ref frames 1
DESCRIPTION
Describes gallilean transformations of reference frames.TRANSCRIPT
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FRAMES OF REFERENCE –IIROTATING FRAMES
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FRAMES OF REFERENCE –IIROTATING FRAMES
1 Galilean Transformations
2 Galilean Relativity
3 Non-inertial Frames
Uniformly accelerated frames
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Motion in Rotating Frames
First, an important result:
Frames of Reference Rotating frames
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Motion in Rotating Frames
First, an important result:
* 2* 2* 0* .05* 0.1* 0.15001* 0.20001* 0.25002* 0.30002* 0.35002* 0.400
Frames of Reference Rotating frames
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Motion in Rotating Frames
First, an important result:
* 2* 2* 0* .05* 0.1* 0.15001* 0.20001* 0.25002* 0.30002* 0.35002* 0.400Frames of Reference Rotating frames
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Motion in Rotating Frames
First, an important result:
* 2* 2* 0* .05* 0.1* 0.15001* 0.20001* 0.25002* 0.30002* 0.35002* 0.400Frames of Reference Rotating frames
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Motion in Rotating Frames
First, an important result:
* 2* 2* 0* .05* 0.1* 0.15001* 0.20001* 0.25002* 0.30002* 0.35002* 0.400Frames of Reference Rotating frames
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Motion in Rotating Frames
First, an important result:
* 2* 2* 0* .05* 0.1* 0.15001* 0.20001* 0.25002* 0.30002* 0.35002* 0.400Frames of Reference Rotating frames
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Motion in Rotating Frames
First, an important result:
* 2* 2* 0* .05* 0.1* 0.15001* 0.20001* 0.25002* 0.30002* 0.35002* 0.400Frames of Reference Rotating frames
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Motion in Rotating Frames
First, an important result:
* 2* 2* 0* .05* 0.1* 0.15001* 0.20001* 0.25002* 0.30002* 0.35002* 0.400Frames of Reference Rotating frames
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Motion in Rotating Frames
First, an important result:
* 2* 2* 0* .05* 0.1* 0.15001* 0.20001* 0.25002* 0.30002* 0.35002* 0.400Frames of Reference Rotating frames
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Motion in Rotating Frames
First, an important result:
* 2* 2* 0* .05* 0.1* 0.15001* 0.20001* 0.25002* 0.30002* 0.35002* 0.400Frames of Reference Rotating frames
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Motion in Rotating Frames
Rotating frame (x, y, z), coincident with inertial frame (x,y,z) at t
rotate0.pdf
Frames of Reference Rotating frames
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Motion in Rotating Frames
Rotating frame (x, y, z), coincident with inertial frame (x,y,z) at t
Vector
−→
B (t) on x-z plane (Say)
rotate0.pdf
Frames of Reference Rotating frames
M i i R i F
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Motion in Rotating Frames
Rotating frame (x, y, z), coincident with inertial frame (x,y,z) at t
Vector
−→
B (t) on x-z plane (Say)At t + ∆t,
rotateS0.pdf
Frames of Reference Rotating frames
M ti i R t ti F
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Motion in Rotating Frames
Rotating frame (x, y, z), coincident with inertial frame (x,y,z) at t
Vector
−→
B (t) on x-z plane (Say)At t + ∆t,
In inertial frame,
∆−→B =
−→B (t + ∆t) −
−→B (t)
rotateS1.pdf
Frames of Reference Rotating frames
M ti i R t ti F
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Motion in Rotating Frames
Rotating frame (x, y, z), coincident with inertial frame (x,y,z) at t
Vector
−→
B (t) on x-z plane (Say)At t + ∆t,
In inertial frame,
∆−→B =
−→B (t + ∆t) −
−→B (t)
In rotating frame, rotateS’0.pdf
Frames of Reference Rotating frames
M ti i R t ti F
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Motion in Rotating Frames
