PHY2048GENERAL PHYSICS 1

CHAPTER 1 Introduction

CHAPTER 2 Motion in one dimension

CHAPTER 3 Motion in two and three dimensions

CHAPTER 4 Newton’s Laws

CHAPTER 5 Applications of Newton’s Laws

CHAPTER 6 Work and Energy

CHAPTER 7 Conservation of energy

CHAPTER 8 Systems of particles and conservationof momentum

CHAPTER 9 Rotation

CHAPTER 10 Conservation of angular momentum

CHAPTER 11 The gravitational field

CHAPTER 12 Static equilibrium and elasticity

CHAPTER 13 Fluids

CHAPTER 14 Simple harmonic motion

? CHAPTER 15 Waves ?

CHAPTER 1

INTRODUCTION

• Units and standards of measurement • length • time • mass

• Conversion of units *

• Dimensional analysis

• Scientific notation *

• Prefixes

• Estimates and creative thinking • Fermi problems *

* Notes on the website

UNITS are critical to all of science and engineering and everyday life:

However, different countries/continents may have different systems of units

Europe: Meters, kilometers, grams, kilograms,

liters, !C ...

USA: Feet, miles, pounds, tons, pints, !F ...

Just like coinage ... Euros and Dollars

But like coinage there must be accepted conversions and agreements ... otherwise ...

“... two spacecraft teams [in Colorado and California] ... unknowingly were exchanging some vital information in different units on measurement.”

Mars Climate Orbiter: launched December 1998, crashed September 23, 1999.

From the Sun Sentinel ...

Origin of standardized units:LENGTH

FOOT:• Introduced by the Holy Roman Emperor Charlemagne (~800 CE) the foot was defined as the length of the king’s foot! In France it was called ... “le pied du roi”.

• Would vary from country to country and from one king to his successor.

METER:In 1670 Gabriel Mouton (a French vicar) proposed a unit of length based on the size of the Earth rather than the human anatomy. In 1791-1793 the meter was defined.

In 1736-1737 Pierre Louis Moreau de Maupertuis led an expedition to Lapland to measure the length of a degree along the meridian.

In 1793 the meter was defined as 110,000,000 of the pole

to equator quadrant of the Earth’s meridian passing close to Dunkirk, Paris and Barcelona.

Origin of standardized units:

LENGTH

Between 1795 - 1927 a “fallback” unit of length was based on the length of metal bars. For example, in 1889 the meter was defined as the distance between two lines on a standard bar of an alloy of platinum with 10% iridium measured at the melting point of ice. The bar was kept at the International Bureau of Weights and Measures near Paris. The estimated uncertainty in

measurement was 0.1→ 0.2 µm ( 1 µm = 10−6 m).

Origin of standardized units:LENGTH

• Today 1 meter is the distance light travels in

1

299,792,458th of a second.

Origin of standardized units:LENGTH

• Today 1 meter is the distance light travels in

1

299,792,458th of a second.

Origin of standardized units:TIME

SECOND:

Originally defined as 1

60 th of a minute, which was

1

60 th of an hour, which was 1

24th of a day. Basically,

it was determined by the rate of rotation of the Earth.

But there’s a problem ... of course!

Origin of standardized units:TIME

The length of the day ( 86,400 seconds) is not constant! It varies by a few-thousanths of a second each 24 hour period.

× 10−3s

1973 1980 1990 2000 2010

4

3

2

1

0

−1

Origin of standardized units:TIME

With the NBS-6 clock the second is defined as the time that passes during 9,192,631,770 atomic vibrations of a cesium atom. This translates to an accuracy of <1 s in 300,000 years.

NIST-F1, which began operation in 1999, has an accuracy of 1 s in 20 million years.

The NBS-6 “atomic clock” introduced in 1975

Origin of standardized units:TIME

Metrication gone mad? ...

Following the proposal of the metric system during the French Revolution, in 1792 the 24 hour day was re-defined to a 10-hour day. But, of course, the idea didn’t catch on! Some clocks produced in France during this period like this one combined a 10 hour clock (inner circle) and a 24 hour clock (outer circle).

