2.e) Acceleration Coordinate systems and motions

3.a) Polar basis Coordinate systems and motions

Velocity and acceleration in polar coordinates

3.b) velocity-acceleration in polar basis

Velocity and acceleration in cylindric coordinates

Velocity and acceleration in Spherical coordinates

4.34M

Category:

physics

Institut de Physique et Ingénierie

1. Mechanics-L1

2. Kinematics

•Coordinate systems and motions
1)Units
2)Position-velocity-acceleration-cartesian coordinates
3)Polar and Cylindric coordinates
4)Introduction to Ellipse -Examples
5)Spherical coordinates
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3. Length

1. Units
Length
Time
Coordinate systems and motions
Historical definition
1 meter: 1/10000 of the quarter
of Earth’s meridian
1 seconde: Fraction 1/86400 of
a day (Earth ‘s revolution on its
axis) 1 day =24*60*60=86400 s
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4. 1. Units

Coordinate systems and motions
Modern definition with the speed of light:
c= 299 792 458 m/s
1 meter: distance that travels light during 1/c
seconds =3.34… ns
1 second: duration of 9 192 631 770 periods of the
radiation corresponding to the transition between
the two hyperfine levels of the ground state of
the cesium 133 atom”
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5. Kinematics

•Coordinate systems and motions
1)Units
2)Position-velocity-acceleration-cartesian
coordinates
3)Polar and Cylindric coordinates
4)Introduction to Ellipse -Examples
5)Spherical coordinates
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6.

2.a) Position
Position-vector
Coordinate systems and motions
of a point M:
1 dimensions
An origin: O
A direction : axis Ox
A unit vector:
whose norm is 1: A
component :
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7. Position-vector of a point M:

2.a) Position
Position-vector
Coordinate systems and motions
of a point M:
2 dimensions
An origin: O
Two directions : axis Ox and Oy
Two unit vectors:
and
whose
norms are 1
and
Two components:
Orthonormal basis
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8. Position-vector of a point M:

2.a) Position
Coordinate systems and motions
Position-vector of a point M:
3 dimensions
An origin: O
Three directions : axis Ox, Oy and Oz
Three unit vectors :
and
,
Three components:
and
,
Orthonormal basis
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9. Where is Wally ?

Interlude
Coordinate systems and motions
Where is Wally ?
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10.

Interlude
Coordinate systems and motions
Where is Wally ?
y
yw
Here !!
(xw ;yw)
O
XW
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x

11.

Interlude
Coordinate systems and motions
Where is Wally ?
But
also
here !!
(xw1 ; yw1)
O1
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12. Norm of a vector:

2.b) Norm
Coordinate systems and motions
Norm of a vector:
A(4,5)
O(0,0)
B(6,-2)
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13. Norm of a vector:

2.b) Norm
Coordinate systems and motions
Norm of a vector:
A(4,5)
Coordinates
O(0,0)
B(6,-2)
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14.

2.b) Norm
Coordinate systems and motions
Norm of a vector:
Pythagoras
A(4,5)
Coordinates
Norm (scalar quantity)
O(0,0)
B(6,-2)
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15. Norm of a vector:

2.b) Norm
Coordinate systems and motions
Norm of a vector:
A(4,5)
O(0,0)
B(6,-2)
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16. Norm of a vector:

2.b) Norm
Coordinate systems and motions
Norm of a vector:
A(4,5)
O(0,0)
B(6,-2)
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17. Norm of a vector:

2.b) Norm
Coordinate systems and motions
Norm of a vector:
A(4,5)
O(0,0)
B(6,-2)
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18. 2.c) Vectorial manipulations

Coordinate systems and motions
Norm of a vector 3 dimensions:
Scalar (number)
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19. 2.c) Vectorial manipulations

Coordinate systems and motions
Norm of a vector 3 dimensions:
Scalar (number)
Scalar product:
Scalar
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20. 2.c) Vectorial manipulations

Coordinate systems and motions
Norm of a vector 3 dimensions:
Scalar (number)
Scalar product:
Scalar
Vectorial cross product:
Vector
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21. 2.d) Velocity

Coordinate systems and motions
…..when going from A to B
Average velocity:
Velocity = Distance
Time
(m/s)
• Average velocity
over path AB
B
A
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22. 2.d) Velocity

Coordinate systems and motions
…..when going from A to B
Average velocity:
Velocity = Distance
Time
Tram
(m/s)
Car Foot
• Average velocity over path AB
B
A
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23. 2.d) Velocity

Coordinate systems and motions
…..when going from A to B
Average velocity:
Velocity = Distance
Time
Tram
(m/s)
Car Foot
• Average velocity over path AB
B
1.7 km ≠ 3.1 km ≠ 1.4 km
A
8 min ≠ 14 min ≠ 19 min
12.75 km/h
12.4 km/h
5.01 km/h
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24. 2.d) Velocity

