9/5 Objectives (A day)• Introduction of AP physics

• Lab safety

• Sign in lab safety attendance sheet

• Classical Mechanics

• Coordinate Systems

• Units of Measurement

• Changing Units

• Dimensional Analysis

Lab safety General guidelines

1. Conduct yourself in a responsible manner.

2. Perform only those experiments and activities for which you have received instruction and permission.

3. Be alert, notify the instructor immediately of any unsafe conditions you observe.

4. Work area must be kept clean.

5. Dress properly during a laboratory activity. Long hair should be tied back, jackets, ties, and other loose garments and jewelry should be removed.

6. When removing an electrical plug from its socket, grasp the plug, not the electrical cord. Hand must be completely dry before touching an electrical switch, plug, or outlet.

7. Report damaged electrical equipment immediately. Look for things such as frayed cords, exposed wires, and loose connections. Do not use damaged electrical equipment.

Please sign in lab safety attendance sheet

Classical Mechanics• Mechanics is a study of motion and its causes.• We shall concern ourselves with the motion of a

particle. This motion is described by giving its position as a function of time.

Specific position & time → eventPosition (time) → velocity (time) → acceleration

• Ideal particle– Classical physics concept– Point like object / no size– Has mass

• Measurements of position, time and mass completely describe this ideal classical particle.

• We can ignore the charge, spin of elementary particles.

Position• If a particle moves along a

straight line → 1-coordinate curve/surface → 2-coordinate Volume → 3-coordinate

• General description requires a coordinate system with an origin.– Fixed reference point, origin– A set of axes or directions– Instruction on labeling a point relative to origin, the

directions of axes and the unit of axes.– The unit vector

Rectangular coordinates - Cartesian

Simplest system, easiest to visualize.

To describe point P, we use three coordinates: (x, y, z)

Spherical coordinate• Nice system for motion

on a spherical surface

• need 3 numbers to completely specify location: (r, Φ, θ)

− r: distance between point to origin

− Φ: angle between line OP and z: latitude = π/2 - Φ.

− θ: angle in xy plane with x – longitude.

Time • Time is absolute. The rate at which time

elapse is independent of position and velocity.

Unit of measurement Quantity/dimension

being measuredSI unit (symbol) British (derived)

unit

Length/[L] meter (m) Foot (ft)

Time/[T] second (s) Second (s)

Mass/[M] kilogram (kg) Slug

Ele. Current/[I] ampere (A)

Temperature[Θ] Kelvin (K)

Amount of substance[N] Mole (mol)

Luminous intensity[J] Candela (cd)

International system of units (SI) consists of 7 base units. All other units can be expressed by combinations of these base units. The combined base units is called derived units

Physical Dimensions

• The dimension of a physical quantity specifies what sort of quantity it is—space, time, energy, etc.

• We find that the dimensions of all physical quantities can be expressed as combinations of a few fundamental dimensions: length [L], mass [M], time [T].

• For example, – Energy: E = ML2/T2

– Speed: V = L/T

Derived units

• Like derived dimensions, when we combine basic unit to describe a quantity, we call the combined unit a derived unit.

• Example:– Volume = L3 (m3)– Velocity = length / time = LT-1 (m/s)– Density = mass / volume = ML3 (kg/m3)

SI prefixes• SI prefixes are prefixes (such as k, m, c,

G) combined with SI base units to form new units that are larger or smaller than the base units by a multiple or sub-multiple of 10.

• Example: km – where k is prefix, m is base unit for length.

• 1 km = 103 m = 1000 m, where 103 is in scientific notation using powers of 10

SI uses prefixes for extremes prefixes for power of ten

Prefix Symbol Notation

tera T 1012

giga G 109

mega M 106

kilo k 103

deci d 10-1

centi c 10-2

milli m 10-3

micro μ 10-6

nano n 10-9

pico p 10-12

Example: Convert the following

5 Tg ___________ kg

2 μm ___________ m

6 cg ___________ kg

7 nm ___________ m

4 Gg ___________ kg

5 x 109

2 x 10-6

6 x 10-5

7 x 10-9

4 x 106

Unit conversionsNote: the units are a part of the measurement as important as the number. They must always be kept together.

