HANDOUT #1
PHYS 1- Measurement and Vectors
PART 1- Measurement
 Physics, like other sciences, is a creative endeavor. It is not simply a collection of facts. Important
theories are created with the idea of explaining observations. To be accepted, theories are "tested” by
comparing their predictions with the results of actual experiments.
 Scientists often devise models of physical phenomena. A model is a kind of picture or analogy that
helps to describe the phenomena in terms of something we already know.
 A theory, often developed from a model, is usually deeper and more complex than a simple model.
 A scientific law is a concise statement, often expressed in the form of an equation, which
quantitatively describes a wide range of phenomena.
 Measurements play a crucial role in physics, but can never be perfectly precise. It is important to
specify the uncertainty of a measurement either by stating it directly using the ± notation, and/or by keeping
only the correct number of significant figures.
 Precision in a strict sense refers to the repeatability of the measurement using a given instrument.
For example, if you measure the width of a board many times, getting results like 8.81 cm, 8.85cm,
8.78cm, 8.82cm (estimating between the 0.1-cm marks as best as possible each time), you could
say the measurements give a precision a bit better than 0.1 cm.
 Accuracy refers to how close a measurement is to the true value. For example, if the ruler was
manufactured with a 2% error, the accuracy of its measurement of the board’s width (let say about
8.8 cm) would be about 2% of 8.8cm, or about ± 0.2 cm.
 Estimated uncertainty is meant to lake both accuracy and precision into account.
Illustration:
A friend asks to borrow your precious diamond for a day to show her family. You are a bit worried,
so you carefully have your diamond weighed on a scale which reads 8.17 grams. The scale s
accuracy is claimed to be ± 0.05 gram. The next day you weigh the returned diamond again,
getting 8.09 grams. Is this your diamond?
 Significant digits (Taken from http://www.nku.edu/~intsci/sci110/worksheets/rules_for_significant_figures.html)
 Non-zero digits are always significant
 Any zeros between two significant digits are significant
 A final zero or trailing zeros in the decimal portion ONLY are significant
 Addition/ subtraction:
o Count the number of significant figures in the decimal portion ONLY of
each number in the problem
o Add or subtract in the normal fashion
o
Your final answer may have no more significant figures to the right of
the decimal than the LEAST number of significant figures in any number
in the problem
 Multiplication/ Division:
o The LEAST number of significant figures in any number of the problem
determines the number of significant figures in the answer.
o
 Physical quantities are always specified relative to a particular standard or unit, and the unit used
should always be stated. The commonly accepted set of units today is the Systeme International (SI), in
which the standard units of length, mass, and time are the meter, kilogram, and second.
 When converting units, check all conversion factors for correct cancellation of units.
Illustration:
The posted speed limit is 55 miles per hour (mi/h or mph). What is this speed (a) in meters per
second (m/s) and (b) in kilometers per hour (km/h)?
PROBLEMS:
1. Global positioning satellites (GPS) can be used to deter-mine positions with great accuracy. The
system works by determining the distance between the observer and each of several satellites orbiting
Earth If one of the satellites is at a distance of 20.000 km from you, what percent accuracy in the
distance is required if we desire a 2-meter uncertainty? How many significant figures do we need to
have in the distance?
2. (a) How many seconds are there in 1.00 year? (b) How many nanoseconds are there in 1.00 year? (c)
How many years are there in 1.00 second?
3. One hectare is defined as 104 m2. One acre is 4 x 104 ft2. How many acres are in one hectare?
PART 2- Vectors
 A quantity such as velocity, that has both a magnitude and a direction, is called a vector.
 A quantity such as mass, that has only a magnitude, is called a scalar.
 Addition of vectors can be done graphically by placing the tail of each successive arrow at the tip of
the previous one. The sum, or resultant vector, is the arrow drawn from the tail of the first vector to the
tip of the last vector.
 Two vectors can also be added using the parallelogram method.
 Vectors can be added more accurately by adding their components along chosen axes with the aid of
trigonometric functions. A vector of magnitude V making an angle θ with the x axis has components
 Given the components, we can find a vector's magnitude and direction from
√
EXAMPLE (Adding vectors by components):
A rural mail carrier leaves the post office and drives 10.0 km in a northerly direction. She then drives in
a direction 45.0˚ south of east for 13.0 km. What is her displacement from the post office?
PROBLEMS:
1. A delivery truck travels 18 blocks north, 10 blocks east, and 16 blocks south. What is its final
displacement from the origin? Assume the blocks are equal length.
2. Vector ⃗ 1 is 6.6 units long and points along the negative x axis. Vector ⃗ 2 is 8.5 units long and
points at +45° to the positive x axis. (a) What are the x and y components of each vector? (b)
Determine the sum ⃗ 1 + ⃗ 2 (magnitude and angle).
3. Determine the vector
- , given the vectors
and in Fig 1.
4. For the vectors given in Fig. 1, determine (a)
- ⃗ + , (b) + ⃗ - , and (c)
5. For the vectors shown in Fig. 1, determine (a) ⃗ - 2 , (b) 2 - 3 ⃗ + 2 .
-
- ⃗.
Figure 1.
Note: All the materials (except those taken from the web) presented in this handout were taken from the book “PHYSICS: PRINCIPLES AND
APPLICATIONS” 6th edition by Douglas C. Giancoli.