Transcript The Realm of Physics - Houston Independent School District

Objectives:
By the end of this chapter you should be able to
• appreciate the order of magnitude of various quantities;
• perform simple order-of-magnitude calculations mentally;
• state the fundamental units of the SI system.
Try to answer the following questions?
1. How many molecules are there in the sun?
2. How much is 1 Newton of force?
3. How long will it take light to cross the nucleus of the
hydrogen atom?
SI (Le Système International d’Unités)
Fundamental units
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meter (m)
kilogram (kg)
second (s)
ampere (A)
Kelvin (K)
mole (mol)
candela (cd)
 The ampere (A). unit of electric current.
It is defined as that current which, when flowing in two
parallel conductors 1 m apart, produces a force of 2 x 10-7
N on a length of 1 m of the conductors.
 The Kelvin (K). This is the unit temperature.
 The mole (mol). One mole of a substance contains as many as
there are atoms in 12 g of carbon-12. This special number of
molecules is called the avogadro’s number ans is
approximately 6.02 x 10^23.
The candela (cd). This is the unit of luminous intensity.
** The details of these definitions should not be memorized.
Any unit made up of two or more fundamental units
Example:
 m/s ------> ms-1
 m/s2 ---- > ms-2
 Newton= kg m/s2 ----> kg m s-2
 Joule (J) = Kg m = kg m s-2 m = kg m2 s-2
 Watt = Joule/s = kg m2 s-3
 Coulomb (C) - A s
Rules
1.
Zeroes between other nonzero
digits are significant
Example
a. 40.3 m ( 3 s.f.)
b. 23.005 s ( 5 s.f.)
1.
Zeroes before a nonzero digit
are not significant
a. 0.04 g (1 s.f)
b. 0.0092 s (2 s.f)
3.
Zeroes that are at the end of a
a number and also to the right
of the decimal are significant
a. 56.00 kg (4 s.f.)
b. 1.000 s (4 s.f.)
4.
Zeroes at the end of a number
but to the left of a decimal are
are significant if they have been
measured or are the first estimated
digits; otherwise they are not
significant
a. 1000 kg ( 1 s.f.)
b. 30 m (1 s.f.)
The final answer should be rounded off so that it has the same
number of decimal places (to the right of the decimal point)
as the quantity in the calculation having the least number of
decimal places.
Ex. 1. Add
Ex. 2. Subtract
4.2 cm
45.488 cm
5.76 cm
- 11.84 cm
+ 14.905 cm
_______
_______________
33.648 cm
24.865
rounded off to 24.9
rounded off to 33.65
cm
The final answer should contain the same number of
significant figure as the quantity in the calculation having
the least number of significant figure.
Ex. 1. 12.68 cm x 4.5 cm = 57.06 cm2 ≈ 57 cm2
Ex. 2. 25.8 g / 3.1 ml = 8.32258 g/ml ≈ 8.3 g/ml
OBJECTIVES
At the end of this chapter you should be able to:
1. State the various types of errors that may arise in the
measurement of a physical quantities;
2. State the different between accuracy and precision;
3. Draw a line of best fit;
4. Appreciate the importance of significant digits.
Describes the degree of exactness of a measurement
Precision depends on the instrument used to make the
measurement
The device that has the finest division on the scale produces
the most precise measurement
The precision of a measurement is one-half the smallest
division of the instrument
Describes how well the results of an experiment agree with
the standard value
 Measurements are accurate if the systematic error is small.
They are precise if the random error is small.
Accurate shot means we are close to (and hit) the target
1. Random Error




The measurements fluctuate about some value
Revealed by repeated measurement
Can be reduced by averaging over repeated
measurements
Depends in part on the skill of the experimenter
2. Systematic Error



Due to incorrect calibration of instrument
Cannot be reduced by repeated measurement
Also arise if instrument has a zero error
Systematic errors are consistent but shifted away from the
theoretical line.
Random uncertainties reveal a statistical scatter.
Mistakes or personal error are clearly inconsistent with the
rest of the data
Raw or Absolute Uncertainty : ±Δx
Ex.
Length of wire = 14.5 cm
absolute uncertainty = ±0.2 cm
Length and its uncertainty :
L ± ΔL = 14.5 cm ± 0.2 cm = (14.5 ± 0.2) cm
Lmin = 14.5 – 0.2 = 14.3 cm
Lmax = 14.5 + 0.2 = 14.7 cm

*best measurement is called absolute value of the
measure quantity. (L)
When adding experimental uncertainty to measured or
calculated values, uncertainties should be rounded to one
significant figure.
Ex. 1. ± 4 or ± 0.02 (correct)
2. ± 12.3 or ± 0.025 (wrong)