Suppose we wish to convert 2 miles into meters. (1 miles = 1760 yards, 1 yd = 0.9144 m)

x milesyd

mile myd

x1

1760

1

0.91442 = 3218 m

example• Convert 80 km/hr to m/s.

• Given: 1 km = 1000 m; 1 hr = 3600 s

ms80

km

hrx

1 km1000 m x

3600 s1 hr = 22

Units obey same rules as algebraic variables and numbers!!

Dimensional analysis• We can check for error in an equation or expression by

checking the dimensions. Quantities on the opposite sides of an equal sign must have the same dimensions. Quantities of different dimensions can be multiplied but not added together.

• For example, a proposed equation of motion, relating distance traveled (x) to the acceleration (a) and elapsed time (t).

2

2

1atx

Dimensionally, this looks like

At least, the equation is dimensionally correct; it may still be wrong on other grounds, of course.

T2L =L

T2= L

Another example

d = v / t

use dimensional analysis to check if the equation is correct.

L = (L ∕ T ) ∕ T

[L] = L ∕ T2

Significant Figures (Digits)• Instruments cannot perform measurements to arbitrary

precision. A meter stick commonly has markings 1 millimeter (mm) apart, so distances shorter than that cannot be measured accurately with a meter stick.

• We report only significant digits—those whose values we feel sure are accurately measured. There are two basic rules: – (i) the last significant digit is the first uncertain digit– (ii) when multiply/divide numbers, the result has no more

significant digits than the least precise of the original numbers.

The tests and exercises in the textbook assume there are 3 significant digits.

Scientific Notation and Significant Digits

• Scientific notation is simply a way of writing very large or very small numbers in a compact way.

• The uncertainty can be shown in scientific notation simply by the number of digits displayed in the mantissa

9

8

10088.18780000000010.0

10998.2299792485

3105.1 2 digits, the 5 is uncertain.

3 digits, the 0 is uncertain.31050.1

Percent error• Measurements made during laboratory work

yield an experimental value • Accepted value is the measurements

determined by scientists and published in the reference table.

• The difference between and experimental value and the published accepted value is called the absolute error.

• The percent error of a measurement can be calculated by

Percent error = accepted value

X 100%experimental value – accepted value

(absolute error)

Lab period

• Lab report format

Class work

Homework1. Read and sign the Lab Requirement Letter –

be sure to include both your signature and your parent or guardian’s signature.

2. Read and sign the Student Safety Agreement – both your signature and your guardian’s signature.

3. Reading assignment: 1.1 – 1.6,

p. 29: #1.1, 1.3, 1.9

9/6 do now• The micrometer (1 μm) is often called the micron. How

many microns make up 1.0 km?

• Homework questions?• Quiz tomorrow – on homework assignments

Objectives (B day)

• Sign up on mastering physics – do assignments

• Math review – class work

Register Mastering Physics

(See instructions at mastering physics sign up info)• Go to http://www.masteringphysics.com• Register with the access code in the front of theaccess kit in your new text, or pay with a creditcard if you bought a used book.• WRITE DOWN YOUR NAME AND PASSWORD• Log on to masteringphysics.com with your newname and password.• The VC zip code is 12549• The Course ID: MPLABARBERA1010

Mastering physics due by 11:00 pm tonight

9/7 do now (A day)

• quiz

9/7 Objectives (A day)

• Vector review

• Lab report requirement

There are two kinds of quantities…

• Vectors have both magnitude and direction • displacement, velocity,

acceleration

• Scalars have magnitude only • distance, speed, time, mass

x

y

o

p(x1, y1)y1

x1

Two ways to represent vectors

Vectors are symbolized graphically as arrows, in text by bold-face type or with a line/arrow on top.

Magnitude: the size of the arrow

Direction: degree from East

Vectors are represent in a coordinate system, e.g. Cartesian x, y, z. The system must be an inertial coordinate system, which means it is non-accelerated.

Geometric approach

Algebraic approach

Aθ

Magnitude: |R| = √x12 +y1

2

Direction: θ = tan-1(y1/x1)

Aθ

θ

Equal and Inverse Vectors

Equal vectors have the same length and direction.

Inverse vectors have the same length, but opposite direction.

