Chapter_10_Notes_new

Classical Mechanics

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Unformatted text preview: 1 PHYS620 Fall 2009 Notes for Chapter 10 Dynamics of rigid bodies The motion of a rigid body is often most easily analyzed by separating it into translational motion of the CM and rotational motion of the rigid body about the CM. The rotational inertia tensor The total kinetic energy of the rigid body is the sum of CM kinetic energy and the rotational kinetic energy due to rotation about the CM. The rotational contribution is ( ) ( ) ( )( ) ( )( ) Body Body Body 1 2 1 2 1 . 2 rot rot rot T dm dm dm = ⋅ = × ⋅ × = ⋅ ⋅- ⋅ ⋅ ∫ ∫ ∫ v v ω r ω r ω ω r r ω r ω r (10.1.1) Here ω is the instantaneous angular velocity and r is the position vector of mass element dm relative to the CM. Choose a co-ordinate system fixed to the body with origin at the CM. In this co-ordinate system, called a body system , ( ) ( ) Body Body 1 2 1 . 2 rot i i k k i i j j i j k k ij i j T r r r r dm r r rr dm ωω ω ω ωω δ =- =- ∫ ∫ (10.1.2) Here we are using a summation convention. It is implied that there is a sum from 1 to 3 for all repeated indices, e.g. k k r r is the same as 3 1 . k k k r r = ∑ ij δ is the Kronecker delta. We can write 1 , 2 rot i ij j T I ω ω = (10.1.3) where the set of 9 quantities ( ) Body ij k k ij i j I r r rr dm δ =- ∫ (10.1.4) are the elements of the inertia tensor . The result in equation (10.1.3) is the generalization of the familiar expression for the kinetic energy of a body rotating about a fixed axis. From equation (10.1.4), we see that the inertia tensor is symmetric, i.e. . ji ij I I = Thus there are only 6 independent elements. 2 Note that although to give physical meaning to the inertia tensor, we chose the origin of co-ordinates to be at the CM, the inertia tensor can be evaluated for any choice of origin. This is useful if a point of the body is fixed in a particular inertial frame. Angular momentum The angular momentum of the body about a point fixed in the body system is Body . dm = × ∫ L r v (10.1.5) The best choice of fixed point depends on the particular problem. If one or more points of the body are fixed, then it is convenient to choose one of the fixed points as the origin. If no points of the body are fixed, then it is most convenient to choose the CM as the origin. Relative to the body system (i.e. the position vector r specifies a particular point of the body), . = × v ω r (10.1.6) Hence ( ) ( ) ( ) Body Body . dm dm = × × = ⋅- ⋅ ∫ ∫ L r ω r r r ω r ω r (10.1.7) In component form, this gives ( ) ( ) ( ) Body Body . i k k i j j i j k k ij i j L r r r r dm r r rr dm ω ω ω δ =- =- ∫ ∫ (10.1.8) Hence . i ij j L I ω = (10.1.9) Also note that 1 1 1 ....
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Chapter_10_Notes_new - 1 PHYS620 Fall 2009 Notes for...

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