Fundamental Equations of Dynamics KINEMATICS Particle Rectilinear Motion Variable a Constant a = ac dv v = v0 + act a= dt ds s = s0 + v0t + 12 act 2 v = dt a ds = v dv v2 = v20 + 2ac(s - s0) Equations of Motion Particle "F = ma Rigid Body "Fx = m(aG)x (Plane Motion) "Fy = m(aG)y "MG = IGa or "MP = "(mk)P Principle of Work and Energy T 1 + "U1 - 2 = T 2 Kinetic Energy Particle T = 21mv2 Rigid Body (Plane Motion) T = 1mvG2 + 1IG v2 Particle Curvilinear Motion x, y, z Coordinates r, u, z Coordinates # ## # # ## ax = x vx = x Work vr = r ar = r - r u 2 # ## # ## # Variable force # vy = y ay = y vu = r u au = r u + 2r u # ## vz = z az = z n, t, b Coordinates # v = s Relative Motion vB = vA + vB/A # vz = z ## az = z dv # at = v = v ds [1 + (dy >dx)2]3>2 v2 an = r = r 0 d2y >dx 2 0 aB = aA + aB/A Rigid Body Motion About a Fixed Axis Variable a Constant a = ac dv a = v = v0 + act dt du v = u = u0 + v0t + 21act 2 dt v dv = a du v2 = v20 + 2ac(u - u0) For Point P s = ur v = vr at = ar an = v2r Relative General Plane Motion—Translating Axes vB = vA + vB>A (pin) aB = aA + aB>A (pin) Relative General Plane Motion—Trans. and Rot. Axis vB = vA + ! * rB>A + (vB>A )xyz # aB = aA + ! * rB>A + ! * (! * rB>A ) + 2! * (vB>A )xyz + (aB>A )xyz KINETICS r2 dm L I = IG + md2 I k = Am Mass Moment of Inertia I = Parallel-Axis Theorem Radius of Gyration 2 UF = 2 F cos u ds L (Fc cos u) # s - W #y - 1 21 ks 22 - 21 ks 21 2 M#u Constant force UF = Weight UW = Spring Us = Couple moment UM = Power and Efficiency Pout Uout dU e = = P= = F#v dt Pin Uin Conservation of Energy Theorem T1 + V 1 = T2 + V 2 Potential Energy V = V g + V e, where V g = {W y, V e = + 21 ks 2 Principle of Linear Impulse and Momentum Particle mv1 + " L F dt = mv2 F dt = m(vG)2 L Conservation of Linear Momentum "(syst. mv)1 = "(syst. mv)2 (vB)2 - (vA)2 Coefficient of Restitution e = (vA)1 - (vB)1 Principle of Angular Impulse and Momentum Rigid Body Particle m(vG)1 + " MO dt = (HO)2 L where HO = (d)(mv) (HO)1 + " MG dt = (HG)2 L where HG = IGv (HG)1 + " Rigid Body (Plane motion) MO dt = (HO)2 L where HO = IOv Conservation of Angular Momentum "(syst. H)1 = "(syst. H)2 (HO)1 + "