A

-A

Head and tail method Parallelogram method

Graphical Addition of Vectors: “Head and tail ” & “parallelogram”

C is called the resultant vector!

E E

E is called the equilibrant vector!

Vector Addition Laws

Commutative Law: a + b = b + a

Associative Law: (a + b) + c = c + (b + a)

Subtract vectors: adding a negative vector

Component Addition of Vectors

1) Resolve each vector into its x- and y-components.Ax = Acos Ay = Asin

Bx = Bcos By = Bsin etc.

2) Add the x-components together to get Rx and the y-components to get Ry.

Component Addition of Vectors

3) Calculate the magnitude of the resultant with the Pythagorean Theorem (|R| = Rx

2 + Ry2).

4) Determine the angle with the equation = tan-1 Ry/Rx.

Algebraic Addition of Vectors

A

BAx

Ay

Bx

By

R

Rx

Ry

Ax = AcosA Ay = AsinA

Bx = BcosB By = BsinB

θA

θB

Rx = Ax + Bx

Ry = Ay + By

|R| = Rx2 + Ry

2.

= tan-1 Ry/Rx

θ

• Homework

1. Reading assignment: 1.7 – 1.9• p. 30 #31, 41, 43

Lab Period

• Lab 1: Vector Addition

• Objective: To compare the experimental value of a resultant of several vectors to the values obtained through graphical and analytical methods.

• Equipment: A force table set

9/10 do now vi vf

The direction of the change in velocity is best shown by

A B

C B

E

Objectives (B day)

• Quiz corrections – count as a grade

• Homework questions?

• Unit vector

Unit vectors • A unit vector is a vector that has a magnitude of 1, with

no units. Its only purpose is to point, or describe a direction in space.

• Unit vector is denoted by “^” symbol.• For example:

– represents a unit vector that points in the direction of the + x-axis

– unit vector points in the + y-axis

– unit vector points in the + z-axisk

j

i

ij

x

y

zk

• Any vector can be represented in terms of unit vectors, i, j, k

Vector A has components:

Ax, Ay, Az

A = Axi + Ayj + Azk

In two dimensions:

A = Axi + Ayj

Magnitude and direction of the vector

The magnitude of the vector is

|A| = √Ax2 + Ay

2 + Az2

The magnitude of the vector is

|A| = √Ax2 + Ay

2

In two dimensions:

The direction of the vector is

θ = tan-1(Ay/Ax)

In three dimensions:

Adding Vectors By Component

s = a + bWhere a = axi + ayj & b = bxi + byjs = (ax + bx)i + (ay + by)jsx = ax + bx; sy = ay + by s = sxi + syjs2 = sx

2 + sy2

tansy / sx

example

• Determine 1. The x, y, z component of A

2. the magnitude A

3. the direction of vector A with +x

A =(3i + 4j ) mGiven vector:

example

a. Is the vector A = i + j + k a unit vector?

b. Can a unit vector have any components with magnitude greater than unity? Can it have any negative components?

c. If A = a (3.0 i + 4.0 j ) where a is a constant, determine the value of a that makes A a unit vector.

Example – using unit vectors

• Its magnitude = (√ 82 + 112 + 102 ) m = 17 m

B =(4 i - 5 j + 8 k ) mA =(6 i + 3 j - k ) m

• Find the magnitude of the displacement 2A - B

2A - B=(8 i + 11 j - 10 k ) m

Given the two displacement

• Class work

• Homework– Read 1.9 #1.49; 1.69; 1.75– Mastering physics – due Thu. 9/13, 11:00 pm

– Chapter 1 test is Friday

9/11 do now

F2

F1

o

a. b.

c. d.

e.

Two forces F1 and F2 are acting at a point O, as

shown below.

Which of the fooling is very nearly the resultant vector of F2 – F1?

Objectives (A day)

• Homework questions?

• Multiplication of vectors

Multiplication of Vectors:

Scalar or Vector

Product of a Scalar and a Vector

Vector

Product of Two Vectors

Multiplication of Vector by Scalar produce a vector

• Examples:• momentum p = mv• Net force F = ma• Result

• A vector with the same direction, a different magnitude and perhaps different units.

Multiplication of vector by vector produce a scalar

• Scalar product or dot product, yields a result that is a scalar quantity

• Examples:

• work W = F d• Result

• A scalar with magnitude and no direction.

