Physics 218: Mechanics Instructor: Dr. Tatiana Erukhimova

Lectures 2,3

Mechanics

• Various forms of motion:

- mechanical

- electromagnetic

- thermal, etc.

Mechanical form of motion is connected with displacements of various bodies relative to each other and with changes of the shapes of the bodies

Historical Notes

• History of mechanics linked with history of human culture

• Aristotle (384-322 B.C.); Physics

• Archimedes (3rd century B.C.), the law of lever, the law of equilibrium for floating bodies

• Galileo Galilei (1564-1624), the basic law of motion

Archimedes (3rd century B.C.), the law of lever, the law of equilibrium for floating bodies

GIVE ME A PLACE TO STAND AND I WILL MOVE THE EARTH

Galileo Galilei

1564-1642

“Father of modern science”

Was the first to apply a scientific method: Put forward a hypothesis, verify it by experiment, describe it with a mathematical model

Insisted that language of mathematics should describe the laws of nature and experiments should prove it.

No place for arguments based on beauty, religion etc.

Stephen Hawking: Galileo probably bears more of the responsibility for the birth of modern science than anybody else.

Albert Einstein

Achievements in physicsVerified that free-fall acceleration is independent on masses of bodies. This fact inspired Einstein’s General Relativity.

Formulated the Principle of Relativity, which laid the framework for Newton’s laws and inspired Einstein’s Special Relativity.

Proposed the Principle of Inertia, which was used (borrowed?) by Newton as his First Law.

Found that the period of a pendulum is independent on its amplitude.

He discovered it by observing swings of a bronze chandelier in the Cathedral of Pisa and using his pulse to measure the time!

Imagine you drop a light feather and a heavy coin from Albritton Bell Tower (138 ft.) Will they reach the ground at the same time?

384-322 B.C. Aristotle says : “No! The coin will land first because heavier objects fall faster than the lighter ones, in direct proportion to weight”.

1800 years later Galileo says : “Yes! A coin and a feather will land together if there is no air resistance!”

1564-1642

Free fall

g-positive!

On planet Earth, if you neglect

air resistance, any body which is dropped will experience a constant acceleration, called g, independent of its size or weight.

g=9.8 m/s2=32 ft/s2

a

v

a = g = const for all bodies independently on their masses

Galileo Galilei (1564-1624), the basic law of motion

22 /32/8.9 sftsmg

Galileo's “Law of Falling Bodies” distance (S) is proportional to time (T) squared

Galileo’s notes

Free fall

Falling with air resistance

A New Era of Science

Newton’s law of gravitation

Clockwork universe

1905 Albert Einstein

"Gravitation cannot be held responsible for people falling in love.“ Albert Einstein

•Derivatives

•Indefinite integrals

•Definite integrals

•Examples

Overview of Today’s Class

Quiz

1. If f(x)=2x3 what is the derivative of f(x) with respect to x?

•6x

•8x2

•I don’t know how to start

•6x2

1. If f(x)=2x3 evaluate dxxf )(

•2x4

•I don’t know how to start

•0.5x4+Const

•2x2/3

1. If f(x)=2x evaluate

3

2

)( dxxf

•2x3/3

•I don’t know how to start

•5

•x2+Const

Derivatives

A derivative of a function at a point is a slope of a tangent of this function at this point.

x

xfxxf

dx

dfx

)()(lim

0

1)( nn

nkxdx

kxd

dx

df

dx

dg

dx

df

dx

gfd

)(

Derivatives

x

xfxxf

dx

xdfx

)()(lim

)(0

,)( 1 n

n

nktdt

ktd

dt

dx

dt

dg

dt

df

dt

gfd

)(

Derivatives

dt

dgf

dt

dfg

dt

fgd

)(

t

txttx

dt

tdxt

)()(lim

)(0

or

0)(

dt

Constd

Function x(t) is a machine: you plug in the value of argument t and it spits out the value of function x(t).Derivative d/dt is another machine: you plug in the function x(t) and it spits out another function V(t) = dx/dt