Origin of standardized units:MASS

KILOGRAM:The mass of a block of a platinum-iridium alloy stored, since 1889, in a vault near Paris belonging to the International Bureau of Weights and Measures. 40 identical replicas were made a distributed to countries around the world.

A new definition based on fundamental natural constants is currently being sought.

Dimensional analysis

Most physical quantities have “dimension”, i.e., some ratio of length, time and mass.

** Do not confuse dimensions with units **

• Area: [L]× [L]⇒ [L]2. Possible units: m2, ft2, etc.

• Volume: [L]× [L]× [L]⇒ [L]3.Possible units: m3, ft3, etc.

• Speed:

distancetime

⇒[L][T]

⇒ [L][T]−1.

Possible units: m/s, ft/s, mi/h, etc.

• Acceleration:

speedtime

⇒[L][T][T]

⇒ [L][T]−2.

Possible units: m/s2, ft/s2, etc.

Dimension UnitLength ⇒ [L] m, ft, km, miTime ⇒ [T] s, min, hMass ⇒ [M] kg, lb, g

Dimensional analysis

Most physical quantities in this course have “dimension”,

i.e., some ratio of length, time and mass.

TABLE 1Dimensions of some physical quantities

Area [L ]2

Volume [L ]3

Velocity and speed [L ][T]−1

Acceleration [L ][T]−2

Frequency [T]−1

Force [M ][L][T]−2

Pressure Force

Area( ) [M ][L]−1[T]−2

Density Mass

Volume( ) [M ][L]−3

Energy [M ][L]2[T]−2

Power Energy

Time( ) [M ][L]2[T]−3

Momentum [M ][L][T]−1

Angular momentum [M ][L]2[T]−1

DISCUSSION PROBLEM 1.1:

Can you think of any physical quantities that do not have dimension?

Example 1: What is the dimension of 2πℓg

?

ℓ⇒ [L], g⇒ [L][T]−2 and 2π ⇒ dimensionless

∴ℓg⇒

[L][L][T]−2 ⇒ [T]2 ⇒ [T]

This is the periodic time of a simple pendulum.

Example 2: Use dimensional analysis (not units) to determine what physical quantity is described by

Force × speed?(Use Table 1 to get dimensions.)

Force × speed ⇒

[M][L][T]2

×[L][T]

=[M][L]2

[T]3

⇒ Power .Note: an equation must be dimensionally consistent, i.e., the quantities of the left hand side must have the same dimension as quantities on the right hand side.

Question 1.1: The force (F) acting on an object that is

being swung around in a circle at the end of a string,

depends on the mass (m) and speed (v) of the object and

the radius (r) of the circle. How are these quantities

related?

Put

Because an equation must be dimensionally consistent,

then using Table 1, we have

Equating exponents we find

Hence,

We will see later that this is the expression for a

centripetal force.

F ∝mavbrc.

[M][L][T]−2 ∝[M]a[L]b[T]−b[L]c.

a = 1, b + c = 1 and b = 2.

∴F ∝ mv2

r.

c = −1.

Question 1.2: The periodic time of the swing of a

compound pendulum is

where I is the moment of inertia of the pendulum, m is its

mass, g is the acceleration of gravity and d is the distance

from the pivot point to the center of mass of the

pendulum. What is the dimension of I?

T = 2π I

mgd,

Since T = 2π Imgd

,

I = T2

4π2mgd ⇒ [T]2[M][L][T]−2[L]

= [M][L]2.

TABLE 2

In addition to the basic units, e.g., meter, kilogram and second, there are

sub-units, such as millimeters and nanoseconds. The prefixes milli- and

nano- denote multipliers of the basic units based on various powers of ten.

For example, 1 millimeter (1 mm) is 1× 10−3m.

Important common prefixes

Power Prefix Abbreviation

10−15 femto- f

10−12 pico- p

10−9 nano- n

10−6 micro- µ

10−3 milli- m

10−2 centi- c

10−1 deci- d

103 kilo- k

106 mega- M

109 giga - G

1012 terra- T

DISCUSSION PROBLEM 1.3:

Estimate how many quarts of milk (all kinds) are consumed each day at breakfast in the U.S.

DISCUSSION PROBLEM 1.2:

DISCUSSION PROBLEM 1.3: