This document covers several topics related to measurement and error analysis in physics experiments. It begins by defining accuracy and precision, distinguishing between the two concepts. Accuracy refers to how close a measurement is to the true value, while precision describes the degree of variation in repeated measurements of the same quantity. It then discusses random and systematic errors, explaining that random errors vary unpredictably while systematic errors remain constant. The document provides examples of different types of systematic errors like instrumental, environmental, and observational errors. Finally, it introduces concepts like absolute error, relative error, and percentage error to quantify the uncertainty in measurements.

2. 1. The effect of instruments on
measurements
2. Uncertainties and deviations in
measurement
3. Sources and types of error
4. Vectors and vector addition
1. Position, time, distance, displacement,
speed, average velocity, instantaneous
velocity
2. Average acceleration, and instantaneous
acceleration
3. Uniformly accelerated linear motion
4. Free-fall motion
5. 1D Uniform Acceleration Problems

3. Relative motion
1. Position, distance, displacement,
speed, average velocity,
instantaneous velocity, average
acceleration, and instantaneous
acceleration in 2- and 3-
dimensions
2. Projectile Motion 3. Circular
Motion
1. Newton’s Law’s of Motion
2. Inertial Reference Frames
3. Action at a distance forces
4. Types of contact forces: tension, normal
force, kinetic and static friction, fluid resistance
5. Action -Reaction Pairs
6. Free -Body Diagrams

4. 7. Applications of Newton’s Laws to
single -body and multibody
dynamics
8. Problem solving using Newton’s
Laws
1. Dot or Scalar Product
2. Work done by a force
3. Work -energy relation
4. Kinetic energy
5. Power
6. Conservative and non-conservative forces
7. Gravitational potential energy
8. Elastic potential energy
9. Equilibria and potential energy diagrams
10. Energy Conservation, Work, and Power
Problems

5. 1. Center of mass
2. Momentum
3. Impulse
4. Impulse -momentum relation
5. Law of conservation of momentum
6. Collisions
7. Center of Mass, Impulse, Momentum,
and Collision Problems

6. ✔ Physical Quantities
✔ Conversion of Units
✔ Scientific Notation
Module 1: Lesson 1
Units of Measurements

7. 1. define physical quantity;
2. differentiate fundamental and derived
quantity;
3. differentiate metric and English system of
measurement;
4. convert units of measurement;
5. express number in scientific notation; and
6. solve measurement problems involving
conversion of units and expression in
scientific notation
Students should be able to:

8. ✔ Physical Quantities

9. Physicists, like other scientists, make
observations and ask basic questions.
How big is an
object?
How much mass
does it have?
How far did it
travel?
They make
measurements with
various instruments
(e.g., meter stick,
balance, stopwatch,
etc.).

10. Physical Quantities
All physical quantities in the International System of
Units (SI) are expressed in terms of combinations of
seven fundamental physical units, which are units for:
length, mass, time, electric current, temperature,
amount of a substance, and luminous intensity.

11. SI units
There are two major systems of units
used in the world:
acronym for the French Le
Systeme International d’ Unites,
also known as the metric
system),
English units
(also known as the
imperial system).
United States is the only
country that still uses
English units extensively
Virtually every other country in the world now uses
the metric system, which is the standard system
agreed upon by scientists and mathematicians.

12. In physics, there are seven fundamental physical
quantities that are measured in base or physical
fundamental units:
Length
Mass
Time
Electric current
Temperature
Amount of substance
Luminous intensity
meter
kilogram
seconds
Kelvin
Ampere
Candela
mole
m
K
A
s
kg
Cd
Mol

15. British Sytem:
1 inch = 2.54 cm
1 lb = 4.448221615260 N

16. ✔Conversion of Units

17. Unit Conversion and
Dimensional Analysis
A conversion factor
relating meters to
kilometers.
A conversion factor is
simply a fraction which
equals 1.
A conversion factor is a
ratio expressing how
many of one unit are
equal to another unit

18. 150 cm to m = ?
20 ft to m = ?
1ft = 12 in.
1 in. = 2.54 cm.
1 m = 100 cm

19. b. 67.21
Convert 6 721 millimeters
to meters
c. 672 100
d. 6 721 000
a. 6.721

20.  Scientific Notation

21. Scientific notation is a way of writing numbers that are too large
or small to be conveniently written as a decimal.
Scientific notation follows this general format
X x 10 y