Derivative is the rate at which something is changing

Velocity: rate at which position changes with timeVdt

dx

adt

dV Acceleration: rate at which velocity changes with time

Force: rate at which potential energy changes with position

Fdx

dU

Derivative is the rate at which something is changing-Size of pizza with respect to the price-Population of dolphins with respect to the sea temperature…………………

GDP per capita

dt

d )GDP(

Quiz

6x2+5

If what is the derivative of x(t) with respect to t?2

2

1)( attx

If f(x)=2x3+5x what is the derivative of f(x) with respect to x?

at

Indefinite integral(anti-derivative)

A function F is an “anti-derivative” or an indefinite integral of the function f

dxxfF )(

if

)(xfdx

dF

Indefinite integral(anti-derivative)

)()(

xfdxdx

xdf Const

dx

xdfConstxf

dx

d )())((

Constkxn

dxkxdxxf nn

1

1

1)()(

nkxxf )(

)1

1( 1 Constkxndx

d n

)()1

1( 1 Const

dx

dkx

ndx

d n

01

1 nkxn

nnkx

n – integer except n=-1

Definite integral

)()()( AFBFFdxxfB

A

BA

F is indefinite integral

Definite integral

Integrals

Indefinite integral:

Constkxn

dxkx nn

1

1

1)(

n – any number except -1

Definite integral:

11

1

1

1

1

nn

B

A

n kAn

kBn

dxkx

Gottfried Leibniz

dxxfF )(

)(xfdx

dF

These are Leibniz’ notations: Integral sign as an elongated S from “Summa” and d as a differential (infinitely small increment).

1646-1716

Leibniz-Newton calculus priority dispute

Motion in One Dimension (Chapter 2)

We consider a particle

)(tx - as time goes, the position of the particle changes

Velocity is the rate at which the position changes with time

Average velocity:

t

x

tt

txtxv

initialfinal

initialfinalave

)()(

dt

tdxv

)(

You travel from CS to Houston. First 20 miles to Navasota you cover in 20 min. You make a 10 min stop in Navasota and continue for another 20 min until you reach Hempstead which is 20 miles from Navasota. There you make a 15 min stop for lunch. Then you continue the remain 50 miles to Houston and reach it in 35 min. Find your average velocity.

Acceleration is the rate at which the velocity changes with time

Average acceleration

t

v

tt

tvtva

initialfinal

initialfinalave

)()(

dt

tdva

)(

)0()0(2

1)( 2 xtvtatx c

)0()( vtatv c

))()((2)()( 1212

22 txtxatvtv c

dt

tdvta

dt

tdxtv

)()(

)()(

€

x(t) = v(t)dt∫v(t) = a(t)dt∫

or

x(t) = x0 + v(t)dt0

t

∫

v(t) = v0 + a(t)dt0

t

∫

If a=ac=Const:

)0()0(2

1)( 2 xtvtatx c

)0()( vtatv c

))()((2)()( 1212

22 txtxatvtv c

A “police car” problem

x=0 x1 x2

V3=20m/sa=0

ap=kt

x2 – x1 = 3.5 km

V1=30m/s V2=40m/s

You start moving from rest with constant acceleration. There is a police car hiding behind the tree. The policeman has a metric radar. He measures your velocity to be 30 m/s. While the policeman is converting m/s to mph, you continue accelerating. You meet another police car. This policeman measures your velocity to be 40 m/s. You also notice the police, drop your velocity to 20 m/s and start moving with a constant velocity. However, it is too late. This police car starts chasing you with acceleration kt (k is a constant). After some distance he catches you.

a=const

V(t=0)=0

A “police car” problem

x=0 x1 x2

V3=20m/sa=0

ap=kt

x2 – x1 = 3.5 km

V1=30m/s V2=40m/s

1. What was your acceleration before you meet the second police car?2. How long did you travel from x1 to x2? 3. Find x1

4. At which distance does the police car catch you?5.Convert the velocity from m/s to mph

a=const

V(t=0)=0

Have a great day!

Reading: Chapter 1