1. The scientific method could be the most efficient problem-solving tool
ever devised. There are 6 steps in the scientific method: (I) Define the
problem, (II) Gather information, (III) State your hypothesis, (IV) Test
the hypothesis, (V) Form your conclusion, (VI) Publish the results. The
scientific method is the process by which scientists, collectively and over
time, endeavor to construct an accurate (that is, reliable, consistent and
non-arbitrary) representation of the natural world.
2. The Metric System (Systeme International, SI) was first introduced by
the French Academy of Science in 1795 as an attempt to unify existing
systems. The SI contains the basic units for length (meter, m), mass
(kilogram, kg), and time (second, s), with speed (m/s), volume (m3),
and density (kg/m3) as some derived units.
3. All calculations must be done observing significant digits and scientific
notation. When a number is expressed in scientific notation, the number
of significant figures is the number of digits needed to express the number
to within the uncertainty of measurement. We always round to the least
precise measurement. Here is an example problem, if you have to multiply
2.5 cm x 1.23 cm the result is 3.1 cm2.
4. The number of significant figures of a product or quotient of two or more
quantities is equal to the smallest number of significant figures for the
quantities involved. For example, if you multiply 5.2 x 3.751 x 6.43, your
answer must be written using only two significant digits as 130.
For addition or subtraction, the number of significant figures is determined
with the smallest significant figure of all the quantities involved. For example,
the sum 10.234 + 5.2 + 100.3234 is 115.7574, but should be written 115.8
(with rounding), since the quantity 5.2 is significant only to ± 0.1.
for answers to certain questions. An order of magnitude calculation is an
estimate to determine if a more precise calculation is necessary. We round
off or guess at various inputs to obtain a result that is reliable to within a
factor of 10. Specifically, to get the order of magnitude of a given quantity,
we round off to the closest power of 10 (ex: 75 kg is expressed as 102 kg).
Another ex., the average distance from the Earth to the Sun is 93,000,000
miles. In scientific notation it is 9.3x107 miles. But since 9.3 is closest to 101,
we would express the order of magnitude as 108 miles.
6. A frame of reference is a coordinate system for specifying the precise
location of objects in space. You make a convenient choice for the origin of
the system. Maybe you have heard the expression, "It depends on your
frame of reference."
7. Accuracy refers to the agreement of a measured value with an accepted
value. Percent error measures accuracy. Precision is the agreement of a set
of measured values with each other. It can be measured by average deviation.
8. Each instrument has an inherent amount of uncertainty in its measurement.
Even the most precise measuring device cannot give the actual value because
to do so would require an infinitely precise instrument.
9. A measure of the precision of an instrument is given by its uncertainty.
As a good rule of thumb, the uncertainty of a measuring device is 20%
of the least count. Recall that the least count is the smallest subdivision given
on the measuring device. The uncertainty of the measurement should be given
with the actual measurement, for example, 41.64 ± 0.02cm.
10. Here are some typical uncertainties of various laboratory instruments:
Here's an example. The uncertainty of all measurements made with a meter
stick whose smallest division (least count) is one millimeter is 20% of 1mm
or 0.02 cm. Say you use the meter stick to measure a metal rod and find that
the rod is between 10.2 cm and 10.3 cm. You may think that the rod is closer
to 10.2 cm than it is to 10.3 cm, so you make your best guess that the rod
is 10.23 cm in length. Since the uncertainty in the measurement is 0.02cm,
you report the length of the rod as 10.23 ± 0.02 cm (0.1023 ± 0.0002 m).
When a quantity is graphed, it is common for the uncertainty of that quantity
to be represented by error bars.
11. Graphs are plotted with the independent (control) variable on the x-axis,
and the dependent (measured) variable on the y-axis.
12. Graphs can show direct (linear), inverse (hyperbolic), periodic (sinusoidal),
quadratic (parabolic), or chaotic relationships.
13. All equations must be dimensionally correct. We use dimensional analysis
(factor labeling) to determine if equations are correct.
14. The number of atoms in one mole of any element or compound is 6.02x1023
which we know as Avogadro's number.
15. The density of a substance is defined as its mass per unit volume. We use
the Greek letter rho, ρ, for density and the equation is ρ = m / V.
16. Some equations that you may remember from Mathematics are A = πr2,
C = 2πr, A = 4πr2, V = πr2h, V = 4/3 πr3, and d = vt. Notice that variables in
Physics are case sensitive. For example, A is area, but a is acceleration.
Another example, T is temperature, but t is time.
17. We also need to remember SOHCAHTOA to compute the value of unknown
sides and angles of right triangles.
18. We need these steps to solve any problem in Physics:
(i) read the problem and identify the given variables
(ii) determine what you are asked to solve for
(iii) find the correct formula to use
(iv) isolate the unknown
(v) substitute-in the given information and simplify.
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Website Homework Assignment #1. Show all work.
