graph to interpolate or extrapolate values.

22, 23

 

Homework Problems

 

1.      How does the last digit differ from the other digits in a measurement?  Why should this concept be applied to essentially all numerical values that represent some physical quantity?  Explain.

2.      Copy the following values on your paper, draw one circle around the significant digits, and in parentheses write the approximate level of implied uncertainty (e.g. ± 10 L).

a. 350 L           b. 305 L          c. 0.0350 L          d. 0.0305 L

e. 350.0 L        f. 350.0350 L          g. 3 ´ 1010 L              h. 3.00 ´ 108 L

3.      Perform the following calculations.  Copy the problem.  Round your answer to the correct number of significant digits. 

a. (360 g) ¸ (219.0 g/L)                  b. (13.0 m) ´ (1.5 m)

c. (250 m/s) ´ (0.0080 s)        d. (90.00 km) ¸ (1000 km/h)

e. (6.67 x 10-11 m3/kg s2) ´ (5.974 x 1024 kg) ¸ (6.378 x 106 m)2  

4.      Perform the following calculations.  Copy the problem.  Round your answer to the correct number of significant digits. 

a. (54.7 km) - (23.66 km)       b. (55 m) + (65 m)

c. (93 L) + (7.0 L)             d. (0.0138 L) - (0.0108 L)

e. (5.24 m/s) - (5.2 m/s)

5.      A certain right triangle has sides that measure:  45.7 mm, 53.8 mm, and 70.6 mm. 
(a) Determine the perimeter of the triangle.  (b) Determine the area of the triangle.

6.      A hole in a piece of metal has a diameter of 1.00 mm.  (a) Find the area of the hole.  (b) Find the circumference of the hole.

7.      Suppose you measure the length of an object using a certain ruler.  What will affect the precision of your measurement?  What will affect the accuracy of your measurement?  Explain.

8.      Jane Doe measured the speed of light in an experiment and reported a value of: 
2.999 x 108 m/s ± 0.006 ´ 108 m/s.  John Q. Public measured the speed of light with a different method and got:  3.001 ´ 108 m/s ± 0.001 ´ 108 m/s.  (a) Explain which is most precise.  (b) Explain which is most accurate.  You may need to look in your book.  An appendix maybe?

9.      The distance between two mountain peaks was measured repeatedly to get the following set of measures:  25.63 km, 25.32 km, 25.51 km, 25.40 km, and 25.35 km.  (a) Find the best value for this distance.  (b) Find the average absolute and relative deviations for the measurement set.  (c) On a later date the same distance measured to be 25.25 km.  State whether the distance had changed and support your answer.

10.  Planck's constant is determined by an experimenter to be 6.67 ´ 10-34 J/Hz.  Using the value in your book located in the front cover, (a) calculate the absolute error and (b) calculate the relative error.

11.  A team of physicists performs an experiment to determine the rest mass of the proton resulting in the following set of values:  1.683 ´ 10-27 kg, 1.685 ´ 10-27 kg, 1.685 ´ 10-27 kg,
1.681 ´ 10-27 kg, 1.684 ´ 10-27 kg, and 1.684 ´ 10-27 kg.  (a) Find the best value for the mass using this data.  (b) Determine the average relative deviation.  (c) Using the best value, find the absolute and relative error with respect to the value given in your book.  (d) What kind of error is apparent?  Support your answer.  (e) Write a brief statement describing the accuracy and precision of this set of values.

12.  Use metric prefixes to change the following measures so that they read numerically between 1 and 1000.  Remember - Show work.

(a) 50200 m   (b) 0.0000045 g    (c) 3.6 x 1010 m      (d) 0.030 L

(e) 4.2 x 10-7 m      (f) 2.6 x 104 g

13.  Change units of measure as indicated.  Use scientific notation when proper. 
Remember - Show work.

(a) 640 mg to g      (b) 88 Mm to km      (c) 452 kL to L      (d) 0.0800 mL to mL

(e) 0.924 Mg to kg      (f) 6265 nm to mm (g) 25.0 Gg to kg

14.  Make the following conversions. Remember - Show work.
Some useful definitions:  1 in. = 2.54 cm, 5280 ft. = 1 mi., 1 mL = 1 cm3 

(a) 10.0 miles to km      (b) 2.5 yards to m      (c) 180 hours to s

(d) 55.0 mph to m/s      (e) 350 in.3 to L      (f) 0.75 m2/s to ft2/min

(g) 1.50 L to cm3              (h) 17 m3 to L             (i) 0.64 mL to m3

15.  Solve using unit analysis.  Find the distance a bike travels in 17 s, if it is traveling at a constant velocity of 11 m/s.

16.  Solve using unit analysis.  How long would it take a car to travel 6000 m if its speed is a constant 30 km/h?

17.  Air resistance or “drag” is measured in units of newtons where one newton is equal to one kilogram-meter per seconds squared (1 N = 1 kg m/s2).  Air resistance can be calculated using a formula that involves the density (kg/m3) of the air, the relative speed (m/s) of the air, and the cross sectional area (m2) of the object.  Based on analysis of the units, what is a likely equation for calculating drag, D, in terms of density, ρ, speed, v, and area, A?

18.  Solve using unit analysis.  A falling acorn accelerates 9.80 m/s2.  How much time is required for the acorn to reach 25.0 m/s (about 55 mph)?

