Kinematics and Dynamics and Relativity

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13 Νοε 2013 (πριν από 3 χρόνια και 7 μήνες)

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Kinematics and

Dynamics and Relativity

Carlos Silva

September 30
th

2009

Isaac Newton (1643
-
1727)

Philosophie naturalis principia mathematica


Gravitational force


Motion laws


Principles of Mechanics:

Motion of bodies


Developed Calculus

Area of mathematics that deals with limits, derivatives, integrals


Newton tree (?)

(Cambridge, UK)

Kinematics and Dynamics

Kinematics


How to describe a motion of rigid bodies

Linear motion

Circular motion


Dynamics


How forces affect motion of rigid bodies

Force

Torque

Momentum conservation

Newton’s laws of motion


KINEMATICS

Linear motion

Time

General definition (t)

Clock reading (day, hour, minute, second)


Physical definition (
Δ
t
)


Elapsed time measured in seconds between two events

Δ t = time at event B


time at event A (always positive)

Position, change in position and distance

Position (x)

Coordinates of a point in space


Change in position (
Δ

x)

Difference betwen coordinates of two different positions


Δ x = x
B



x
A

(signal gives the direction)



Distance

Length of the path taken between position A and position
B (always positive)


Euclidean Distance (norm):








2
2
y
x
D




A

B

d

y

x

Δ
y

Δ
x

D

Velocity and Acceleration

Velocity

Rate of position change:

(derivative of position change in order to time)


Acceleration

Rate of velocity change:

(derivative of velocity change in order to time)


For constant acceleration and simple linear motions



t
v
a



t
x
v



2
0
2
1
t
a
t
v
x



Determine the motion

Constant speed / null acceleration

Motion with changing speed / constant acceleration









Describe what type of motion did the
object had in x:

KINEMATICS

Circular motion

Angle, angular velocity and acceleration

Angle (
θ
)

Measured in radians (360º = 2
π

rad
)


Angular velocity (
ω
)

Measured in rad s
-
1

Angular frequency is the magnitude of angular velocity


Frequency

(Hz=s
-
1
)

Number

of events per second

In this case is usual to measure in rpm (rotation per minute)


Linear quantities of a particle

Linear velocity

Linear acceleration






T



2
f
2



DYNAMICS

Free fall objects motion

Force

Aristotle proposed “force” has the reason why an object puts another in motion


Newton proposed that force is always a two object relation

Gravity force is an interaction between Earth and another object

Different forces may be acting on a single object

Forces act at the distance


Types of force

Gravitational force

Friction force (between two objects)

Electrical Force

Magnetic Force


Free fall motion

Velocity graph of a falling object (experiment by Galileo)








The acceleration is constant, regardless the mass! (9,8ms
-
2
)

This is the acceleration caused by
Gravitation Force

Gravitational acceleration g = 9,8ms
-
2

Gravity always “pulls” down

Weight is the quantity of the force that attracts us to the ground

(Weight in N, Mass in kg)


Projectile launch

Object launched with
horizontal speed





It always fall due to gravity



To achieve the longest
distance, launch at 45º



Mass, Center of Mass, Inertia

Mass

Property of a body (how much matter does a body has)

It becomes different form weight in places where the gravitational force is different
from
g

(moon)


Inertia

All corps maintain their state of motion (rest or constant velocity) if no force is applied



Center of Mass /Gravity

Average of every position of a body weighted by their mass

Point whose motion describes the object motion if all mass was concentrated in a
single point

Different from geometric center


Newton’s Laws

First law

If the sum of acting forces is zero, the center of mass continues in the same state of
motion




Second law

If the acting forces are not zero, the acceleration of the body is proportional to the
force





Third law

For each force, there is always an equal and opposite force




ma
F

Momentum

Conservative quantity of body (Ns)

If no external force is acting on a body, the body maintains its momentum

Product of mass by its velocity

This explains several phenomena:










mv
p

Ballerina spinning


I
M

Torque (Moment of Force)

