刚体转动惯量

9999 ROTATIONROTATIONROTATIONROTATION OFOFOFOF RIGIDRIGIDRIGIDRIGID BODIESBODIESBODIESBODIES 9.1 Angular Velocity and Acceleration Angular Velocity (角速度) Angular Velocity As a Vector (角速度方向) 右手判定角速度方向,四指指向转动方向，大拇指指向角速度方向 An gular Acceleration (角加速) 0 lim z z z t d t dt ω ω α ∆ → ∆ = = ∆ 2 2z d d d d t d t d t θ θ α = = Angular Acceleration as a Vector If the object rotates around the fixed z-axis, then α �� has only a z-component; the quantity z α is just that component. In this case, α �� is in the same direction as ω �� if the rotation is slowing down. 0 lim z t d t dt θ θ ω ∆ → ∆ = = ∆ 9.2 Rotation with Constant Angular Acceleration Comparison of Linear and Angular Motion with Constant Acceleration 0 2 0 0 2 2 0 0 0 0 =constant 1 2 2 ( ) 1 ( ) 2 z z z z z z z z z z z t t t t α ω ω α θ θ ω α ω ω α θ θ θ θ ω ω = + = + + = + − − = + 0 2 0 0 2 2 0 0 0 0 constant 1 2 2 ( ) 1 ( ) 2 x x x x x x x x x x x a v v a t x x v t a t v v a x x x x v v t = = + = + + = + − − = + 9.3 Relating Liner and Angular Kinematics Linear Speed in Rigid-Body Rotation (转动刚体的线速度) v rω= (relationship between linear and angular speeds) Linear Acceleration in Rigid-Body Rotation(刚体转动的线性加速度) We can represent the acceleration of a particle moving in a circle in terms of its centripetal and tangential components, rad a and tana . The tangentialtangentialtangentialtangential componentcomponentcomponentcomponent ofofofof accelerationaccelerationaccelerationacceleration（切线方向加速度）（切线方向加速度）（切线方向加速度）（切线方向加速度） tana , the component parallel to the instantaneous velocity(瞬时速度 ), acts to change the magnitudemagnitudemagnitudemagnitude of the particle’s velocity. tan dv d a r r dt dt ω α= = = The component of the particle’s acceleration directed toward the rotation axis, the centripetalcentripetalcentripetalcentripetal componentcomponentcomponentcomponent ofofofof accelerationaccelerationaccelerationacceleration((((向心加速度向心加速度向心加速度向心加速度)))) tana ,,,, is association with the change of directiondirectiondirectiondirection of the particle’s velocity. 2 2 rad v a r r ω= = The vectorvectorvectorvector sumsumsumsum of the centripetalcentripetalcentripetalcentripetal and tangentialtangentialtangentialtangential components of acceleration of a particle in a rotating body is the linearlinearlinearlinear accelerationaccelerationaccelerationacceleration a � . （线性加速度）（线性加速度）（线性加速度）（线性加速度） 9.4 Energy in Rotational Motion The kinetic energy of the ith particle can be expressed as 2 2 21 1 2 2i i i i m v m r ω= The total kinetic energy of the body is the sum of the kinetic energies of all its particles 2 2 2 2 2 2 1 1 2 2 1 1 1 2 2 2 i i i K m r m r m rω ω ω= + + ⋅ ⋅ ⋅ = ∑ 2 2 2 2 2 1 1 2 2 1 1 ( ) ( ) 2 2 i i i K m r m r m rω ω= + + ⋅⋅⋅ = ∑ The quantity in parentheses, obtained by multiplying the mass of each particle by the square of its distance from the axis of rotation and adding these products, is denoted by I and is called the momentmomentmomentmoment ofofofof inertiainertiainertiainertia((((转动惯量转动惯量转动惯量转动惯量)))) of the body for this rotation axis 2 2 2 1 1 2 2 i i i I m r m r m r= + + ⋅ ⋅ ⋅ = ∑ In terms of moment of inertia I ，the rotationalrotationalrotationalrotational kinetickinetickinetickinetic energyenergyenergyenergy K （转动动量转动动量转动动量转动动量）of a rigid body is 21 2 K Iω= 9.5 Parallel-Axis Theorem（平行轴定理） A body doesn’t have just one moment of inertia. In fact, it has infinitely many, because there are infinitely many axes about which it might rotate. But there is a simple relationship between the moment of inertia cm I of a body of mass M about an axis through its center of mass and the moment of inertia P I about any other axis parallel to the original one but displaced from it by a distance d .This relationship is called the parallel-axisparallel-axisparallel-axisparallel-axis theoremtheoremtheoremtheorem （ cm I 是以经过重心的轴为旋转中心的转动惯量， P I 是以任意平行与之前的轴为旋转中心的转 动惯量， d 是两轴间的水平距离) 2 P cm I I Md= + 9.6 Moment-of-Inertia Calculations If a rigid body is a continuous distribution of mass, imagine dividing the body into elements of mass dm that are very small, and all points in a particular element are at essentially the same perpendicular distance from the axis of rotation. We call this distance r , as before. Then the moment of inertia is 2 I r d m= ∫ For a three-dimensional object it is usually easiest to express dm in terms of an element of volume dV and the density ρ of the body. dm dv ρ = , so we may also write it as 2 I r dVρ= ∫ This expression tells us that a body’s moment of inertia depends on how its density varies within its volume. IfIfIfIf thethethethe bodybodybodybody isisisis uniformuniformuniformuniform inininin densitydensitydensitydensity, then 2 I r dVρ= ∫ To use this equation, we have to express the volume element dV in terms of the differential of the integration variables, such as dV dxdydz= . (page295) (page 296)