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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:

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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


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PHY 107 Home page:

http://www.physics.buffalo.edu/phy107/


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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


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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]


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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.


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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)


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Physics

Classical (before 1900) (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:

  • 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)


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I

-

+

V

  • 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

Example: Ohm’s law The ratio V/I for a conductor is a constant known as the resistance R

(1-3)


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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)


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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)


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A

earth

C

equator

B

The meter

1 meter  AB/107

(1-6)


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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)


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The kilogram (kg)

It is defined as the mass equal to the mass of a cylinder made of platinum-iridium made by the International Bureau of Weights and Measures. All other standards are made as copies of this cylinder

(1-8)


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N

earth

S

(1-9)

The second (s)

The second is defined as the duration of the mean solar day divided by 86400

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


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Most common

10-12 please correct!

Examples: 1 km = 1000 m 1 ms = 10-3 s

(1-10)


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.

.

m

t

h

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

h = gt2/2 Solve this equation for g

g = 2h/t2

Units of left Units of right hand side hand side

Units (g) = units (2h/t2) = m/s2

Note: The number 2 has no units (1-11)

g

=


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m

a

F

Example 2:Find the units of force

Newton’s second law

F = ma

Units of left Units of right hand side 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)

=


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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



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L2

L1

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

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.

(1-15)


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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)


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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:

.

.

m

t

h

g

h = gt2/2 [LHS] = [RHS]

[LHS] = [h] = [L]

[RHS] = [gt2/2] = [g][t2] = [L][T]-2[T]2 = [L]

Indeed [LHS] = [RHS] = [L] as might be expected from the equation h = gt2/2 which we know to be true.

(1-17)


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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.

.

.

m

t

h

Example: Lets try the (incorrect) equation

h = gt2/3 [LHS] = [L]

[RHS] = [g][t2] = [L]

Even though [LHS] = [RHS] equation

h = gt2/3 is wrong!

(1-18)

g


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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)


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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: 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:or as: a

3. A vector is represented by an arrow whose length is proportional to the vector’s magnitude. The arrow has the same direction as the vector

(1-20)

a


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  • Vector addition (geometric method)

  • Recipe for determining R = A + B (see fig.a)

  • 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

A + B

B + A

(1-21)


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  • 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)


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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)


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  • Vector Subtraction (geometric method)

  • We are given vectors A and B are are asked to determine T = A – B

  • Determine –B from B

  • Add vector (-B) to vector A using the recipe of page (1-21)

A

B

(1-24)


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  • 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 oppositedirection of B

(1-25)


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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)


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Vector Components

Any vector V in the xy-planecan be written as a sum of two other vectors, one along the x-axis and the other along the y-axis. These are called the components of V

V = Vxi + Vyj Here i and j are the unit vectors along the 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)


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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)


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y

B

A

x

O

Vector addition (algebraic method)

(1-29)


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y

B

A

x

O

Vector subtraction (algebraic method)

(1-30)


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(1-31)

B

xyz is a right handed coordinate system

A

O

C


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  • 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)


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