Chapter 1Kinematics of a Particle1.1 Introduction1.1.1 Position, Velocity, and AccelerationThe position of a particle P relative to a given reference frame with origin O is givenby the position vector r from point O to point P, as shown in Fig. 1.1. If the particler(t + t)P (t + t)r(t + t) r(t)OP (t)r(t)path−ΔΔΔFig. 1.1 Position of a particle PP is in motion relative to the reference frame, the position vector r is a function oftime t, Fig. 1.1, and can be expressed asr = r(t).The velocity of the particle P relative to the reference frame at time t is defined byv =drdt=˙r = lim∆t→0r(t + ∆t)−r(t)∆t, (1.1)12 1 Kinematics of a Particlewhere the vector r(t + ∆t) −r(t) is the change in position, or displacement of P,during the interval of time ∆t, Fig. 1.1. The velocity is the rate of change of theposition of the particle P. The magnitude of the velocity v is the speed v = |v|. Thedimensions of v are (distance)/(time). The position and velocity of a particle can bespecified only relative to a reference frame.The acceleration of the particle P relative to the given reference frame at time tis defined bya =dvdt=˙v = lim∆t→0v(t + ∆t)−v(t)∆t, (1.2)where v(t + ∆t) −v(t) is the change in the velocity of P during the interval oftime ∆t, Fig. 1.1. The acceleration is the rate of change of the velocity of P attime t (the second time derivative of the displacement), and its dimensions are(distance)/(time)2.1.1.2 Angular Motion of a LineThe angular motion of the line L, in a plane, relative to a reference line L0, in theplane, is given by an angle θ , Fig. 1.2. The angular velocity of L relative to L0isdefined byω =dθdt=˙θ , (1.3)and the angular acceleration of L relative to L0is defined byα =dωdt=d2θdt2=˙ω =¨θ . (1.4)θL0LFig. 1.2 Angular motion of line L relative to a reference line L01.1 Introduction 3The dimensions of the angular position, angular velocity, and angular accelerationare [rad], [rad/s], and [rad/s2], respectively. The scalar coordinate θ can be positiveor negative. The counterclockwise (ccw) direction is considered positive.1.1.3 Rotating Unit VectorThe angular motion of a unit vector u in a plane can be described as the angularmotion of a line. The direction of u relative to a reference line L0, is specified by theangle θ in Fig. 1.3(a), and the rate of rotation of u relative to L0is defined by theangular velocityω =dθdt=˙θ .u(t +Δt)nΔuΔθθ(t)L0u(t)L0ududtθ(a)(b)uFig. 1.3 Angular motion of a unit vector u in plane4 1 Kinematics of a ParticleThe time derivative of u is specified bydudt= lim∆t→0u(t + ∆t)−u(t)∆t.Figure 1.3(a) shows the vector u at time t and at time t + ∆t. The change in uduring this interval is ∆ u = u(t + ∆t)u(t), and the angle through which u rotates is∆ θ = θ (t + ∆t)−θ (t). The triangle in Fig. 1.3(a) is isosceles, so the magnitude of∆ u is|∆u|= 2|u|sin(∆θ /2) = 2 sin(∆ θ /2).The vector ∆u is∆ u = |∆ u|n = 2sin(∆θ/2)n,where n is a unit vector that points in the direction of ∆ u, Fig. 1.3(a). The timederivative of u isdudt= lim∆t→0∆ u∆t= lim∆t→02sin(∆θ/2)n∆t= lim∆t→0sin(∆θ/2)∆ θ/2∆ θ∆tn =lim∆t→0sin(∆θ/2)∆ θ/2∆ θ∆tn = lim∆t→0∆ θ∆tn =dθdtn,where lim∆t→0sin(∆θ/2)∆ θ/2= 1 and lim∆t→0∆ θ∆t=dθdt.So the time derivative of the unit vector u isdudt=dθdtn =˙θ n = ω n,where n is a unit vector that is perpendicular to u, n ⊥ u, and points in the positiveθ direction, Fig. 1.3(b).1.2 Rectilinear MotionThe position of a particle P on a straight line relative to a reference point O canbe indicated by the coordinate s measured along the line from O to P, as shown inFig. 1.4. In this case the the reference frame is the straight line and the origin of therusOPsFig. 1.4 Straight line motion of P1.3 Curvilinear Motion 5the reference frame is the point O. The reference frame and its origin are used todescribe the position of particle P. The coordinate s is considered to be positive tothe right of the origin O and is considered to be negative to the left of the origin.Let u be a unit vector parallel to the straight line and pointing in the positive s,Fig. 1.4. The position vector of the point P relative to the origin O isr = su.The velocity of the particle P relative to the origin O isv =drdt=dsdtu = ˙su.The magnitude v of the velocity vector v = vu is the speed (velocity scalar)v =dsdt= ˙s.The speed v of the particle P is equal to the slope at time t of the line tangent to thegraph of s as a function of time.The acceleration of the particle P relative to O isa =dvdt=ddt(vu) =dvdtu = ˙vu = ¨su.The magnitude a of the acceleration vector a = au is the acceleration scalara =dvdt=d2sdt2.The acceleration a is equal to the slope at time t of the line tangent to the graph of vas a function of time.1.3 Curvilinear MotionThe motion of the particle P along a curvilinear path, relative to a reference frame,can be specified in terms of its position, velocity, and acceleration vectors. The di-rections and magnitudes of the position, velocity, and acceleration vectors do notdepend on the particular coordinate system used to express them. The represen-tations of the position, velocity, and acceleration vectors are different in differentcoordinate systems.6 1 Kinematics of a Particle1.3.1 Cartesian CoordinatesLet r be the position vector of a particle P relative to the origin O of a cartesianreference frame, Fig. 1.5. The components of r are the x, y, and z coordinates of theparticle Pr = xı + yj + zk. (1.5)The velocity of the particle P relative to the reference frame isv =drdt=˙r =dxdtı +dydtj +dzdtk = ˙xı + ˙yj + ˙zk. (1.6)The velocity in terms of scalar components isv = vxı + vyj + vzk, (1.7)Three scalar equations can be obtainedvx=dxdt= ˙x, vy=dydt= ˙y, vz=dzdt= ˙z. (1.8)The acceleration of the particle P relative to the reference frame isa =dvdt=˙v =¨r =dvxdtı +dvydtj +dvzdtk = ˙vxı + ˙vyj + ˙vzk = ¨xı + ¨yj + ¨zk.Expressing the acceleration in terms of scalar componentsa = axı + ayj + azk, (1.9)three scalar equations can be obtainedOrP(x, y, z)ıjkxyzvpathFig. 1.5 Position vector of a particle P in a cartesian reference frame1.3 Curvilinear Motion 7ax=dvxdt= ˙vx= ¨x, ay=dvydt= ˙vy= ¨y, az=dvzdt= ˙vz= ¨z. (1.10)Equations (1.8) and (1.10) describe the motion of a particle relative to a cartesiancoordinate system.1.3.2 Normal and Tangential CoordinatesThe position,