理论力学第四章讲义

CATALOG OF CHAPTER 4§4.1 SPACIAL ROTATIONAL FRAME OF REFERENCE§4.2 THE EFFECTS INDUCED BY THE EARTH’S ROTATIONCH4 ROTATIONAL FRAME OF REFERENCE §4.1 Spacial Rotational Frame ofReference（一）Kinematicsthe stationary frame of reference S：Setting up a stationary coordinate system o-ξηζ；the spacial rotational frame of reference S'：Setting up a moving coordinate system o-xyz；⇳The origin is coincident with that of the stationarycoordinate system ，∴r r '= ozηξζxy P⇳The angular velocity is always through the originpoint O ；ωIn the stationary frame , the time derivative of anyphysical quantity isGG k dtdG j dt dG i dt dG z y x ⨯+++=ωˆˆˆ)ˆˆˆ(k G j G i G dtd dt G d z y x ++= the component expressionfor the vector Gthe moving framein the ozηξζxy PωG dtG d dt G d⨯+=ω*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.The expression can be written asozηξζxy PωThe velocity of any point P in space relative to thestationary framedt r d v =rdtr d ⨯+=ω*rv ⨯+'=ωThe acceleration of any point P in space relative to the stationary framedt v d a =vdtv d ⨯+=ω*Substituting the formula of velocity into it:)()(***r dtr d r dt r d dt d a ⨯+⨯+⨯+=ωωω)(***2*2r dtr d dt r d r dt d dt r d ⨯⨯+⨯+⨯+⨯+=ωωωωωcombinationdtr d r r dt d dt r d a**2*22)(⨯+⨯⨯+⨯+=⇒ωωωωωωωω⨯+=dt d dt d *dtd ω*=v r r a a '⨯+⨯⨯+⨯+'=⇒ ωωωω2)(relative accelerationconvected accelerationCoriolis’ acceleration)1.4.4(rotating with constant angular velocityv r r a a '⨯+⨯⨯+⨯+'= ωωωω2)(v R a a '⨯+-'=∴ ωω22orωR//R )()]([//2//R R R R +-+⋅=ωωω()RR R 2//2//ωωωω--=R R R 2//2//2ωωω--=R 2ω-=r r r2)()(ωωωωω-⋅=⨯⨯)2.4.4(the planar rotational frame of reference S '(such as a flat ）ozηζxyS ' The origin is coincident with that of the stationary coordi-nate system:RotatING around the axis perpendicular to itself.A particle is moving on the flat:jy i x r ˆˆ+= r r '= kz ˆωω=P（二）Dynamics 1. Dynamical Equations()v m r m r m F a m '⨯-⨯⨯-⨯-='⇒ ωωωω2主①②③Multiplying the formula (4.4.1) by m ：()v m r m r m a m a m '⨯+⨯⨯+⨯+'=⇒ ωωωω2v r r a a '⨯+⨯⨯+⨯+'= ωωωω2)(There are three kinds of inertial forces induced by therotation of the frame S ’.()v m r m r m r m F a m '⨯-+⋅-⨯-='⇒ ωωωωω22主①②③rm 2ω+：centrifugal inertia force：Coriolis forcev m '⨯-ω2v m R m F a m '⨯-+='⇒ ωω22主constant angular velocity v m R m a m a m '⨯+-'=⇒ ωω22v R a a '⨯+-'= ωω22over2. Relative Equilibrium()v m r m r m F a m '⨯-⨯⨯-⨯-=' ωωωω2主v m R m F a m '⨯-+=' ωω22主constant angular velocityA particle is in the state of equilibrium relative to thenon-inertial frame, then:0,0='='a v()0=⨯⨯-⨯-r m r m Fωωω主02=+R m Fω主⎩⎨⎧⇒constant angular velocity§4.2 THE EFFECTS INDUCED BY THE ERATH’S ROTATION （一）Centrifugal Inertia Force ωNSoTRm 2ω引F gm Assuming thata.The angular velocity is constant vector along the earth’s axis;b.A particle is at rest relative to the earth.2=+R m F ω主02=++⇒R m T F ω引Rm F g m 2ω+=⇒引Then ：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.ωNSoTRm 2ω引F gm（二）Coriolis ForceAssuming thata.The angular velocity is a constant vector along the earth’s axis;the unit vectors kj i ˆ,ˆ,ˆ* Adhere to the earth’s surface* Point to south horizontally, east horizontally, and upwards vertically respectively ωNSi ˆjˆkˆW Eλki z x ˆˆωωω+=k i ˆsin ˆcos λωλω+-=kˆi ˆωλλωNSi ˆjˆkˆWEλ引主F F =k mg ˆ-=gm =Substituting them into the dynamical equations :b.The effects of the centrifugal inertia force is neglected :2≈R m ωc.Only the gravitational force amongthe initiative forces is considered;)ˆˆˆ()ˆsin ˆcos (2ˆk zj y i x k i m k mg ++⨯+---=λωλωv m R m F a m '⨯-+=' ωω22主Coriolis forceωNSi ˆjˆkˆWEλ⎪⎭⎪⎬⎫+-=+-==λωλλωλωcos 2)cos sin (2sin 2y m mg z m z x m y m y m x m Working out the formula above, it givesthe Northern Hemisphere :0sin ,0cos >>λλ0sin ,0cos <>λλthe Southern Hemisphere ：)1.2.4(赤道附近空气因热上升，在高空向两极推进；两极地带空气因冷下降，在地面附近向赤道推进;→形成对流，称“贸易风”受科氏力的影响，原本南北向的气流向东西向偏转。

由动力学方程(4.2.1)用以上动力学方程解释科氏力的影响I. Trade Wind 贸易风南北向(x 轴)速度产生东西向(y 轴)的科氏力；ωNSW E热热冷冷由(4.2.1) 式，科氏力指向西方；河流：ωNSWEv '科F自北向南运动：0,0==>z y x 故：右岸的冲刷甚于左岸，比较陡峭。