Welcome

Course: AP Physics

Room: 207

Teacher: Mrs. LaBarberaEmail:diana.labarbera@valleycentralschools.org

Post session: Tue. – Fri.

Objectives• Introduction of AP physics curriculum

• Lab safety

• Sign in lab safety attendance sheet

• Chapter 1 - units, physical quantities, and vectors

Chapter 1- units, physical quantities, and vectors

1. Know the fundamental quantities and units of mechanics.

2. Be able to determine the number of significant figures in calculations.

3. Differentiate between vectors and scalars

4. Be able to add and subtract vectors graphically.

5. Be able to determine the components of vectors and to use them in calculations

6. Know the unit vector and be able to use them with components to describe vectors

7. Know the two ways of multiplying vectors.

1.1 The Nature of Physics

• Physics is an experimental science.

• Theories are formed through observation and experiments. However, no theory is ever regarded as the final and ultimate truth.

• All theories can be revised by new observations.

• All theories have a range of validity.

Percent error• Measurements made during laboratory work

yield an experimental value • Accepted value are 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)

• Identify the relevant concepts – determine target variable and the given quantities.

• Set up the problem – choose equations based on the known and unknown from Identify step.

• Execute the solution – “do the math”

• Evaluate your answer – “Does the answer make sense?”

1.2 Solving Physics Problems

I SEE

1.3 Standards and Units

Quantity Standards SI unit (symbol)

Time Time required for 9,192,631,770 cycles of cesium microwave radiation

second (s)

Length Distance light travels in vacuum in 1/299,792,458 seconds

Meter (m)

Mass The mass of a particular cylinder of platinum-iridium alloy kept at International Bureau of Weights and Measures a Servres, France

kilogram (kg)

All other units can be expressed by combinations of these fundamental (base) units. The combined base units is called derived units.

• SI Fundamental Quantities And Units Of Mechanics

Derived units• Like derived dimensions, when we

combine base 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 tenPrefix 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

• The British System– Length: 1 inch = 2.54 cm– Force: 1 pound = 4.448221615260

N

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, – The dimension for Energy: E = ML2/T2

– The dimension for Impulse: J = ML/T

1.4 Unit consistency and conversions

• 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.

L =L

T2= L

Example

d = v / t

use dimensional analysis to check if the equation is correct.

L = (L ∕ T ) ∕ T

[L] ≠ L ∕ T2

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

Example: we wish to convert 2 miles into meters. (given conversion factors:1 miles = 1760 yards, 1 yd = 0.9144 m)

Conversion Strategies: I SEE•Identify the target units and the known conversion factors•Setup the problem using the given units and conversion factors to determine the unknown. Note units can be multiplied or divided like numbers. •Execute: do the math•Evaluate: “Does the answer make sense?”

myard

m

mile

yardmile 3218

1

9144.0

1

17602

Example 1.1

• The official world land speed record is 1228.0 km/h, set on 10/15/1997, by Andy Green in the jet engine car Thrust SSC. Express this speed in m/s.

Example 1.2

• The world’s largest cut diamond is the First Star of Africa. Its volume is 1.84 cubic inches. What is tis volume in cubic centimeers? In cubic meters?

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!!

Example

Suppose we want to convert 65 mph to ft/s or m/s.

Dimensional Analysis is simply a technique you can use to convert from one unit to another. The main thing you have to remember is that the GIVEN UNIT MUST CANCEL OUT.

hour

miles65

sec60

min1

min60

1

hour

mile

ft

1

5280

160601

52801165

s

ft95

s

ft95

ft

meter

281.3

1

281.31

195 sm /29

1.5 Uncertainty and Significant Figures

• 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

Example 1.3

The rest energy E of an object with rest mass m is given by Einstein’s equation

E = mc2

Where c is the speed of the light in vacuum (c = 2.99792458 x 108 m/s). Find E for an object with m = 9.11 x 10-31 kg.

Test Your Understanding 1.5

• The density of a material is equal to its mass divided by its volume. What is the density (in kg/m3) of a rock of mass 1.80 kg and volume 6.0 x 10-4 m3

1. 3 x 103 kg/m3

2. 3.0 x 103 kg/m3

3. 3.00x 103 kg/m3

4. 3.000x 103 kg/m3

5. Any of these

1.6 Estimates and orders of magnitude

Estimation of an answer is often done by rounding any data used in a calculation.

Comparison of an estimate to an actual calculation can “head off” errors in final results.

Example 1.4

• You are writing an adventure novel in which the hero escapes across the border with a billion dollars’ worth of gold in his suitcase. Is this possible? Would that amount of gold fit in a suitcase? Would it be too heavy to carry? (given 1 g of gold ≈ $10.00 and density of gold ≈ 1 g/cm3)

Test Your Understanding 1.6

• What is approximate number of teeth in all the mouths of everyone at VC?

1.7 vectors and vector additions

• There are two kinds of quantities…• Vectors have both magnitude and direction

• displacement, velocity, acceleration• Scalars have magnitude only

• distance, speed, time, mass

Vectors• Vectors show magnitude and direction, drawn as a ray.

Equal and Inverse Vectors

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

Vector addition• Vectors may be added graphically, “head to tail.” or

“parallegram

)()( CBACBACBAR

Commutative properties of vector addition

A

B

R

A + B = R

Resultant and equilibrant

R is called the resultant vector!

E is called the equilibrant vector!

E

Subtract vectors: adding a negative vector

example• At time t = t1, and object’s velocity is given by the vector v1 a

short time later, at t = t2, the object’s velocity is the vector v2. If the magnitude of v1 = the magnitude of v2, which one of the following vectors best illustrates the object’s average acceleration between t = t1 and t = t2

v1 v2

A B C D E

v2

-v1

v2 -v1

v1

v2 v2 –v1

Example 1.5• A cross-country skier

skies 1.00 km north and then 2.00 km east on a horizontal snow field. How far and in what direction is she from the starting point?

