Physics for Scientists and Engineers I
Dr. Beatriz Roldán Cuenya
University of Central Florida, Physics Department, Orlando, FL PHY 2048, Section 4
Chapter 0 - Introduction
I. General
II. International System of Units III. Conversion of units
IV. Dimensional Analysis
V. Problem Solving Strategies
I. Objectives of Physics
- Find the limited number of fundamental laws that govern natural phenomena.
- Use these laws to develop theories that can predict the results of future experiments.
-Express the laws in the language of mathematics.
- Physics is divided into six major areas:
1. Classical Mechanics (PHY2048) 2. Relativity
3. Thermodynamics
4. Electromagnetism (PHY2049) 5. Optics (PHY2049)
6. Quantum Mechanics
II. International System of Units
K Kelvin
Temperature
W = J/s Watt
Power
J = Nm Joule
Energy
Pa = N/m
2Pascal
Pressure
N Newton
Force
m/s
2Acceleration
m/s Speed
kg kilogram
Mass
s second
Time
m meter
Length
UNIT SYMBOL UNIT NAME
QUANTITY
f femto
10
-15p pico
10
-12n nano
10
-9micro µ 10
-6m milli
10
-3c centi
10
-2D deci
10
-1da deka
10
1h hecto
10
2k kilo
10
3M mega
10
6G giga
10
9T tera
10
12P peta
10
15ABBREVIATION PREFIX
POWER
Example: 316 feet/h m/s
III. Conversion of units
Chain-link conversion method: The original data are multiplied successively by conversion factors written as unity. Units can be treated like algebraic quantities that can cancel each other out.
IV. Dimensional Analysis
Dimension of a quantity: indicates the type of quantity it is; length [L], mass [M], time [T]
Example: x=x 0 +v 0 t+at 2 /2
s feet m
m s
h h
feet 0 . 027 /
28 . 3
1 3600
316 1 =
⋅
⋅
[ ] [ ] [ ]
[ ] [ ] [ ]
[ ] [ ] T [ ] [ ] [ ] L L L T
T L T L L
L = + + 2 2 = + +
Dimensional consistency: both sides of the equation must have the same dimensions.
Note: There are no dimensions for the constant (1/2)
Significant figure one that is reliably known.
Zeros may or may not be significant:
- Those used to position the decimal point are not significant.
- To remove ambiguity, use scientific notation.
Ex: 2.56 m/s has 3 significant figures, 2 decimal places.
0.000256 m/s has 3 significant figures and 6 decimal places.
10.0 m has 3 significant figures.
1500 m is ambiguous 1.5 x 10 3 (2 figures), 1.50 x 10 3 (3 fig.), 1.500 x 10 3 (4 figs.)
Order of magnitude the power of 10 that applies.
V. Problem solving tactics
• Explain the problem with your own words.
• Make a good picture describing the problem.
• Write down the given data with their units. Convert all data into S.I. system.
• Identify the unknowns.
• Find the connections between the unknowns and the data.
• Write the physical equations that can be applied to the problem.
• Solve those equations.
• Always include units for every quantity. Carry the units through the entire calculation.
• Check if the values obtained are reasonable order of magnitude and units.
Chapter 1 - Vectors
I. Definition
II. Arithmetic operations involving vectors A) Addition and subtraction
- Graphical method
- Analytical method Vector components
B) Multiplication
Review of angle reference system
Origin of angle reference system θ
10º<θ
1<90º
90º<θ
2<180º θ
2180º<θ
3<270º
θ
3θ
4
270º<θ
4<360º 90º
180º
270º
0º
Θ
4=300º=-60º Angle origin
I. Definition
Vector quantity: quantity with a magnitude and a direction. It can be represented by a vector.
Examples: displacement, velocity, acceleration.
Same displacement Displacement does not describe the object’s path.
Scalar quantity: quantity with magnitude, no direction.
Examples: temperature, pressure
Rules:
) 1 . 3 ( ) ( commutativ e law a
b b a
+
= +
) 2 . 3 ( ) (
) ( )
( a b c a b c associativ e law +
+
= + +
II. Arithmetic operations involving vectors
- Geometrical method a b
b a s
= + Vector addition: s a b
= +
Vector subtraction: d a b a ( b ) ( 3 . 3 )
− +
=
−
=
Vector component: projection of the vector on an axis.
θ θ
sin
) 4 . 3 ( cos
a a
a a
y x
=
=
y x
a a a
a = +
) 5 . 3 (
2
2 Vector magnitude
a of components
Scalar
Unit vector: Vector with magnitude 1.
No dimensions, no units.
axes z y x of direction positive
in vectors unit k j
i ˆ , ˆ , ˆ → , ,
) 6 . 3 ˆ (
ˆ a j i a a = x + y
Vector component
- Analytical method: adding vectors by components.
Vector addition:
) 7 . 3 ˆ (
) ˆ (
)
( a b i a b j b
a
r = + = x + x + y + y
Vectors & Physics:
-The relationships among vectors do not depend on the location of the origin of the coordinate system or on the orientation of the axes.
