理论力学第四章讲义

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§4.1 SPACIAL ROTATIONAL FRAME OF REFERENCE
In the stationary frame , the time derivative of any physical quantity G is
dG d ˆ ˆ (Gxi G y ˆ Gz k ) j dt dt
：centrifugal inertia force m r 2m v ：Coriolis force
2
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§4.1 SPACIAL ROTATIONAL FRAME OF REFERENCE
constant angular velocity
2 a a R 2 v 2 ma ma m R 2m v
A particle is in the state of equilibrium relative to the non-inertial frame, then: v 0, a 0
r m r 0 F主 m 2 F主 m R 0 constant angular velocity
The expression can be written as
* dG d G G dt dt
z

y P

x

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The first term：“the relative rate of change”, which represents the variation with time relative to frame S’； The second term：“the convected rate of change”, which represents the variation with time due to the motion of the frame S itself.
r ( r ) 2 v a a
r m r 2m v ma ma m
r m r 2m v ma F主 m
the stationary frame of reference S： Setting up a stationary coordinate system o-； the spacial rotational frame of reference S： Setting up a moving coordinate system o-xyz；
CATALOG OF CHAPTER 4
§4.1 SPACIAL ROTATIONAL FRAME OF REFERENCE §4.2 THE EFFECTS INDUCED BY THE EARTH’S ROTATION
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CH4 ROTATIONAL FRAME OF REFERENCE
§4.1 Spacial Rotational Frame of （一）Kinematics Reference

ˆ j A particle is moving on the flat: r xi yˆ
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§4.1 SPACIAL ROTATIONAL FRAME OF REFERENCE
（二）Dynamics 1. Dynamical Equations Multiplying the formula (4.4.1) by m：
2*
combination
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§4.1 SPACIAL ROTATIONAL FRAME OF REFERENCE
* * d r d r d a 2 r ( r ) 2 dt dt dt
2*
d d d dt dt dt
v r
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§4.1 SPACIAL ROTATIONAL FRAME OF REFERENCE
The acceleration of any point P in space relative to the stationary frame
* dv d v a v dt dt
2 ma F主 m R 2m v
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§4.1 SPACIAL ROTATIONAL FRAME OF REFERENCE
2. Relative Equilibrium
r m r 2m v ma F主 m 2 F主 m R 2m v constant angular velocity ma
rotating with constant angular velocity r ( r ) 2 v a a
2 ( r ) ( r ) r 2 [ ( R// R)] ( R// R) 2 2 R// R// R 2 2 2 2 R// R// R R
b. A particle is at rest relative to the earth.
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Leabharlann Baidu
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F引
m R
2
mg
2 Then： F主 m R 0
S
2 2 F引 T m R 0 mg F引 m R
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§4.2 THE EFFECTS INDUCED BY THE EARTH’S ROTATION
（二）Coriolis Force
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ˆ j ˆ the unit vectors i , ˆ, k
* Adhere to the earth’s surface * Point to south horizontally, east W horizontally, and upwards vertically respectively Assuming that a. The angular velocity is a constant vector along the earth’s axis;
* *
Substituting the formula of velocity into it:
* d r d d r a ( r) ( r) dt dt dt
* * * d r d d r d r 2 r ( r ) dt dt dt dt
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§4.1 SPACIAL ROTATIONAL FRAME OF REFERENCE
The velocity of any point P in space relative to the
stationary frame
* dr dr v r dt dt
over
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§4.2 THE EFFECTS INDUCED BY THE ERATH’S ROTATION （一）Centrifugal Inertia Force Assuming that
a. The angular velocity is constant vector along the earth’s axis;
*


*

r ( r ) 2 v a a
relative acceleration
(4.4.1)
Coriolis’ acceleration
convected acceleration
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§4.1 SPACIAL ROTATIONAL FRAME OF REFERENCE
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ˆ k
ˆ j

iˆ
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ˆ k
iˆ

ˆ ˆ xiˆ z k cos iˆ sin k
2 a a R 2 v


R//

R
r
o
(4.4.2)
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§4.1 SPACIAL ROTATIONAL FRAME OF REFERENCE
the planar rotational frame of reference S (such as a flat）
the component expression for the vector G in the the moving frame
z

y P

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dGx ˆ dGy ˆ dGz ˆ i j k G dt dt dt
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§4.1 SPACIAL ROTATIONAL FRAME OF REFERENCE
y

S P
x

The origin is coincident with that of the stationary coordinate system:
z
o
r r
RotatING around the axis perpendicular to itself.
ˆ z k
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§4.1 SPACIAL ROTATIONAL FRAME OF REFERENCE
z

y P

x

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The origin is coincident with that of the stationary coordinate system， r r The angular velocity is always through the origin point O；
§4.2 THE EFFECTS INDUCED BY THE EARTH’S ROTATION
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T
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F引
m R
2
mg
S
Due to the effect of the centrifugal inertia force. the gravity is the resultant force of the gravitational force and the centrifugal force. The latitude is higher, the gravity is bigger; at the poles the gravity is equal to the gravitational force.

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§4.1 SPACIAL ROTATIONAL FRAME OF REFERENCE
r m r m 2 r 2m v ma F主 m

There are three kinds of inertial forces induced by the rotation of the frame S’.