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MIT16_07F09_Lec10

# MIT16_07F09_Lec10 - S Widnall J Peraire 16.07 Dynamics Fall...

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Unformatted text preview: S. Widnall, J. Peraire 16.07 Dynamics Fall 2008 Version 2.0 Lecture L10- Angular Impulse and Momentum for a Particle In addition to the equations of linear impulse and momentum considered in the previous lecture, there is a parallel set of equations that relate the angular impulse and momentum. Angular Momentum We consider a particle of mass m , with velocity v , moving under the inﬂuence of a force F . The angular momentum about point O is defined as the “moment” of the particle’s linear momentum, L , about O . Thus, the particle’s angular momentum is given by, H O = r × m v = r × L . (1) The units for the angular momentum are kg m 2 /s in the SI system, and slug ft 2 /s in the English system. · · 1 It is clear from its definition that the angular momentum is a vector which is perpendicular to the plane defined by r and v . Thus, on some occasions it may be more convenient to determine the direction of H O from the right hand rule, and its modulus directly from the definition of the vector product, H O = mvr sin α , where α is the angle between r and v . In other situations, it may be convenient to directly calculate the angular momentum in component form. For instance, using a right handed cartesian coordinate system, the components of the angular momentum are calculated as H O = H x i + H y j + H z k = i j k x y z mv x mv y mv z = m ( v z y − v y z ) i + m ( v x z − v z x ) j + m ( v y x − v x y ) k . Similarly, in cylindrical coordinates we have H O = H r e r + H θ e θ + H z k = e r e θ k r z mv r mv θ mv z = − mv θ z e r + m ( v r z − v z r ) e θ + mv θ r k . Rate of Change of Angular Momentum We now want to examine how the angular momentum changes with time. We examine this in two different coordinate systems: system a) is about a fixed point O ; system b) is about the center of mass of the particle. Of course system b) is rather trivial for a point mass, but its later extensions to finite bodies will be extremely important. Even at this trivial level, we will obtain an important result. About a Fixed Point O The angular momentum about the fixed point O is H O = r × m v (2) 2 Taking a time derivative of this expression , we have H ˙ O = r ˙ × m v + r × m v ˙ . Here, we have assumed that m is constant. If O is a fixed point, then r ˙ = v and r ˙ × m v = 0. Thus, we end up with, H ˙ O = r × m v ˙ = r × m a . Applying Newton’s second law to the right hand side of the above equation, we have that r × m a = r × F = M O , where M O is the moment of the force F about point O . The equation expressing the rate of change of angular momentum is then written as M O = H ˙ O . (3) We note that this expression is valid whenever point O is fixed. The above equation is analogous to the equation derived in the previous lecture expressing the rate of change of linear momentum. It states that the rate of change of linear momentum about a fixed point O is equal to the moment about O due to the resultant force acting...
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MIT16_07F09_Lec10 - S Widnall J Peraire 16.07 Dynamics Fall...

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