EGR 280 Mechanics 16 – Newton’s Second Law for Rigid Bodies

Kinetics of rigid bodies in plane motion – Newton’s Second Law The motion of the mass center G with respect to the fixed frame XYZ ∑F = ma G The motion about the mass center with respect to the centroidal reference frame xyz ∑M G = d(H G )/dt These equations are always valid. G X Y Z x y z F1F1 F2F2

Angular momentum of a rigid body in plane motion H G = ∫(r´× v´dm) = ∫[r´× (ω × r´)dm] = ω ∫[(r´) 2 dm] k But ∫(r´) 2 dm = I G, the mass moment of inertia about the mass center, so H G = ω I G d(H G )/dt = α I G G r´r´ dm v´v´

Plane motion of a body: ∑F x = ma Gx ∑F y = ma Gy ∑M G = I G α Special cases: Translation: α = 0 Centroidal rotation:a G = 0: a Gx = a Gy = 0

Non-centroidal rotation Let a body rotate about a fixed point O that is not its mass center. The acceleration of its mass center is a G = a Gt + a Gn = dα e t + ω 2 d e n Taking moments about point O gives ∑M O = ∑M G + (mdα)d = I G α + md 2 α = I O α where I O = I G + md 2 is the mass moment of inertia about the fixed point O. d G anan atat O

Rolling motion When a disk rolls on a plane surface, and its mass center coincides with the center of rotation, then Rolling without sliding: f ≤ μ s N; a G = rα i Rolling, sliding impending f = μ s N; a G = rα i Sliding and rotating f = μ k N; a G, α are independent If the disk is unbalanced, G does not coincide with the center of rotation, a O = rα a G = a O + a G/O W P r N f ω,αω,α G G