22. X x 10 y
x is the value of the measurement with all placeholder zeros
removed
10y is the factor

23. Consider the number
840,000,000,000,000
8.40 × 10 14
Example
Consider the number
0.0000045
4.5 × 10 -6

24. ● Transform the following scientific notation to standard
notation
9 x 10 5
900 000
3.1 x 10 -6
0.0000031

25. ● Problem Solving
The sun is about 9.3 x 10 7 miles away from the
earth.
a. How far is the sun from the earth in
meters?
1mile= 1.609344 km
1 km = 1 000m

26. ● Problem Solving
The sun is about 9.3 x 10 7 miles away from the
earth.
b. How many minutes does it take for
sunlight to reach the surface of the earth?
(the speed of light is about 3 x 10 8 m/s)
v = d/t

27. CONGRATULATIONS you
completed Module 1

28. Module 2

29. ACCURACY and
PRECISION

30. After going through this module, you are expected to:
1. define accuracy and precision;
2. differentiate accuracy and precision; and
3. illustrate an example of accuracy and precision

31. Accuracy
It is how close a measurement is to the correct value for that
measurement.
11.1 inches
11.2 inches
10.9 inches
measuring the length of a
standard piece of bond
paper
11 inches 12 inches

32. Precision
However, if the measured values had
been 10.9 inches, 11.1 inches, and 11.9
inches, then the measurements would
not be very precise.
It states how well repeated measurements of something generate the
same or similar results.
In the case of the printer paper
measurements, the lowest value was
10.9 inches and the highest value was
11.2 inches.
Therefore, the precision
of measurements refers
to how close together
the measurements are
when you measure the
same thing several times.
Precise

33. Which group of measurements is most precise?
a. 0.005 g, 0.0049 g, 0.0051 g
b. 1.23 cm3, 2.21 cm3, 9.92 cm3
c. 23.4 mm, 12.4 mm, 50.2 mm
d. 2.3 x 10-2 kg, 2.31 x 102 kg, 2.29 x 1012 kg
a

34. The volume of a liquid is 20.5 ml. Which of the
following sets of measurement the value with
good accuracy?
a. 18.6 ml, 17.6 ml, 19.6 ml, 17.2 ml
b. 18.8 ml, 19.0 ml, 19.2 ml, 18.8 ml
c. 19.3 ml, 19.2 ml, 18.6 ml, 18.7 ml
d. 20.2 ml, 20.5 ml, 20.3 ml 20.1 ml
d

35. The mass of unknown substance is 2.86 g.
Which of the following sets of measurement
represents the value with both accuracy and
precision?
a. 1.78 g, 1.80 g, 1.76 g, 1.81 g
b. 1.95 g, 2.02 g, 1.96 g, 2.01 g
c. 2.81 g, 1.98 g, 2.40 g, 2.78 g
d. 2.85 g, 2.86 g, 2.84 g, 2.81 g
d

36. The mass of a sample of a copper nitrate is 3.82 g.
A student measures the mass and finds it to be
3.81 g, 3.82 g, 3.79 g and 3.80 g in the first, second,
third and fourth trial, respectively. Which of the
following statements is true for his measurements?
a. They have good accuracy but poor precision.
b. They have poor accuracy but good precision.
c. They are neither precise nor accurate.
d. They have good accuracy and precision.
d

37. Looking at the above rifle target, how would you
describe the shooting of this contestant?
a. accurate and imprecise
b. accurate and precise
c. inaccurate and precise
d. inaccurate and imprecise
c

39. RANDOM ERRORS and
SYSTEMATIC ERRORS

40. Random errors
It usually results from the experimenter’s inability to take
the same measurement in exactly the same way to get
exactly the same number.
The uncertain disturbances occurring in the experiment
is known as the random errors. Such types of errors
remain in the experiment even after the removal of the
systematic error. The magnitude of error varies from one
reading to another. The random errors are inconsistent
and occur in both the directions.

41. The presence of random errors is determined only when the different
readings are obtained for the measurement of the same quantity under the
same condition.