19.  Solve using unit analysis.  What cross sectional area (m2) must a heating duct have if air moving 3.0 m/s along it can replenish the air in a room of 350 m3 volume every 12 minutes?

DIRECTIONS FOR ALL GRAPHING PROBLEMS ALL YEAR:  All graphs must be on graph paper to receive credit.  Work the entire problem right on the graph paper -- this means everything -- calculations, equations, answers, etc.

20.  A certain volume of a particular substance was placed in a container and then massed on a balance in order to produce the data below.  Determine which variable is dependent and which is independent. 

      Mass (kg)      Volume (m3)

       35.3          0.010
       39.4          0.015
       45.0          0.020
       48.9          0.025
       56.1          0.030

(a) Graph the data.  Draw the best fit line.
(b) Determine the equation.
(c) What physical characteristics do the slope and y-intercept represent?
(d) According to the graph what volume of the substance would correlate with a mass of zero?  Does this make sense? 

21.  A student does an experiment to determine the effect of driving faster on her commute time.  She drives at different speeds and measures the time it takes to get to work.

      Speed (mph)      Time (min.)
         30.0           10.5
         40.0            7.4
         50.0            6.0
         60.0            4.9
         70.0            4.3
         80.0            3.8

(a) Graph the data and draw the curve that best fits all points.
(b) Find the equation that describes the relationship shown by the graph.
(c) How much commute time in seconds can be saved by increasing her speed from 55 mph to 65 mph?  from 65 mph to 75 mph?

22.  The total distance traveled by a moving object was measured at 1.0 second intervals and the following data were obtained:

      Time (s)      Distance (m)
        0.0               0.0
        1.0               3.2
        2.0              12.6
        3.0              30.0
        4.0              51.1
        5.0              80.1

(a) Graph the data and draw the curve that best fits all points.
(b) Find the equation that describes the relationship shown by the graph.
(c) What distance had the object traveled after 2.5 s?
(d) What amount of time did it take the object to move 100 m?
(e) Of the two previous questions which is interpolation?  Which is extrapolation?  Which is likely more accurate?  Why?

23.  An above ground swimming pool is drained by a spigot located at the bottom.  It is found that there is a relation between the speed of the water shooting out of the spigot and the depth of the water remaining in the pool.  The following data is collected as the pool drains:

      Pool Depth (m)      Water Speed (m/s)
          1.20                       4.80
          1.00                       4.53
          0.80                       3.98
          0.60                       3.43
          0.40                       2.82
          0.20                       1.90

The theories of fluid dynamics predict that the speed of the escaping water is proportional to the square root of the pool's depth.  Plotting this data on your calculator reveals that this seems to be the case -- when this data is graphed "as is" a parabolic curve is formed.  A common practice in science is to "straighten the curve" when graphing data with a nonlinear relation.  To do this, first prepare a column of values equal to the square root of the pool's depth.
(a) Now, graph water speed versus the square root of the pool's depth.
(b) Draw the line of best fit and find the corresponding equation.
(c) What are the advantages of this method of analyzing nonlinear data?

Answers to Selected Problems

1.

2. a.

    b.

    c.
    d.

    e.

    f.

    g.

    h.

3. a. 1.6 L

    b. 20 m2

    c. 2.0 m

    d. 0.0900 h (5.40 min.)

    e. 9.80 m/s2

4. a. 31.0 km

    b. 120 m

    c. 100 L

    d. 0.0030 L

    e. 0.0 m/s

5. a. 170.1 mm

    b. 1230 mm2

6. a. 0.785 mm2

    b. 3.14 mm

7.

8.  a.

     b.

9.  a. 25.44 km

     b. 0.10 km, 0.39%

     c.

10. a. 4 ´ 10-36 J/Hz

      b. 0.7%

11. a. 1.684 ´ 10-27 kg

      b. 0.06%

      c. 1.1 ´ 10-29 kg, 0.66%

      d.

      e.

12. a. 50.2 km

      b. 4.5 mg

      c. 36 Gm

      d. 30 mL

      e. 420 nm

      f. 26 kg

13. a. 0.64 g

      b. 88,000 km

      c. 4.52 ´ 105 L

      d. 80.0 mL

      e. 924 kg

      f. 6.265 mm

      g. 2.50 ´ 107 kg

14. a. 16.1 km

      b. 2.3 m

      c. 6.5 ´ 105 s

      d. 24.6 m/s

      e. 5.7 L

      f. 480 ft2/min

      g. 1500 cm3

      h. 1.7 ´ 104 L

      i. 6.4 ´ 10-7 m3

15. 190 m

16. 0.2 h

17.

18. 2.55 s

19. 0.16 m2

20. a. graph (line)

      b. M = (1020 kg/m3)V + (24.5 kg)

          where M = mass, V = volume

      c.

      d. -0.024 m3, explain

21. a. graph (hyperbola)

      b. t = (300 mph min)/v

          where t = time, v = speed

      c. 51 s, 37 s

22. a.

      b. d = (3.2 m/s2) t2

          where  d = distance, t = time

      c. 20 m

      d. 5.6 s

      e.

23. a. graph (line)

      b. v = (4.5 m˝/s) h˝ - (0.09 m/s)
          where  v = speed, h = depth

      c.