Rotating frame (x, y, z), coincident with inertial frame (x,y,z) at t
Vector
−→
B (t) on x-z plane (Say)At t + ∆t,
In inertial frame,
∆−→B =
−→B (t + ∆t) −
−→B (t)
In rotating frame, rotateS’1.pdf
Frames of Reference Rotating frames
Motion in Rotating Frames
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Motion in Rotating Frames
Rotating frame (x, y, z), coincident with inertial frame (x,y,z) at t
Vector
−→
B (t) on x-z plane (Say)At t + ∆t,
In inertial frame,
∆−→B =
−→B (t + ∆t) −
−→B (t)
In rotating frame,
∆
−→
B
=
−→
B (t + ∆t) −
−→
B
(t)
rotateS’1.pdf
Frames of Reference Rotating frames
Motion in Rotating Frames
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Motion in Rotating Frames
Rotating frame (x, y, z), coincident with inertial frame (x,y,z) at t
Vector
−→
B (t) on x-z plane (Say)At t + ∆t,
In inertial frame,
∆−→B =
−→B (t + ∆t) −
−→B (t)
In rotating frame,
∆
−→
B
=
−→
B (t + ∆t) −
−→
B
(t)
∴ ∆−→B = ∆
−→B
+ [−→B
(t) −−→B (t)]
rotateS’1.pdf
Frames of Reference Rotating frames
Motion in Rotating Frames
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Motion in Rotating Frames
Rotating frame (x, y, z), coincident with inertial frame (x,y,z) at t
Vector
−→
B (t) on x-z plane (Say)At t + ∆t,
In inertial frame,
∆−→B =
−→B (t + ∆t) −
−→B (t)
In rotating frame,
∆
−→
B
=
−→
B (t + ∆t) −
−→
B
(t)
∴ ∆−→B = ∆
−→B
+ [−→B
(t) −−→B (t)]
−→B rotated with frame
rotate5.pdf
Frames of Reference Rotating frames
Motion in Rotating Frames
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Motion in Rotating Frames
Rotating frame (x, y, z), coincident with inertial frame (x,y,z) at t
Vector
−→B
(t) on
x-z
plane (Say)At t + ∆t,
In inertial frame,
∆−→B =
−→B (t + ∆t) −
−→B (t)
In rotating frame,
∆
−→
B
=
−→
B (t + ∆t) −
−→
B
(t)
∴ ∆−→B = ∆
−→B
+ [−→B
(t) −−→B (t)]
= (−→Ω ×
−→B )∆t
rotate6.pdf
Frames of Reference Rotating frames
Motion in Rotating Frames
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Motion in Rotating Frames
Rotating frame (x, y, z), coincident with inertial frame (x,y,z) at t
Vector−→B
(t)
on x-z plane (Say)
At t + ∆t,
In inertial frame,
∆−→B =
−→B (t + ∆t) −
−→B (t)
In rotating frame,
∆
−→
B
=
−→
B (t + ∆t) −
−→
B
(t)
∴ ∆−→B = ∆
−→B
+ [−→B
(t) −−→B (t)]
= (−→Ω ×
−→B )∆t
∆−→B
∆t
= ∆−→B
∆t
+ Ω ×−→B
rotate7.pdf
Frames of Reference Rotating frames
Velocity and Acceleration in Rotating Frames
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Velocity and Acceleration in Rotating Frames
Frames of Reference Rotating frames
Velocity and Acceleration in Rotating Frames
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Velocity and Acceleration in Rotating Frames
d−→B
dt
in
= d−→B
dt
rot
+−→Ω ×
−→B
Frames of Reference Rotating frames
Velocity and Acceleration in Rotating Frames
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Velocity and Acceleration in Rotating Frames
d−→B
dt
in
= d−→B
dt
rot
+−→Ω ×
−→B
Put−→B = r, vin = vrot +
−→Ω × r
Frames of Reference Rotating frames
Velocity and Acceleration in Rotating Frames
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Velocity and Acceleration in Rotating Frames
d−→B
dt
in
= d−→B
dt
rot
+−→Ω ×
−→B
Put−→B = r, vin = vrot +
−→Ω × r
ain = dv
dt
in
=
Frames of Reference Rotating frames
Velocity and Acceleration in Rotating Frames
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Velocity and Acceleration in Rotating Frames
d−→B
dt
in
= d−→B
dt
rot
+−→Ω ×
−→B
Put−→B = r, vin = vrot +
−→Ω × r
ain = dv
dt
in
= d
dt(vrot +
−→Ω × r)
rot
+−→Ω × (vrot +
−→Ω
Frames of Reference Rotating frames
Velocity and Acceleration in Rotating Frames
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Velocity and Acceleration in Rotating Frames
d−→B
dt
in
= d−→B
dt
rot
+−→Ω ×
−→B
Put−→B = r, vin = vrot +
−→Ω × r
ain = dv
dt
in
= d
dt(vrot +
−→Ω × r)
rot
+−→Ω × (vrot +
−→Ω
Assume Ω constant,
