# chapt9 - Plane Motion of Rigid Bodies Forces and...

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Plane Motion of Rigid Bodies: Forces and Accelerations

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Introduction • In this chapter and in Chapters 17 and 18, we will be concerned with the kinetics of rigid bodies, i.e., relations between the forces acting on a rigid body, the shape and mass of the body, and the motion produced. • Our approach will be to consider rigid bodies as made of large numbers of particles and to use the results of Chapter 14 for the motion of systems of particles. Specifically, G G H M a m F ± G G G G = = and • Results of this chapter will be restricted to: - plane motion of rigid bodies, and - rigid bodies consisting of plane slabs or bodies which are symmetrical with respect to the reference plane. • D’Alembert’s principle is applied to prove that the external forces acting on a rigid body are equivalent a vector attached to the mass center and a couple of moment a m G . α I
Equations of Motion for a Rigid Body • Consider a rigid body acted upon by several external forces. • Assume that the body is made of a large number of particles. • For the motion of the mass center G of the body with respect to the Newtonian frame Oxyz , a m F G G = • For the motion of the body with respect to the centroidal frame Gx’y’z’ , G G H M ± G G = • System of external forces is equipollent to the system consisting of . and G H a m ± G G

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Angular Momentum of a Rigid Body in Plane Motion • Consider a rigid slab in plane motion. • Angular momentum of the slab may be computed by () [] ω G G G G G G G G I m r m r r m v r H i i n i i i i n i i i i G = = × × = × = = = Δ Δ Δ 2 1 1 • After differentiation, αω G ± G ± G I I H G = = • Results are also valid for plane motion of bodies which are symmetrical with respect to the reference plane. • Results are not valid for asymmetrical bodies or three-dimensional motion.
Plane Motion of a Rigid Body: D’Alembert’s Principle α I M a m F a m F G y y x x = = = • Motion of a rigid body in plane motion is completely defined by the resultant and moment resultant about G of the external forces. • The external forces and the collective effective forces of the slab particles are equipollent (reduce to the same resultant and moment resultant) and equivalent (have the same effect on the body). • The most general motion of a rigid body that is symmetrical with respect to the reference plane can be replaced by the sum of a translation and a centroidal rotation. d’Alembert’s Principle : The external forces acting on a rigid body are equivalent to the effective forces of the various particles forming the body.

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Problems Involving the Motion of a Rigid Body • The fundamental relation between the forces acting on a rigid body in plane motion and the acceleration of its mass center and the angular acceleration of the body is illustrated in a free-body-diagram equation.
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chapt9 - Plane Motion of Rigid Bodies Forces and...

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