The last significant figure in any stated answer should be
of the same order of magnitude (same decimal position)
as the uncertainty.
Ex. 1. 124 ± 5 (correct);
2.05 ± 0.01 (correct)
2. 124 ± 0.5 (wrong)
2.5 ± 0.02 (wrong)
Determining the Range of
Uncertainty
1) Analogue scales
(rulers,thermometers meters with needles)
±
half of the smallest division
Since the smallest division on the
cylinder is 10 ml, the reading
would be 32 ± 5 ml
50
40
30
20
10
2) Digital scales
±
the smallest division on the readout
If the digital scale reads 5.052g, then
the uncertainty would be ± 0.001g
Absolute Uncertainty- has units of the measurement
Range of Uncertainty (cont.)
3. Significant Figures
If you are given a value without an
uncertainty, assume its uncertainty is
±1 of the last significant figure
Examples:
•The measurement is 14.742 g, the
uncertainty of the measurement is
14.742 ± .001 g
•The measurement is 50ml, the
uncertainty of the measurement is
50 ± 1 ml
 Analogue scales
Ex. Bathroom scale, thermometer, ruler
 ± half of the smallest division
 since the smallest division on the graduated
cylinder is 1 ml, the reading would be
(43.0 ± 0.5) ml
 Digital scale  Ex. Stopwatch
 Digital readouts are not scales but are displays of integers,
such as 1234, or 0.002.
 uncertainty is ± the smallest unit of measure
 If start time is 12’ 00’’ 00
stop time is 12’ 58’’ 36
T ± ΔT = 58.36 ± 0.01
Ex. Data for three trials of a toy car going down a 1-m ramp
Distance s/m
Time
t/s
Trial 1
Trial 2
Trial 3
1
1
1
1.25
1.37
1.33
Uncertainties in calculated results
1.2.10 State uncertainties as absolute, fractional and percentage uncertainties.
1.2.11 Determine the uncertainties in results.
Uncertainties in calculated results
Absolute, fractional and percentage uncertainties.
Absolute uncertainty is the raw uncertainty of your measurement.
EXAMPLE:
A student measures the length of a wire with a meter stick to be 15 mm  1
mm. What is the absolute uncertainty in his measurement?
SOLUTION:
 The  number is the absolute or raw uncertainty , ± Δx
Thus 1 mm is the absolute uncertainty.
Uncertainties in calculated results
Absolute, fractional and percentage uncertainties.
Fractional uncertainty is given by
Absolute uncertainty
Measured Value
Fractional Uncertainty =
EXAMPLE: A student measures the length of a wire with a meter stick to be
15 mm  1 mm. What is the fractional uncertainty in her measurement?
SOLUTION:
Fractional error =
1
15
= 0.07
Uncertainties in calculated results
Absolute, fractional and percentage uncertainties.
Percentage uncertainty is given by
Absolute uncertainty
Measured Value
Percentage Uncertainty =
· 100%
EXAMPLE: A student measures the length of a wire with a meter stick to be 15
mm  1 mm. What is the percentage uncertainty in his measurement?
SOLUTION:
Percentage error =
1
15
·100%
= 7%
Uncertainties in calculated results
To find the uncertainty in a sum or difference you just add the
uncertainties of all the .
In formula form we have
Sum
Difference
(A±ΔA) + (B±ΔB) = (A+B) ±(ΔA±ΔB)
(A±ΔA) - (B±ΔB) = (A+B) ±(ΔA±ΔB)
FYI
Note that whether or not the calculation has a + or a -, the uncertainties are
ADDED.
Uncertainties DO NOT EVER REDUCE ONE ANOTHER.
Mathematical Representation
of Uncertainty
Example for Addition and Subtraction:
•Determine the thickness of a pipe if the
external radius is 5.0 ± 0.1 cm and the internal
radius is 2.6 ± 0.1 cm
External radius = 5.0 ± 0.1 cm
Internal radius = 2.6 ± 0.1 cm
Thickness of pipe: 5.0 cm – 2.6 cm = 2.4 cm
Uncertainty = 0.1 cm + 0.1 cm = 0.2 cm
Thickness with uncertainty: 2.4 ± 0.2 cm OR 2.4 cm ± 8%
Mathematical Representation
of Uncertainty
Multiplication and Division:
The overall uncertainty is approximately equal to
the sum of the percentage (or fractional)
uncertainties of each quantity.
Total percentage/
fractional uncertainty
Dy = Da + Db + Dc
y
a
b
c
Fractional Uncertainties
of each quantity
Denominators
represent best
values
Mathematical Representation
of Uncertainty
Example for Multiplication and Division:
Given the following info: mass was 12 g ± 1 g and its
volume is 3.5 ± 0.2 cm3, find the density with correct
uncertainty. Density = 3.4 g cm-3
Dy = Da + Db =
1 + 0.2 = 0.08 + 0.06 = .14
y
a
b
12
3.5 ( this means 14%)
14% of 3.4 g cm-3 is 0.5 g cm-3
The result of this calculation with uncertainty is:
3.4 ± 0.5 g cm-3 or 3.4 g cm-3 ± 14%
Mathematical Representation
of Uncertainty
For exponential calculations , x2 or x3:
Just multiply the exponent by the percentage
(or fractional) uncertainty of the number.
Example:
Cube- each side is 5.0 ± 0.1 cm
Volume = (5 cm)3 = 125 cm3
Percent
= 0.1 x 100 % = 2.0%
uncertainty
5
Uncertainty
= 3 (2%) = ± 6% (or 8 cm3)
Therefore the volume is 125 ± 8 cm3