Frames of Reference Rotating frames
Velocity and Acceleration in Rotating Frames
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Velocity and Acceleration in Rotating Frames
d−→B
dt
in
= d−→B
dt
rot
+−→Ω ×
−→B
Put−→B = r, vin = vrot +
−→Ω × r
ain = dv
dt
in
= d
dt(vrot +
−→Ω × r)
rot
+−→Ω × (vrot +
−→Ω
Assume Ω constant,
ain =
Frames of Reference Rotating frames
Velocity and Acceleration in Rotating Frames
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e oc ty a d cce e at o otat g a es
d−→B
dt
in
= d−→B
dt
rot
+−→Ω ×
−→B
Put−→B = r, vin = vrot +
−→Ω × r
ain = dv
dt
in
= d
dt(vrot +
−→Ω × r)
rot
+−→Ω × (vrot +
−→Ω
Assume Ω constant,
ain = dv
dt
rot
Frames of Reference Rotating frames
Velocity and Acceleration in Rotating Frames
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y g
d−→B
dt
in
= d−→B
dt
rot
+−→Ω ×
−→B
Put−→B = r, vin = vrot +
−→Ω × r
ain = dv
dt
in
= d
dt(vrot +
−→Ω × r)
rot
+−→Ω × (vrot +
−→Ω
Assume Ω constant,
ain = dv
dt
rot
+−→Ω ×
dr
dt
rot
Frames of Reference Rotating frames
Velocity and Acceleration in Rotating Frames
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y g
d−→B
dt
in
= d−→B
dt
rot
+−→Ω ×
−→B
Put−→B = r, vin = vrot +
−→Ω × r
ain = dv
dt
in
= d
dt(vrot +
−→Ω × r)
rot
+−→Ω × (vrot +
−→Ω
Assume Ω constant,
ain = dv
dt
rot
+−→Ω ×
dr
dt
rot
+−→Ω × vrot
Frames of Reference Rotating frames
Velocity and Acceleration in Rotating Frames
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y g
d−→B
dt
in
= d−→B
dt
rot
+−→Ω ×
−→B
Put−→B = r, vin = vrot +
−→Ω × r
ain = dv
dt
in
= d
dt(vrot +
−→Ω × r)
rot
+−→Ω × (vrot +
−→Ω
Assume Ω constant,
ain = dv
dt
rot
+−→Ω ×
dr
dt
rot
+−→Ω × vrot +
−→Ω × (
−→Ω ×
Frames of Reference Rotating frames
Velocity and Acceleration in Rotating Frames
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y g
d−→B
dt
in
= d−→B
dt
rot
+−→Ω ×
−→B
Put−→B = r, vin = vrot +
−→Ω × r
ain = dv
dt
in
= d
dt(vrot +
−→Ω × r)
rot
+−→Ω × (vrot +
−→Ω
Assume Ω constant,
ain = dv
dt
rot
+−→Ω ×
dr
dt
rot
+−→Ω × vrot +
−→Ω × (
−→Ω ×
= arot +
Frames of Reference Rotating frames
Velocity and Acceleration in Rotating Frames
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d−→B
dt
in
= d−→B
dt
rot
+−→Ω ×
−→B
Put−→B = r, vin = vrot +
−→Ω × r
ain = dv
dt
in
= d
dt(vrot +
−→Ω × r)
rot
+−→Ω × (vrot +
−→Ω
Assume Ω constant,
ain = dv
dt
rot
+−→Ω ×
dr
dt
rot
+−→Ω × vrot +
−→Ω × (
−→Ω ×
= arot + 2−→Ω × vrot
Frames of Reference Rotating frames
Velocity and Acceleration in Rotating Frames
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d−→B
dt
in
= d−→B
dt
rot
+−→Ω ×
−→B
Put−→B = r, vin = vrot +
−→Ω × r
ain = dv
dt
in
= d
dt(vrot +
−→Ω × r)
rot
+−→Ω × (vrot +
−→Ω
Assume Ω constant,
ain = dv
dt
rot
+−→Ω ×
dr
dt
rot
+−→Ω × vrot +
−→Ω × (
−→Ω ×
= arot + 2−→Ω × vrot +
−→Ω × (
−→Ω × r)
Frames of Reference Rotating frames
Velocity and Acceleration in Rotating Frames
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d−→B
dt
in
= d−→B
dt
rot
+−→Ω ×
−→B
Put−→B = r, vin = vrot +
−→Ω × r
ain = dv
dt
in
= d
dt(vrot +
−→Ω × r)
rot
+−→Ω × (vrot +
−→Ω
Assume Ω constant,
ain = dv
dt
rot
+−→Ω ×
dr
dt
rot
+−→Ω × vrot +
−→Ω × (
−→Ω ×
= arot + 2−→Ω × vrot +
−→Ω × (
−→Ω × r)
CoriolisFrames of Reference Rotating frames
Velocity and Acceleration in Rotating Frames
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d−→B
dt
in
= d−→B
dt
rot
+−→Ω ×
−→B
Put−→B = r, vin = vrot +
−→Ω × r
ain = dv
dt
in
= d
dt(vrot +
−→Ω × r)
rot
+−→Ω × (vrot +
−→Ω
Assume Ω constant,
ain = dv
dt
rot
+−→Ω ×
dr
dt
rot
+−→Ω × vrot +
−→Ω × (
−→Ω ×
= arot + 2−→Ω × vrot +
−→Ω × (
−→Ω × r)
Coriolis CentrifugalFrames of Reference Rotating frames
Apparent Forces in a rotating system
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Physical force on system:−→F =
−→F in = main
Frames of Reference Rotating frames
Apparent Forces in a rotating system