Coordinate systems and motions
Average velocity
and real velocity …
B
Distance
(km)
B
B
AB =
1.7
k
m
Tram Stop 2
Tram Stop 1
A
∆t=8 min
tA
Velocity
(km/h)
tB
Time
35 km/h
20 km/h
12.75
10 km/h
tA
Tram Stop 1
Tram Stop 2
tB
Time
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A

25. 2.d) Velocity

Coordinate systems and motions
Instantaneous velocity:
…..when going from M(t) to M(t+dt)
Time-derivative of
position-vector
B
A
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26. Instantaneous velocity:

2.d) Velocity
Coordinate systems and motions
Instantaneous velocity: when going from M(t) to M(t+dt)
Time-derivative of
position-vector
B
B
A
A
The instantaneous velocity
Vector is tangent to the
trajectory at point M(t)
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27. 2.d) Velocity

Coordinate systems and motions
Projection onto Oxyz basis
Newton
1643-1727
Leibniz
1646-1716
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28.

2.e) Acceleration
Coordinate systems and motions
Acceleration = Velocity (m/s2)
Time
Average acceleration over path AB
Importance of vectors: example
uniform rotation with
constant.
A
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B

29. 2.e) Acceleration

Coordinate systems and motions
Importance of vectors: example
uniform rotation with
constant.
Acceleration = Velocity (m/s2)
Time
Average acceleration over path AB
A
Instantaneous acceleration at point M
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B

30. 2.e) Acceleration Coordinate systems and motions

Usain Bolt: world record 100 m 9 ’58 Berlin 16/08/2009
AB=100 m
∆t=9.58 s
vB=11.95 m/s
<v>=10.44 m/s
<a>=1.25 m/s2
B
<v> average velocity
A
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31. 2.e) Acceleration

Coordinate systems and motions
Acceleration in cartesian coordinates
Projection onto Oxyz basis
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32. Coordinate systems and motions

Summary with simple example
distance
position
velocity
Time
Time
integration
Time
derivation
velocity
acceleration
Time
Time
integration
Time
derivation
Acceleration
(if constant)
Time
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33. Kinematics

Coordinate systems and motions
1)Units
2)Position-velocity-acceleration-cartesian coordinates
3)Polar and Cylindric coordinates
4)Introduction to Ellipse -Examples
5)Spherical coordinates
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34. 3.a) Polar basis Coordinate systems and motions

Polar basis and time-derivation of unit vectors
!!!
angular velocity
radial, orthoradial
orthonormal direct basis
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35. Velocity and acceleration in polar coordinates

3.b) velocity-acceleration in polar basis
!!!
Coordinate systems and motions
Velocity and acceleration in polar coordinates
Position
Velocity
Acceleration
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36. 3.b) velocity-acceleration in polar basis

Coordinate systems and motions
Example: Karousel
y
1) We have r constant :
a) angular acceleration:
and
x
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37. 3.b) velocity-acceleration in polar basis

Coordinate systems and motions
Example: Karousel
y
1) We have r constant :
a) angular acceleration:
and
x
b) permanent regime
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38. 3.b) velocity-acceleration in polar basis

Coordinate systems and motions
Example: Karousel
y
and
1) We have r constant :
a) angular acceleration:
x
b) permanent regime
2) If r not constant ( motion) along
a) accelerated radial motion
y
x
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39. 3.b) velocity-acceleration in polar basis

Coordinate systems and motions
Example: Karousel
y
1) We have r constant :
a) angular acceleration:
and
x
b) permanent regime
2) If r not constant ( motion along
)
a) accelerated radial motion
y
b) uniform radial motion
y
x
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x

40. Velocity and acceleration in cylindric coordinates

3.c) cylindric coordinates
Coordinate systems and motions
Velocity and acceleration in cylindric coordinates
!!!
symmetry about
Oz axis
Position
Polar position
Cartesian z-position
Velocity
Polar velocity
Acceleration
Cartesian z-velocity
Polar acceleration
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Cartesian z-acceleration

41. Coordinate systems and motions

3.c) cylindric coordinates
Coordinate systems and motions
Example: Karousel
Motion in eletric and magnetic field
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42. Kinematics

43.

4.a) Ellipse
Coordinate systems and motions
Hello Ellipse
M
semi-minor
axis b
Focus F(0,c)
Focus F’(0,-c)
c
eccentricity e
semi-major axis a
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44. 4.a) Ellipse

Coordinate systems and motions
Where can we find ellipses ?
Planet orbits in the solar system
In gardens…..
….and in the metro
Propagation of ellipticaly polarized light
How many ellipses ?
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45. Kinematics

46.

5) Spherical coordinates
Coordinate systems and motions
Spherical basis
radial, orthoradial, azimutal
Position-vector and link with cartesian basis
contribution of new angle
Try at home !
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47. Velocity and acceleration in Spherical coordinates

5) Spherical coordinates
Coordinate systems and motions
Velocity and acceleration
in Spherical coordinates
Velocity
Acceleration
Try at home !
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48. 5) Spherical coordinates : examples

Coordinate systems and motions
Why spherical coordinates ?
Schrödinger equation
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