- The laws of physics are independent of the choice of coordinate system.
φ θ θ = +
+
= +
=
'
) 8 . 3 ( ' '
2 22 2
y x y
x
a a a
a a
Multiplying vectors:
- Vector by a scalar:
- Vector by a vector:
Scalar product = scalar quantity a s
f
⋅
=
) 9 . 3 ( cos a x b x a y b y a z b z
ab b
a ⋅ = φ = + +
(dot product)
) 90 ( 0 cos 0
) 0 ( 1 cos
=
=
←
=
⋅
=
=
←
=
⋅
φ φ
φ φ b
a ab b Rule: a b b a ( 3 . 10 ) a
⋅
=
⋅
0 90 cos 1 1 1 0 cos 1 1
=
⋅
⋅
=
⋅
=
⋅
=
⋅
=
⋅
=
⋅
=
⋅
=
⋅
⋅
=
⋅
=
⋅
=
⋅
j k k j i k k i i j j i
k k j j i i
Multiplying vectors:
- Vector by a vector
Vector product = vector
φ sin
) ˆ ˆ (
) ˆ (
) (
ab c
k a b b a j b a a b i a b b a c b
a y z y z z x z x x y x y
=
− +
−
−
−
=
=
×
(cross product)
Magnitude Angle between two vectors: a b
b a
⋅
= ⋅ ϕ cos
) 12 . 3 ( ) ( a b a b
×
−
= Rule: ×
) 90 ( 1 sin
) 0 ( 0 sin 0
=
=
←
=
×
=
=
←
=
×
φ φ
φ φ ab b a
b a
Direction right hand rule
b a containing plane
to lar perpendicu c
,
1) Place a and b tail to tail without altering their orientations.
2) c will be along a line perpendicular to the plane that contains a and b where they meet.
Vector product
Right-handed coordinate system
x
y z
i k j
Left-handed coordinate system
y
x z
i j
k
0 0 sin 1 1 ⋅ ⋅ =
=
×
=
×
=
×
k k j j i i
j k i i k
i j k k j
k i j j i
k k j j i i
=
×
−
=
×
=
×
−
=
×
=
×
−
=
×
=
×
=
×
=
×
) (
) (
) (
0
P1: If B is added to C = 3i + 4j, the result is a vector in the positive direction of the y axis, with a magnitude equal to that of C. What is the magnitude of B?
ˆ ˆ
2 . 3 1 ˆ 9 3 ˆ 5 ˆ ˆ ) ˆ 4 3 (
5 4 3
) ˆ 4 ˆ 3 ˆ (
2 2
= +
=
→ +
−
=
→
= + +
= +
=
=
=
= + +
= +
B j i B j j i B
D C
j D D j i B C B
Method 1
Method 2
2 . 2 3 sin 2 2
/ sin 2
9 . 36 ) 4 / 3 ( tan
=
=
→
=
=
→
=
θ θ
θ θ
D D B B
D C
B/2
B θ
Isosceles triangle
P2: A fire ant goes through three displacements along level ground: d
1for 0.4m SW, d
20.5m E, d
3=0.6m at 60º North of East. Let the positive x direction be East and the positive y direction be North. (a) What are the
x and y components of d
1, d
2and d
3? (b) What are the x and the y components, the magnitude and the direction of the ant’s net displacement? (c) If the ant is to return directly to the starting point, how far and in what direction should it move?
N
E d
3d
245º d
1D
m d
m d
d m d
m d
m d
y x y x y x
52 . 0 60 sin 6 . 0
30 . 0 60 cos 6 . 0 0 5 . 0
28 . 0 45 sin 4 . 0
28 . 0 45 cos 4 . 0
3 3 2 2 1 1
=
=
=
=
=
=
−
=
−
=
−
=
−
=
(a)
(b)
East of North
m D
m j i j i j i d d D
m j i i j i d d d
8 . 52 24 . 0 24 . tan 0
57 . 0 24 . 0 52 . 0
ˆ ) 24 . ˆ 0 52 . 0 ( ˆ ) 52 . ˆ 0 3 . 0 ( ˆ ) 28 . ˆ 0 22 . 0 (
ˆ ) 28 . ˆ 0 22 . 0 ˆ ( 5 . 0 ˆ ) 28 . ˆ 0 28 . 0 (
1 2 2 3 4
2 1 4
=
=
= +
=
+
= + +
−
= +
=
−
= +
−
−
= +
=
θ
−d
4(c)Return vector negative of net displacement, D=0.57m, directed 25º South of West
P2
k j i d
k j i d
k j i d
2 ˆ 3 ˆ 4 ˆ
3 ˆ 2 ˆ ˆ
6 ˆ 5 ˆ 4 ˆ
3 2 1
+ +
= + +
−
=
− +
=
? , )
(
? )
(
? )
(
? )
(
2 1 2
1 2 1 3 2 1