42. Systematic Errors
The constant error occurs in the experiment because of the
imperfection of the mechanical structure of the apparatus is
known as the systematic error. The systematic errors arise
because of the incorrect calibration of the device.
The error is mainly categorized into three types.
● Instrumental Error
● Environmental Error
● Observational Error

43. Instrumental Error
The instrumental error occurs because of three reasons.
1. Misuse of the apparatus.
2. Imperfection in the mechanical structure of the apparatus.
3. The error occurs because of the loading effect

44. Environmental Error
Happens when some factor in the environment , such as an
uncommon event, leads to error.
For example: If you are trying to measure the mass of an apple on
a scale, and classroom is windy, the wind may cause the scale to
read incorrectly.

45. Observational Error
Occurs due to wrong observation or reading in the instruments.
-errors introduced by the observer(parallax error
while reading a meter, wrong scale selection, habits
of individual observers etc.

46. Estimate Error Using
Variance

47. The accepted value of a measurement is the true or correct value
based on general agreement with a reliable reference.
The experimental value of a measurement is the value that is
measured during the experiment.
The error of an experiment is the difference between the
experimental and accepted values.
Error = experimental value − accepted value
The percent error is the absolute value of the error divided by the
accepted value and multiplied by 100%.

48. Absolute, Relative and
Percentage Error
The Absolute Error is the difference between the actual and
measured value.
The Relative Error is the Absolute Error divided by the actual
measurement.
The Percent Error is the absolute value of the error divided by
the accepted value and multiplied by 100%..
% Error=|experimental value − accepted value | /accepted
value×100%

49. Example: The thermometer measures to the nearest 2 degrees.
The temperature was measured as 38° C
The temperature could be up to 1° either side of 38° (i.e. between
37° and 39°)
Temperature = 38 ±1°
So: Percent Error = 2.63...%
Absolute Error = 1°
And:
Relative Error = 1°/38° = 0.0263...

50. Length = 12.5 ±0.05 m
So:
Absolute Error = 0.05 m
And:
Relative Error = 0.05 m /12.5 m = 0.004
And:
Percent Error = 0.4%

51. Length = 12.5 ±0.05 m
So:
Absolute Error = 0.05 m
And:
Relative Error = 0.05 m 12.5 m = 0.004
And:
Percent Error = 0.4%

52. What is the area of a field with a given width of
326.34 ±0.01m and a given length of 34.23 ±0.02m?
(Give a conservative estimate using percent error)
The area of the field is calculated from the product
of its width and length as given (and we retain all the
digits at first instance):
326.34m x 34.23m = 11170.6182 m2

53. and the corresponding total fractional error in
percent is calculated as:
Percent error =m2 (0.01/326.34 + 0.02/34.23) (100%)
= 0.06149257% m2
The total error based on the total fractional error above is
calculated as follows:
Total error= 11170.6182m2(0.01/326.34 + 0.02/34.23)
=6.096m2 or 6. m2

54. % Error=|experimental value − accepted value | accepted
value×100%
Or
%error = Absolute error/Accepted value x 100%

55. 1. The density of water at 4°C is known to be 1.00g/mL. Kayla
experimentally found the density of water to be 1.075g/mL. What
is her percent error?
Ans: 7.5%

56. 2. John Mark C. wants to buy a card for his wife. John Mark C.
calculate the amount of the card as 4.50 dollar. The actual price
of the card is 4 dollar. What is John Mark’s percent error?
Ans: 12.5%

57. 3. In an experiment, the temperature of a solution is measured by
a student to be 79 degrees, but the true value of the temperature
is 85 degrees. What is the percent error in this measurement?
Ans:

58. 4. A student measured the length of a table to be 65 cm, but the
table was actually 62 cm long. What was the percent error in this
measurement?
Ans:

59. 5. An object has a mass of 35.0 grams. On Anthony’s balance, it
weighs 34.85 grams. What is the percent error of his balance?
Ans:

60. Estimating Uncertainty in
Repeated Measurements

61. xm =
1
5 𝑖=1
𝑁
𝑥𝑖
Mean

62. Standard Deviation
sd=
1
𝑁 𝑖=1
𝑁
(𝑥𝑖− 𝑥𝑚)2

63. Standard Error or the
Standard Deviation of the
Mean
se=
𝑠𝑑
𝑁

64. The mean represents the central tendency of the
measurements and the associated error in
determining the mean is the standard error.

65. Set 1 Set 2 Set 3 Set 4
23.52 2 100.0 50.1
17.20 3 99 55
20.01 1.4 99.1 45
19.2 5 98 38.9
18.00 3.3 99.5 63.2
The Following table list sets of values.

66. CONGRATULATIONS you
completed Module 2