1. Write a paragraph that illustrates how a student would use the scientific
method to determine what college to attend.
2. Find an English-to-Metric System of Measurement conversion table.
Then use it to make the following conversions:
(a) 6.75 inches = ______ centimeters (b) 3.0 miles = _____ kilometers
(c) 400 cubic inches = ______ Liters (d) 100 lbs = ______ kilograms
3. How many significant digits are in each of the following measurements:
(a) 12.375 cm ____ (b) 3.000 m _____ (c) .00075 s _____
(d) 6.0075 in. _____ (e) 93,000,000 mi. ____ (f) 25,000 km ______
4. Express all the measurements given in #3 in scientific notation.
5. Perform the following calculations and express your answers in scientific
notation with the correct amount of significant digits:
(a) 2500 m x 2.75 m = ___________ (b) 2.4x107 cm / 1.45 cm = __________
(c) 4.3 cm + 1.75 cm + .041 cm = ___________
(d) (6.3x108 m x 4x109 m) / 7.11x10-5 m = _____________
6. Express each answer in #5 as an order of magnitude.
7. Solve for the variable in red. Show all of your steps.
(a) F = mv2/r ________ (b) E = mc2 ________
(c) T = 2π√(L ⁄ g) _______ (d) d = ½ at2 _______
(e) 1/p + 1/q = 1/f _______ (f) F = (G m1·m2)/r2 _______
(g) Combine these equations E = mc2 , c = f·λ , E = hf to get
an equation for λ that does not involve E or f .
Website Notes for Scalars and Vectors:
Vectors are used to describe multi-dimensional quantities. Multi-dimensional
quantities are those which require more than one number to completely
describe them. Vectors, unlike scalars, have two characteristics, magnitude
and direction. A vector is indicated by an uppercase letter either in boldface
or with an arrow over the top. For example, A or Â . Examples of vector
quantities are: position in a plane, position in space, velocity, acceleration,
Scalars are used to describe one-dimensional quantities, that is, quantities
which require only one number to completely describe them. They have
magnitude only. Direction does not apply. There are cases where scalars can
be combined mathematically, but we will save that for later. Some examples
of scalar quantities are: temperature, mass, time, volume, density, length,
area, and energy.
III. Vector Diagram.
Any vector can be resolved into perpendicular component vectors using sine
and cosine functions. Actually, for all vector problems just remember
SOHCAHTOA. This allows us to find the x and y components of any vector.
For the horizontal, or x-component of a vector A, we have, A·(cos F ).
Then to find the y-component we use A·(sin F). This can easily be proven
with the Pythagorean Theorem.
IV. Some Properties of Vectors.
Two vectors are equal only if they have the same magnitude and direction.
To find the opposite of a given vector just keep the same magnitude but
point it in the opposite direction. ex. A - B = A + (-B). Vectors can also
be expressed using polar coordinates (r , θ) specifying the length of the
radius vector r , and the angle of rotation, F ("Phi"), from the positive
Additionally, in a two-dimensional coordinate system, vectors can be
denoted using the unit vectors î and ĵ. Each unit vector has magnitude
equal to 1, and they point in the x and y directions, respectively. We can
easily add the third dimension, or z direction using unit vector k.
V. Addition of Vectors
Vectors can be added graphically using the head-to-tail method.
You begin by drawing the first vector in a coordinate system,
and then drawing the second vector from the endpoint of the
first, and so on. Then you can draw a single vector from the
origin to the head of the last vector.
The vector difference works the same as vector addition except
that we multiply the vector we are subtracting by -1. In other
words, to subtract a vector, just add the opposite.
It is much like subtracting two numbers: A - B = A + (-B).
This diagram illustrates vector subtraction in the tip-to-tail style.
The original B vector is shown as a dashed line.
VII. Multiplication of a Vector by a Scalar.
A vector may be multiplied by a scalar by multiplying each of its
components by that number.
Notice above, that the vector does not change direction,
only length. If A = (1,2) then 3A = (3,6).
Problems and Show All Work.
1. Find the x and y components of the following vectors:
a. 240 km at 330º
b. 34 m/s at 210º
2. From the x and y components given, find the direction and magnitude
of the resultant.
a. Fy = 120 N, Fx = 345 N
b. vy = 31 m/s, vx = 8 m/s
3. Add the three vectors below. Use the graphical method to show a picture
of the addition of the vectors. Use the mathematical method to obtain the
magnitude and direction of the resultant vector.
A = 450 N at 20º, B = 250 N at 270º, C = 100 N at 70º
4. A soccerball is kicked with a horizontal velocity of 11.3 m/s and a vertical
velocity of 3.5 m/s. What is the magnitude and direction of the resultant
velocity of the ball?
5. A pole-vaulter applies a force of 415 N to the pole at an angle of 37º.
What are the horizontal and vertical components of this force?