d d of plane in and d to lar perpendicu d of Component d
d along d of Component c
z and r between Angle b
d d d r a
+ +
−
=
θ d
1d
2k j i k j i k j i k j i d d d r
a ) ( 4 ˆ 5 ˆ 6 ˆ ) ( ˆ 2 ˆ 3 ˆ ) ( 4 ˆ 3 ˆ 2 ˆ ) 9 ˆ 6 ˆ 7 ˆ
( =
1−
2+
3= + − − − + + + + + = + −
m r
r k r b
88 . 12 7 6 9
88 123 . 12 cos 7 7 cos ˆ 1 ) (
2 2 2
1
= + +
=
=
= −
→
−
=
⋅
⋅
=
⋅
−θ θ
m d
d m d
d d d d d
d d
d d d
d d
d c
74 . 3 3 2 1
2 . 74 3 . 3 cos 12
cos cos 12 18 10 4 ) (
2 2 2 2
2 1
2 1 1 // 1
1
2 1
2 1 2
1 2
1
= + +
=
−
− =
⋅ =
=
=
= ⋅
→
=
−
=
− +
−
=
⋅
θ
θ
θ d
1//d
1perpm d
m d
d d d
d
perp perp77 . 8 6 5 4
16 . 8 2 . 3 77 . 8 )
(
2 2 2 1
2 2 1 2 1 2
//
1 1
= + +
=
=
−
=
→ +
=
P3
k j i d
k j i d If
ˆ 2 ˆ 5 ˆ
4 ˆ 2 ˆ 3 ˆ
2 1
− +
−
= +
−
=
? ) 4 ( ) ( d
1d
2d
1d
2×
⋅ +
) , ( 4
) ( 4 ) 4 (
) , ( )
(
2 1 2
1 2 1
2 1 2
1
→
=
×
=
×
→
= +
plane d d to lar perpendicu b
d d d d
plane d d in contained a d d
y
x A
B
130º ( a ) Angle between − y and A = 90
+ 50
= 140
90 ) ( ) , (
, ˆ ) ˆ
( , ) (
→
=
−
→
=
×
− xy B A plane
lar perpendicu C because k j angle C B A y Angle b
P4: Vectors A and B lie in an xy plane. A has a magnitude 8.00 and angle 130º; B has components B
x= -7.72, B
y= -9.20. What are the angles between the negative direction of the y axis and (a) the direction of A, (b) the direction of AxB, (c) the direction of Ax(B+3k)? ˆ
k j i k j i E A D
k j i k B E
D k B A Direction c
61 ˆ . ˆ 94 42 . ˆ 15 39 . 18 3 20 . 9 72 . 7
0 13 . 6 14 . 5
ˆ ˆ ˆ
3 ˆ 2 ˆ . ˆ 9 72 . ˆ 7 3
ˆ ) 3 ( )
(
+ +
=
−
−
−
=
×
=
+
−
−
= +
=
= +
×
61 99 . 97 42 . 15 1
ˆ cos
42 . 15 ˆ ) 61 . ˆ 94 42 . ˆ 15 39 . 18 ˆ ( ˆ
61 . 97 61 . 94 42 . 15 39 .
18
2 2 2=
→
= −
⋅
⋅
= −
−
= + +
⋅
−
=
⋅
−
= + +
=
θ
θ D
D j
k j i j D j D
P5: A wheel with a radius of 45 cm rolls without sleeping along a horizontal floor. At time t
1the dot P painted on the rim of the wheel is at the point of contact between the wheel and the floor. At a later time t
2, the wheel has rolled through one-half of a revolution. What are (a) the magnitude and (b) the angle (relative to the floor) of the displacement P during this interval? y
x Vertical displacement:
Horizontal displacement: d
m R 0 . 9 2 =
m R ) 1 . 41 2 2 (
1 π =
5 . 2 32 tan
68 . 1 9 . 0 41 . 1
) ˆ 9 . 0 ˆ ( ) 41 . 1 (
2 2
=
→
=
= +
= +
=
π θ
θ R
R
m r
j m i m r
P6: Vector a has a magnitude of 5.0 m and is directed East. Vector b has a magnitude of 4.0 m and is directed 35º West of North. What are (a) the magnitude and direction of (a+b)?. (b) What are the magnitude and direction of (b-a)?. (c) Draw a vector diagram for each combination.
N
a E b
125º
S W
j i j i b
i a
28 ˆ . ˆ 3 29 . ˆ 2 35 cos ˆ 4 35 sin 4 5 ˆ
+
−
= +
−
=
=
43 . 71 50 . 2 28 . tan 3
25 . 4 28 . 3 71 . 2
28 ˆ . ˆ 3 71 . 2 ) (
2 2
=
→
=
= +
= +
+
= +
θ θ
m b
a
j i b a a
West of North or
m a
b
j i a b a b b
2 . 24 8 . 155 180
8 . 155 ) 2 . 24 ( 180
2 . 29 24 . 7 28 . tan 3
8 28 . 3 29 . 7
28 ˆ . ˆ 3 29 . 7 ) ( ) (
2 2