Test Your Understanding 1.7

• Two displacement vectors, S and T, have magnitudes S = 3 m and T = 4 m. Which of the following could be the magnitude of the difference vector S -T? (there may be more than one correct answer)

1. 9 m

2. 7 m

3. 5 m

4. 1 m

5. 0 m

6. -1 m

1.8 Components of vectors• Manipulating vectors graphically is insightful but difficult when

striving for numeric accuracy. Vector components provide a numeric method of representation.

• Any vector is built from an x component and a y component.

• Any vector may be “decomposed” into its x component using A*cos θ and its y component using A*sin θ (where θ is the angle the vector A sweeps out from 0°).

A

Aysin

A

Aycosyx AAA

The sign of the component depends on the angle from 0o

Y is positiveX is negative

Y is negativeX is negative

Example 1.6.

• a) what are the x and y components of vector D? the magnitude of the vector is D = 3.00 m and the angle α = 45o.

• b) what are the x and y components of vector E? the magnitude of the vector is E = 4.50 m and the angle β = 37.0o.

• Vector addition strategies

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.

Rx = Ax + Bx Ry = Ay + By

3) Calculate the magnitude of the resultant with the Pythagorean Theorem

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

22yx RRR

Doing vector calculations using components

Finding the direction of a vector sum by looking at the individual components

• Multiplying a vector by a scalar

• Multiplying a vector by a positive scalar changes the magnitude (length) of the vector, but not its direction.

A

D =2A2A is twice as long as A

• Multiplying a vector by a negative scalar changes the magnitude (length) of the vector and reverse its

direction.

A

D = -3A

-3A is three times as long as A and points in the opposite

direction.

Dx = 2Ax, Dy = 2Ay

Dx = -3Ax, Dy = -3Ay

Example 1.7

• Three players are brought to the center of a large, flat field, each is given a meter stick, a compass, a calculator, a shovel, and the following three displacements:

– 72. 4 m 32.0o east of north

– 57.3 m 36.0o south of west

– 17.8 m straight south

• The three displacements lead to the point where the keys to a new Porsche are buried. Two players start measuring immediately, but the winner first calculates where to go. What does she calculate?

Example 1.8• After an airplane takes off, it travels 10.4 km west, 8.7

km north, and 2.1 km up. How far us it from the takeoff point?

Test Your Understanding 1.8

• Two vectors A and B both lie in the xy-plane.

a. Is it possible for A to have the same magnitude as B but different components?

b. Is it possible for A to have the same components as B but a different magnitude?

1.9 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:

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

Adding Vectors By Component using unit vector representation

example

a. Is the vector a unit vector?

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

c. If , where a is a constant, determine the value of a that makes A a unit vector.

A = + +i j k

A = a (3.0 + 4.0 )i j

Example 1.9

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

E =(4 - 5 + 8 ) mi j kD =(6 + 3 - ) mi j k

• Find the magnitude of the displacement 2D - E

2D - E =(8 + 11 - 10 ) mi j k

Given the two displacement

Test Your Understanding 1.9

• Arrange the following vectors in order of their magnitude, with the vector of largest magnitude first.

a.A = (3i + 5j – 2k) m

b.B = (-3i + 5j – 2k) m

c.C = (3i – 5j – 2k) m

d.D = (3i + 5j + 2k) m

1. A scalar Product

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

• Example: work W = F d the Result is a scalar with magnitude and no direction.

• Scalar product is commutative:

cosBABAC

1.10 Products of Vectors

C = AxBx + AyBy + AzBz

ABBA

BCACBAC

)(

1.25

cosBABAC

//BAC

BAC //

The sign of the scalar product

• Scalar product of same vectors:

A∙A = |A||A|cos0o = |A|2 AA

• Scalar product of opposite vectors:

A-A

A∙(-A) = |A||A|cos180o = -|A|2

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

Calculating the scalar product using components

Parallel unit vectors

i ∙ i = 1

j ∙ j = 1

k ∙ k = 1

perpendicular unit vectors

i ∙ j = j ∙ i = 0

j ∙ k = k ∙ j = 0

i ∙ k = k ∙ i = 0

zzyyxx BABABABAC

example

A∙j = ?

A∙j = (Axi + Ayj + Azk)∙j = Ay

Component of A along y-Axis

A = Axi + Ayj + Azk

Example 1.10 Calculating a scalar product• Find the vector product A∙B of

the two vectors in the diagram. The magnitudes of the vectors are A = 4.00 and B = 5.00

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

)()(cos

222222zyxzyx

zzyyxx

BBBAAA

BABABA

BA

BA

example

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

A∙B = ?θ = ?

Example 1.11

• Find the angel between the two vectors:A = 2i + 3j + k and B = -4i + 2j - k

The vector product

Termed the “cross product.” Result is a vector with magnitude and a direction perpendicular to the plane established by the other two vectors.

Direction is determined by Right Hand Rule

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

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

B x A = - A x B

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.

BABAC

ABBAC

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

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

direction of +x axis. Vector B has a magnitude of 4 units and lies in the xy-plane, making an angle of 30o with the +x-axis. Find vector product BA

Check Your Understanding 1.10

• Vector A has magnitude 2 and vector B has magnitude 3. the angle φ between A and B is known to be either 0o, 90o, or 180o. For each of the following situations, state what the value of φ must be. (in each situation there may be more than one correct answer.)

1. A∙B = 0

2. A x B = 0

3. A∙B = 6

4. A∙B = - 6

5. │A x B│= 6