Kinematics and
Dynamics and Relativity
Carlos Silva
September 30th 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
Linear motion
KINEMATICS
Time
General definition (t)
Clock reading (day, hour, minute, second)
Physical definition (Δt)
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 = xB – xA (signal gives the direction)
Δx
y
A
d
Δy
Distance
Length of the path taken between position A and position
B (always positive)
Euclidean Distance (norm):
D
x2  y 2
D
B
x
Velocity and Acceleration
Velocity
Rate of position change:
(derivative of position change in order to time)
v
x
t
Acceleration
Rate of velocity change:
(derivative of velocity change in order to time)
a
v
t
For constant acceleration and simple linear motions
1
x  v0t  at 2
2
Determine the motion
Constant speed / null acceleration
Motion with changing speed / constant acceleration
Describe what type of motion did the
object had in x:
Circular motion
KINEMATICS
Angle, angular velocity and acceleration
Angle (θ)
Measured in radians (360º = 2π rad)
Angular velocity (ω)
Measured in rad s-1
Angular frequency is the magnitude of angular velocity
  2  f 
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
2
T
Free fall objects motion
DYNAMICS
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
F  ma
Third law
For each force, there is always an equal and opposite force
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
p  mv
M  I
This explains several phenomena:
Ballerina spinning
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:
Ek 
1 2
mv
2
Ek 
1 2
I
2
Potential Energy
Energy stored in a body that can be transformed into other type of energy
Kinetic, thermal, chemical, elastic
Gravitational potential energy
E p  mgh
Catapult
Examples
DYNAMICS
Force acting on a spring
Hooke’s law
The deformation is proportional to the
applied force that causes deformation
F  kx
Natural frequency

k
m
Elastic Potential Energy
Ep 
1
kx 2
2
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
Motor
Crane
The lever
Based on the application of momentsforce
Breaks
Hand trucks
Spring board
Fishing rod
What would be D1 and D2 in this case?
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)
2 1

1  2
Transforms rotational to linear
motion
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
Pelton wheel
Wind turbine power
1
P  r 2 v 3
2
Betz limit (59%)
Horizontal, vertical
RELATIVITY
Relative velocity
v A  3ms 1
vB  3ms 1
3ms-1
3ms-1
v A B  0ms 1
v A  3ms 1
vB  3ms 1
3ms-1
3ms-1
v A B  6ms 1
v A B  6ms 1
v AC  10ms 1
vC  B  4ms 1
3ms-1
3ms-1
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=mc2)
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
cms-1
v A B  0.9cms 1
v AC  cms 1
0.9cms-1
0ms-1
vC  B  0.1ms 1 ????