Scalar product (Dot Product)

|C| = A B|C| = AB cos|C| = AxBx + AyBy + AzBz

A

B

Commutative property of scalar product

A ∙ B = B ∙ A

• Scalar product of parallel vectors:

A∙A = |A||A|cos0o = |A|2 AA

• Scalar product of anti-parallel vectors:

A-A

A∙(-A) = |A||A|cos180o = -|A|2

The sign of the product as a result of the angle between two vectors

θ < 90o

θ = 90o

θ > 90o

A∙B > 0

A∙B = 0

A∙B < 0

W = F∙d

• When a constant force F is applied to a body that undergoes a displacement d, the work done by the force is given by

The work done by the force is

• positive if the angle between F and d is between 0 and 90o (example: lifting weight)

• Negative if the angle between F and d is between 90o and 180o (example: stop a moving car)

• Zero and F and d are perpendicular to each other (example: waiter holding a tray of food while walk around)

Application of scalar product

Scalar product of unit vectors

Parallel unit vectors

i ∙ i = 1

j ∙ j = 1

k ∙ k = 1

Anti-Parallel unit vectors

i ∙ j = j ∙ i = 0

j ∙ k = k ∙ j = 0

i ∙ k = k ∙ i = 0

Mathematic meaning of scalar product

Aθ

A∙B = A (Bcosθ)

Bcosθ

Component of B along A

Projection of B on A

A∙B = B (Acosθ)

Projection of B on A

Acosθ

- Comp. o

f A lo

ng B

B

Vector multiplications obeys distributive law

(A + B)∙C = A∙C + B∙C

example

A∙j = ?

A∙j = (Axi + Ayj + Azk)∙j = Ay

Component of A along y-Axis

Finding the angles with the scalar product

• Find the dot product and the angle between the two vectors

A · B = |A||B|cosθ=

If cosθ is negative, θ is between 90o and 180o

AxBx + AyBy + AzBz

|A| = √Ax2 + Ay

2 + Az2

|B| = √Bx2 + By

2 + Bz2

|A||B|

A · Bcosθ = =

AxBx + AyBy + AzBz

(√Ax2 + Ay

2 + Az2 )(√Bx

2 + By2 + Bz

2 )

example

A = 3i + 7kB = -i + 2j + k

A∙B = ?θ = ?

example• Find the scalar product A∙B of the two vectors in the

figure. The magnitudes of the vectors are A = 4.00 and B = 5.00

x

y

53.0o

130o

θ

θ = 130o – 53.0o = 77.0o

Ax = (4.00)cos53.0o = 2.407; Ay = (4.00)sin53.0o = 3.195

Az = 0;

Bx = (5.00)cos130o = -3.214; By = (5.00)sin130o = 3.830

Bz = 0

A∙B = (4.00)(5.00)cos 77.0o = 4.50

A∙B = AxBx + AyBy + AzBz = 4.50

AB

Finding the angles with the scalar product

• Find the dot product and the angle between the two vectors

A = 2i + 3j + k

B = -4i +2j - kcosθ = (A∙B) / (|A||B|)

A∙B = AxBx + AyBy + AzBz = -3

|A| = √Ax2 + Ay

2 + Az2 = √14

|B| = √Bx2 + By

2 + Bz2 = √21

cosθ = -0.175

θ = 100o

•Since cosθ is negative, θ is between 90o and 180o

homework

• Read 1.10; p. 31, #53, #55

Lab period

• Force table – unit vector, dot product,

9/12 do now

v1

v2

Q

PA B

C D

E

A particle, as shown below, moving from point to a point Q on a curved path has respectively has v1 and v2

respectively. The direction of the average force on the particle is best given by

Objectives (B day)

• Vector product (Cross product)

– Vector product or cross product, yields another vector

Vector (Cross Product)

•Application•Work = r F•Magnetic force F = qv B

•Result•A vector with magnitude and a direction perpendicular to the plane established by the other two vectors

Vector product (cross product)

• The vector product of two vectors A and B, also called the cross product, is denoted by

C = A x B

• The vector product is a vector. It has a magnitude and direction

Magnitude of C = A B

C = AB sin (magnitude)A

B

Where θ is the angle from A toward B, and θ is the smaller of the two possible angles.