Magnitude of the force applied to a rotational system (Nm)


Equivalent to the Force on circular motion










Power = torque
x
angular speed (Nms
-
1
)


rF


Friction Forces

Static friction force is usually higher that kinetic friction force

Centripetal force (circular motion)

Outward force (1)

It doesn’t exist


Inward force(2)

Conservation of momentum

(First Newton law)



Centripetal

force

Force required to make a body follow a curve path


Kinetic and Potential Energy

Kinetic Energy

The work that it is necessary to bring an object from rest to the present velocity:


Energy that a body possesses due to its motion:




Potential Energy

Energy stored in a body that can be transformed into other type of energy

Kinetic, thermal, chemical, elastic


Gravitational potential energy









2
2
1
mv
E
k

2
2
1

I
E
k

Catapult

mgh
E
p

DYNAMICS

Examples

Force acting on a spring

Hooke’s law

The deformation is proportional to the
applied force that causes deformation



Natural frequency




Elastic Potential Energy

x
k
F



m
k


2
2
1
x
k
E
p


The pulley

Allows to lift large masses into tall
heights


Gain mechanical advantage

2 pulleys, F/2

4 pulleys, F/4


Transfer mechanical forces across
axes

Crane

Motor

The lever

Based on the application of moments
-
force










What would be D
1

and D
2

in this case?

Hand trucks

Breaks

Spring board

Fishing rod

Ramp / inclined plane

Reduce the force applied to lift
at the expense of travelled
distance

Roman inclined plane

(Masada, Israel)

Gears

Transmits rotational forces
between axes (like pulleys)






Transforms rotational to linear
motion

2
1
1
2





Crankshaft

Transmits linear motion into
rotational motion







Crankset +pedal

Flyball governor

Automatically controls the
speed of the motor

Regulating the fuel admission, based on
the rotational acceleration

Watt

Flywheels

Used to store energy

Significant inertia


Used to attenuate peaks



High energy density

130 W∙h/kg, or ~ 500 kJ/kg

Typical capacities range

3

kWh to 133

kWh

Modern flywheel

for storage

(Beacon)

Pump Storage

Used for load balancing


Stores energy in the for of potential energy of water

Turbines (wind and water)

Machine that extracts energy
from a fluid flow







Wind turbine power



Betz limit (59%)

Horizontal, vertical

Pelton wheel

3
2
2
1
v
r
P


RELATIVITY

Relative velocity

3ms
-
1

3ms
-
1

3ms
-
1

3ms
-
1

3ms
-
1

3ms
-
1

1
1
1
ms
6
ms
3
ms
3








B
A
B
A
v
v
v
1
1
1
ms
0
ms
3
ms
3







B
A
B
A
v
v
v
1
1
1
ms
4
ms
10
ms
6











B
C
C
A
B
A
v
v
v
Postulates of Relativity

The Principle of Relativity (Galileo)


The laws by which the states of physical systems undergo change are not affected,
whether these changes of state be referred to the one or the other of two systems
in uniform translatory motion relative to each other.

The Principle of Invariant Light Speed


Light in vacuum propagates with the speed
c

(a fixed constant) in terms of any
system of inertial coordinates, regardless of the state of motion of the light source.

Special principle of relativity
:


If a system of coordinates K is chosen so that, in relation to it, physical laws hold
good in their simplest form, the
same

laws hold good in relation to any other system
of coordinates K' moving in uniform translation relatively to K


Time dilation


Relativity of simultaneity


Velocity of light cannot be exceeded



Mass of an object near speed of light seems to increase


Equivalence of mass and energy (E=mc
2
)


Twins paradox

A twin who makes a journey into space in a high
-
speed rocket will return home
to find he has aged less than his identical twin who stayed on Earth

0.9cms
-
1

0ms
-
1

????
ms
1
.
0
ms
ms
9
.
0
1
1
1











B
C
C
A
B
A
v
c
v
c
v
cms
-
1