Download CH 1

Recipe for PHY 107
1. Study hard (use the grade of test 1 as a gauge)
2. Come to class
3. Do the homework (and more)
4. Do not stay behind
5. If you have questions get them answered
6. If you have a problem tell me about it
PHY 107
Home page:
http://www.physics.buffalo.edu/phy107/
Instructor Coordinates:
Name: Surajit Sen
Office: 325 Fronczak
Telephone: 645-2017 ext.193
E-mail: [email protected]
Office Hours: Mondays/Tuesdays 9-10 am and By
Appointment
Extra help for engineering majors:
Begin at the engineering home page: www.eng.buffalo.edu
From there click on “Freshman Programs:
Then on “Small Groups”
Click on “submit” to sign up
If you cannot sign up contact Bill Wild at:
[email protected]
Text book:
“Physics for scientists and Engineers” Vol.1, 3rd edition, by
Fishbane, Gasiorowicz and Thornton (Prentice Hall)
“Student Solution Manual” for the textbook above. This
contains worked solutions to selected odd-numbered
problems.
Chapter 1
Tooling Up
In this chapter we shall introduce the following
concepts which will be used throughout this
semester (and beyond)
1. Units and systems of units
2. Uncertainties in measurements, propagation of
errors
3. Vectors (vector addition, subtraction, multiplication
of a vector by a scalar, decomposition of a vector
into components)
(1-1)
Classical (before 1900)
Physics
(PHY 107, PHY 108)
Modern (after 1900)
(PHY 207)
In PHY 107 we study mechanics that deals with the motion
of physical bodies using Newton’s equations. These
equations yield accurate results provided that:
1. The bodies in question are macroscopic (roughly
speaking large, e.g. a car, a mouse, a fly)
2. The bodies do not move very fast. How fast? The
yardstick is the speed of light in vacuum.
c = 3108 m/s
(1-2)
The Scientific Method
• Carry out experiments during which we measure physical
parameters such as electric potential V, electric current I, etc
• Form a hypothesis (assumption) which explains the existing
data
• Check the hypothesis by carrying out more experiments to
see if the results agree with the predictions of the hypothesis
conductor
I
+
-
V
(1-3)
Example: Ohm’s law
The ratio V/I for a
conductor is a constant
known as the resistance
R
We must measure !
Example: I step on my bathroom scale and it reads 150
150 what? 150 lb? 150 kg?
For each measurement we need units.
Do we have to define arbitrarily units for each and every
physical parameter?
The answer is no. We need only define arbitrarily units for
the following four parameters:
Length, Mass , Time , Electric Current
In PHY 107 we will need only units for the first three. We will
define the units for electric current in PHY 108 (1, 4)
In this course we shall use the SI (systeme internationale)
system of units as follows:
Parameter
Unit
Symbol
Length
meter
m
Mass
kilogram
kg
Time
second
s
Electric Current
Ampere
A
All other units follow from the arbitrarily defined four units