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Physical force on system:−→F =
−→F in = main
Force observed in rotating frame:
−→F rot = main −2m
−→Ω × vrot −m
−→Ω × (
−→Ω × r)
Frames of Reference Rotating frames
Apparent Forces in a rotating system
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Physical force on system:−→F =
−→F in = main
Force observed in rotating frame:
−→F rot = main −2m
−→Ω × vrot −m
−→Ω × (
−→Ω × r)
=−→F +
−→F fict
Frames of Reference Rotating frames
Apparent Forces in a rotating system
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Physical force on system:−→F =
−→F in = main
Force observed in rotating frame:
−→F rot = main −2m
−→Ω × vrot −m
−→Ω × (
−→Ω × r)
=−→F +
−→F fict
(a) Coriolis force
when the mass is moving in the rotating
frame
Frames of Reference Rotating frames
Apparent Forces in a rotating system
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Physical force on system:−→F =
−→F in = main
Force observed in rotating frame:
−→F rot = main −2m
−→Ω × vrot −m
−→Ω × (
−→Ω × r)
=−→F +
−→F fict
(a) Coriolis force
when the mass is moving in the rotating
frame
(b) Centrifugal Force
evident even when the mas
Frames of Reference Rotating frames
Centrifugal force
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Is familiar to us!
centrifugal0.pdf
Frames of Reference Rotating frames Centrifu
Centrifugal force
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Is familiar to us!
−→Ω × (
−→Ω × r) =
centrifugal1.pdf
Frames of Reference Rotating frames Centrifu
Centrifugal force
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Is familiar to us!
−→Ω × (
−→Ω × r) =
centrifugal2.pdf
Frames of Reference Rotating frames Centrifu
Centrifugal force
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Is familiar to us!
−→Ω × (
−→Ω × r) = −Ω2ρρ
centrifugal3.pdf
Frames of Reference Rotating frames Centrifu
Centrifugal force
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Is familiar to us!
−→Ω × (
−→Ω × r) = −Ω2ρρ
−→F centrifugal = mΩ2ρρ
centrifugal4.pdf
Frames of Reference Rotating frames Centrifu
Centrifugal force
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Is familiar to us!
−→Ω × (
−→Ω × r) = −Ω2ρρ
−→F centrifugal = mΩ2ρρ
directed away from the
axiscentrifugal4.pdf
Frames of Reference Rotating frames Centrifu
Coriolis force
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−→F cor = −2m
−→Ω × v
Gustave_corioli
Frames of Reference Rotating frames Corioli
Coriolis force
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−→F cor = −2m
−→Ω × v
corF0.pdf
Gustave_corioli
Frames of Reference Rotating frames Corioli
Coriolis force
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−→F cor = −2m
−→Ω × v
corF1.pdf
Gustave_corioli
Frames of Reference Rotating frames Corioli
Coriolis force
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−→F cor = −2m
−→Ω × v
corF2.pdf
Gustave_corioli
Frames of Reference Rotating frames Corioli
Coriolis force
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−→F cor = −2m
−→Ω × v
corF3.pdf
Gustave_corioli
Frames of Reference Rotating frames Corioli
Coriolis force
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−→F cor = −2m
−→Ω × v
corF.pdf
Magnitude:
F cor = 2mΩv⊥,
Gustave_corioli
Frames of Reference Rotating frames Corioli
Coriolis force
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−→F cor = −2m
−→Ω × v
corF.pdf
Magnitude:
F cor = 2mΩv⊥,
Direction: er endicular
Gustave_corioli
Frames of Reference Rotating frames Corioli
Coriolis force
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−→F cor = −2m
−→Ω × v
corF.pdf
Magnitude:
F cor = 2mΩv⊥,
Direction: er endicular
Gustave_corioli
On a merry-go-round in the
Coriolis was shaken with friDespite how he walked
’Twas like he was stalked
By some fiend always pu
him right!
– David Morin, Eric Zaslow, E liza be th Ha
Frames of Reference Rotating frames Corioli