Engineering Mechanics: Dynamics Summary of Analysis STATICS DYNAMICS (Equations of Equilibrium) (Equations of Motion) Particle ∑ F = 0 ∑ F = m a (Mass & No Size) Rigid Body ∑ F = 0 ∑ F = m a (Mass & Size) ∑ M = 0 ∑ M = I α Engineering Mechanics: Dynamics Linear Momentum of a Particle • Replacing the acceleration by the derivative of the velocity yields r r dv ∑ F = m dt r d r dL = ()mv = r dt dt L = linear momentum of the particle 12 - 2 Engineering Mechanics: Dynamics Angular Momentum of a Particle r r r • HO = r × mV = moment of momentum or the angular momentum of the particle about O. • Derivative of angular momentum with respect to time, r& r r r r& r r r r H = r& × mV + r × mV = V × mV + r × ma O r = r × ∑ F r = ∑ M O • It follows from Newton’s second law that the sum of the moments about O of the forces acting on the particle is equal to the rate of change of the angular momentum of the particle about O. 12 - 3 Engineering Mechanics: Dynamics Angular Momentum of a Rigid Body in Plane Motion • Angular momentum of the slab may be computed by r n r r HG = ∑()ri′×vi′∆mi i=1 n r r r = ∑[ri′×()ω × ri′ ∆mi ] i=1 r 2 = ω∑ (r′ ∆m ) r i i = Iω • After rdifferentiation, & r& r HG = Iω = Iα • Results are also valid for plane motion of bodies • Consider a rigid slab in which are symmetrical with respect to the plane motion. reference plane. • Results are not valid for asymmetrical bodies or three-dimensional motion. 16 - 4 Engineering Mechanics: Dynamics Equations of Motion for a Rigid Body • For the motion of the mass center G of the body with respect to the Newtonian rframe Oxyzr , ∑F = ma • For the motion of the body with respect to the centroidal frame Gx’y’z’ , r r& ∑MG = HG • Wherer & r& r r HG = Iω = Iα = Angular Momentum H G 16 - 5 Engineering Mechanics: Dynamics Plane Motion of a Rigid Body: Equations of Motion • Motion of a rigid body in plane motion is completely defined by the force resultant and moment resultant about G of the external forces. ∑ F x = m a x ∑ F y = m a y ∑ M G = I α 16 - 6 Engineering Mechanics: Dynamics Constrained Plane Motion: Rolling Motion • For a balanced disk constrained to roll without sliding, x = rθ → a = rα • Rolling, no sliding: F ≤ µs N a = rα Rolling, sliding impending: F = µs N a = rα Rotating and sliding: F = µk N a, rα independent 16 - 7.