listed above
Note: SI used to be called the “MKSA” system of units
(1-5)
The meter
A
earth
C
equator
B
1 meter  AB/107
(1-6)
The standard meter
It is a bar of Platinum-Iridium
kept at a constant temperature
The meter is defined as the
distance between the two scratch
marks
(1-7)
The kilogram (kg)
It is defined as the mass
equal to the mass of a
cylinder made of platinumiridium made by the
International Bureau of
Weights and Measures. All
other standards are made as
copies of this cylinder
(1-8)
(1-9)
The second (s)
The second is defined as the duration of
the mean solar day divided by 86400
N
earth
S
The mean solar day is the average time
it takes the earth to complete one
revolution around its axis
Where does the 86400 come from?
1 day = 24 hours
1 hour = 60 minutes
1 minute = 60 seconds
Thus: 1 day 24  60  60 = 86400 s
Most
common
10-12 please correct!
Examples: 1 km = 1000 m
1 ms = 10-3 s
(1-10)
Question: How do we define all other units?
Answer: Using an equation that connects the parameter
whose units we wish to define with other parameters whose
units are known.
Example 1: Find the units of acceleration
g
..
h = gt2/2 Solve this equation for g
m
g = 2h/t2
t
h
Units of left
hand side
=
Units of right
hand side
Units (g) = units (2h/t2) = m/s2
Note: The number 2 has no units
(1-11)
Example 2: Find the units of force
m
a
Newton’s second law
F = ma
F
Units of left
hand side
=
Units of right
hand side
Units(F) = units(m)  units(a) = kg.m/s2
Note: We call the SI unit of force the “Newton” in honor of
Isaac Newton who formulated the three laws of motion in
mechanics. Symbol: N
(1-12)
Uncertainty in Measurement
(1-13)
There is no such thing as a perfectly accurate measurement.
Each and every measurement has an uncertainty due to: 1. the
observer, 2. the instrument, and 3. the procedure used
How do we express the uncertainty in a measurement?
Assume that we are asked to measure the length L of an object
with the ruler shown on page (1-14). The smallest division on
this ruler is 1 mm. The uncertainty L in L using that
particular ruler is 1 mm. (If one is careful one can reduce it to
0.5 mm). If L is found to be 21.6 cm we write this as:
L = (21.6  0.1)cm This simply means that the real value is
somewhere between 21.5 cm and 21.7 cm. We can give L
using three significant figures
L
0.1
Percentage Uncertainty =
L
 100 
21.6
 100  0.5%
The smallest division of this ruler is equal to 1 mm
millimeters
inches
(1-14)
L2
We are given the ruler shown on page
(1-14) and are asked to measure the width
L1 and height L2 of the rectangle shown
L1 = (21.6  0.1) cm
L2 = (27.9  0.1) cm
Area A = L1L2 = 21.627.9 = 602.6 cm2
A
L1
The uncertainties L1 and L2 in the measurement of L1 and
L2 will results in an error A in the calculated value of the
rectangle area A. This is known as error propagation.
Step 1:
Step 2:
 L1
0.1