Since 0 ≤ θ ≤ 180o, 0 ≤ sinθ ≤ 1, |A x B| is never negative.

Note when A and B are in the same direction or in the opposite direction, sinθ = 0;

The vector product of two parallel or anti-parallel vectors is always zero.

Direction is determined by Right Hand Rule

θ

A x B

A

B

Place the vector tail to tail, they define the plane

A x B is perpendicular to the plane containing the vectors A and B.

θ

A x B

A B

Right-hand rule: we follow the direction of the fingers to go from the A to B, then the thumb points in the direction of A x B

θ

B x AA

BB x A = - A x B

Vector product vs. scalar product

• Vector product: – A x B = ABsinθ (magnitude)– Direction: right-hand rule-perpendicular to the A, B

plane

• Scalar product: – A∙B = ABcosθ (magnitude)– It has no direction.

• When A and B are parallel– AxB is zero– A∙B is maximum

• When A and B are perpendicular to each other– AxB is maximum– A∙B is zero

Calculating the vector product using components

• If we know the components of A and B, we can calculate the components of the vector product.

• The product of any vector with itself is zero＊i x i = 0; j x j = 0; k x k = 0

• Using the right hand rule and A x B = ABsinθ＊i x j = -j x i = k; ＊j x k = -k x j = i;＊k x i = - i x k = j

A x B = (Axi + Ayj + Azk) x (Bxi + Byj + Bzk)

= AxByk - AxBzj

– AyBxk + AyBzi

+ AzBxj - AzByi

A x B = (AyBz – AzBy) i + (AzBx - AxBz) j + (AxBy – AyBx) k

If C = A x B then

Cx = AyBz – AzBy; Cy = AzBx - AxBz; Cz = AxBy – AyBx

The vector product can also be expressed in determinant form as

A x B =

A x B =(AyBz – AzBy) i + (AzBx - AxBz) j + (AxBy – AyBx) k

i j k i j k

Ax Ay Az Ax Ay Az

Bx By Bz Bx By Bz

+ direction- direction

Multiplication of Vector by Vector (Cross Product)

C = A BA

B

i j kC = Ax Ay Az

Bx By Bz

Example 1.12• Vector A has magnitude 6 units and is in the direction of

the + x-axis. Vector B has magnitude 4 units and lies in the xy-plane, making an angle of 30o with the + x-axis (fig. 1.32). Find the vector product C = A x B.

x

y

z

A

B30o

C

C = A x B = ABsinθ = (6)(4)sin30o = 12

From the right-hand rule, the direction of C is along the z-axis, C = 12k

We can also find C using components of A and B

Example 1.12• Vector A has magnitude 6 units and is in the direction of

the + x-axis. Vector B has magnitude 4 units and lies in the xy-plane, making an angle of 30o with the + x-axis (fig. 1.32). Find the vector product C = A x B.

x

y

z

A

B30o

C

We can also find C using components of A and B

A = 6i

B = 3.46i + 2j

C = (6i) x (3.46i + 2j) =12k

example• Find the vector product A X B (expressed in unit vectors)

of the two vectors given in the figure.

70o

A (3.60 m)

B (2.4 m)

30oC = A x B = ABsinθ = (3.60 m)(2.4 m)sin140o

(C = 5.6 k) m

From the right-hand rule, the direction of C is along the z-axis,

homework

• Class work

• Homework– Read: 1.10; p. 31 #1.59, 1.89

9/13 do now

d meters

t sec

P Q

vi vf A. B.

C. D.

E.

• A particle of mass m at point P moves along the x-axis and changes its velocity from vi at P to vf at Q, as shown below. The direction of average acceleration is best given by

9/13 Objectives (B day)

• AP Function Review

• Chapter 1 review

• Lab

• Homework – function sheet

• Chapter 1 test tomorrow

Lab period

• Lab 1 – part 4 – cross product

9/7 do now (A day)

• Antarctica is roughly semicircular, with a radius of 2000 km. The average thickness of its ice cover is 3000 m. How many cubic centimeters of ice does Antarctica contain? (ignore the curvature of Earth)

1.9 x 1022 cm3