 0.005
L1
21.6
A
A

 L1
L1

 L2
L2
 L2
L2
0.1

 0.004
27.9
 0.005  0.004  0.009
 A  0.009 A  0.009  602.6  5 cm 2
A  (603  5) cm 2
(1-15)
Dimensional Analysis
The dimensional analysis of a physical parameter such as
velocity v, acceleration a , etc expresses the parameter as an
algebraic combination of length [L], mass [M], and time [T].
This is because all measurements in mechanics can be
ultimately be reduced to the measurement of length, mass ,
and time.
[L], [M], and [T] are known as primary dimensions
How do we derive the dimensions of a parameter? We use an
equation that involves the particular parameter we are
interested in. For example: v = x/t . In every equation
[Left Hand Side] = [Right Hand Side]
Thus: [v] = [L]/[T] = [L][T]-1
(1-16)
Note: The dimensions of a parameter such as velocity does
not depend on the units. [v] = [L][T]-1 whether v is expressed
in m/s, cm/s, or miles/hour
Dimensional analysis can be used to detect errors in equations
Example:
g
..
m
h = gt2/2
[LHS] = [RHS]
[LHS] = [h] = [L]
t
[RHS] = [gt2/2] = [g][t2] = [L][T]-2[T]2 = [L]
h
Indeed [LHS] = [RHS] = [L] as might be
expected from the equation h = gt2/2 which
we know to be true.
(1-17)
Note 1: If an equation is found to be dimensionally incorrect
then it is incorrect
Note 2: If an equation is dimensionally correct it does not
necessarily means that the equation is correct.
..
g
m
Example: Lets try the (incorrect) equation
h = gt2/3
[LHS] = [L]
[RHS] = [g][t2] = [L]
Even though [LHS] = [RHS] equation
t
h
h = gt2/3 is wrong!
(1-18)
Scalars
Physical Quantities
Vectors
A scalar is completely described by a number. E.g. mass (m),
temperature (T), etc
A vector is completely described by :
• Its magnitude
• Its direction
Example: The displacement vector
magnitude = 30 paces
direction = northeast
(1-19)
Vector notation
1. A vector is denoted either by an arrow on top or by
bold print. Example: The vector of acceleration a is
written either as:
a or as: a
Both methods are used
2. The magnitude of a vector is denoted either by the
symbol:   or by the symbol of the vector written with
regular type. Example: the magnitude of the
acceleration vector can be written either as: a or as: a
3. A vector is represented by an arrow whose length is
proportional to the vector’s magnitude. The arrow has the
a
same direction as the vector
(1-20)
Vector addition (geometric method)
Recipe for determining R = A + B (see fig.a)
1. At the tip of the first vector (A) place the tail of the second vector (B)
2. Join the tail of the first vector (A) with the tip of the second (B)
Note 1: A + B = B + A
(see fig.b)
Note 2: the above recipe can be used for more than two vectors
(1-21)
We are given vectors A, B, and C and are asked to determine
vector S = A + B + C
• At the tip of A we place the tail of B
• At the tip of B we place the tail of C
• To get S we join the tail of A with the tip of C
(1-22)
Negative of a vector
We are given vector B and are asked to determine –B
1. Vector –B has the same magnitude as B
2. Vector –B has the opposite direction
(1-23)
Vector Subtraction (geometric method)
We are given vectors A and B are are asked to determine
T=A–B
1. Determine –B from B
2. Add vector (-B) to vector A using the recipe of page
(1-21)
A
B
(1-24)
Multiplication of a vector B by a scalar b; determine bB
• The magnitude |bB| = |b||B|
• The direction of bB depends on the algebraic sign of b
If b > 0 then bB has the same direction as B
If b < 0 then bB has the opposite direction of B
(1-25)
Unit vector is defined as any vector whose magnitude is
equal to unity
We are given a vector U and are asked to determine a unit
^
vector u
which is parallel to U
Recipe: u^ = U/|U|
^
Vectors U and u are parallel
(1-26)
Vector Components
Any vector V in the xy-plane can be written as a sum of two
other vectors, one along the x-axis and the other along the yaxis. These are called the components of V
^
^
^
^
V = Vxi + Vy j Here i and j are the unit vectors along
x- and y-axes, respectively
Vx is the projection of V along the x-axis
Vy is the projection of V along the y-axis
(1-27)
the
We wish to express Vx and Vy in terms of V and  and vice
versa i.e. express V and  in terms of Vx and Vy
Consider triangle ABC:
Vx = Vcos 
and Vy = Vsin
Also: V = [ Vx2 + Vy2 ]1/2
and
tan = Vy/Vx
C
A
B
(1-28)
Vector addition (algebraic method)
We are given:
y
A  Ax i  Ay j and
B  Bx i  By j
A
Find C  A  B
B
C  Cx i  C y j
x
O
Cx  Ax  Bx ,
C y  Ay  By
Magnitude C  C  C
2
x
2
y
(1-29)
Vector subtraction (algebraic method)
We are given:
y
A  Ax i  Ay j and
B  Bx i  By j
A
Find D  A  B
B
D  Dx i  Dy j
x
O
Dx  Ax  Bx ,
D y  Ay  By
Magnitude D  Dx2  Dy2
(1-30)
(1-31)
B
O
A
C
Vectors in three dimensions
V  Vx i  Vy j  Vz k
Vx  OA, Vy  OB, Vz  OC
xyz is a right handed
coordinate system
How to check whether an xyz coordinate system is
right handed
• Rotate the x-axis in the xy-plane along the shortest angle so
that it coincides with the y-axis. Curl the fingers of the right
hand in the same direction
• The thumb of the right hand must point along the z-axis
(1-32)