dynamics this is a class notes for dynamics that includes kinematics of particles, Lecture notes of Dynamics

Download dynamics this is a class notes for dynamics that includes kinematics of particles and more Lecture notes Dynamics in PDF only on Docsity! INTRODUCTION & RECTILINEAR KINEMATICS: CONTINUOUS MOTION Today’s Objectives: Students will be able to: 1. Find the kinematic quantities (position, displacement, velocity, and acceleration) of a particle traveling along a straight path. In-Class Activities: • Check Homework • Reading Quiz • Applications • Relations between s(t), v(t), and a(t) for general rectilinear motion. • Relations between s(t), v(t), and a(t) when acceleration is constant. • Concept Quiz • Group Problem Solving • Attention Quiz READING QUIZ 1. In dynamics, a particle is assumed to have _________. A) both translation and rotational motions B) only a mass C) a mass but the size and shape cannot be neglected D) no mass or size or shape, it is just a point 2. The average speed is defined as __________. A) Dr/Dt B) Ds/Dt C) sT/Dt D) None of the above. An Overview of Mechanics Statics: The study of bodies in equilibrium. Dynamics: 1. Kinematics – concerned with the geometric aspects of motion 2. Kinetics - concerned with the forces causing the motion Mechanics: The study of how bodies react to forces acting on them. RECTILINEAR KINEMATICS: CONTINIOUS MOTION (Section 12.2) A particle travels along a straight-line path defined by the coordinate axis s. The total distance traveled by the particle, sT, is a positive scalar that represents the total length of the path over which the particle travels. The position of the particle at any instant, relative to the origin, O, is defined by the position vector r, or the scalar s. Scalar s can be positive or negative. Typical units for r and s are meters (m) or feet (ft). The displacement of the particle is defined as its change in position. Vector form: D r = r’ - r Scalar form: D s = s’ - s VELOCITY Velocity is a measure of the rate of change in the position of a particle. It is a vector quantity (it has both magnitude and direction). The magnitude of the velocity is called speed, with units of m/s or ft/s. The average velocity of a particle during a time interval Dt is vavg = Dr / Dt The instantaneous velocity is the time-derivative of position. v = dr / dt Speed is the magnitude of velocity: v = ds / dt Average speed is the total distance traveled divided by elapsed time: (vsp)avg = sT / Dt CONSTANT ACCELERATION The three kinematic equations can be integrated for the special case when acceleration is constant (a = ac) to obtain very useful equations. A common example of constant acceleration is gravity; i.e., a body freely falling toward earth. In this case, ac = g = 9.81 m/s 2 = 32.2 ft/s2 downward. These equations are: tavv co +=yields=  t o c v v dtadv o 2 coo s t(1/2) a t vss ++=yields=  t os dtvds o )s-(s2a)(vv oc 2 o 2 +=yields=  s s c v v oo dsadvv EXAMPLE Plan: Establish the positive coordinate, s, in the direction the particle is traveling. Since the velocity is given as a function of time, take a derivative of it to calculate the acceleration. Conversely, integrate the velocity function to calculate the position. Given: A particle travels along a straight line to the right with a velocity of v = ( 4 t – 3 t2 ) m/s where t is in seconds. Also, s = 0 when t = 0. Find: The position and acceleration of the particle when t = 4 s. EXAMPLE (continued) Solution: 1) Take a derivative of the velocity to determine the acceleration. a = dv / dt = d(4 t – 3 t2) / dt = 4 – 6 t  a = – 20 m/s2 (or in the  direction) when t = 4 s 2) Calculate the distance traveled in 4s by integrating the velocity using so = 0: v = ds / dt  ds = v dt   s – so = 2 t 2 – t3  s – 0 = 2(4)2 – (4)3  s = – 32 m ( or )  = t o s s (4 t – 3 t2) dtds o GROUP PROBLEM SOLVING (continued)Solution: Determine the distance by integrating using s0 = 2. Notice that s = 2 m when t = 0. 1) Since v = ( 4s2)     GROUP PROBLEM SOLVING (continued) m 2) Take a derivative of distance to calculate the velocity and acceleration. m/s m/s2 ATTENTION QUIZ 2. A particle is moving with an initial velocity of v = 12 ft/s and constant acceleration of 3.78 ft/s2 in the same direction as the velocity. Determine the distance the particle has traveled when the velocity reaches 30 ft/s. A) 50 ft B) 100 ft C) 150 ft D) 200 ft 1. A particle has an initial velocity of 3 ft/s to the left at s0 = 0 ft. Determine its position when t = 3 s if the acceleration is 2 ft/s2 to the right. A) 0.0 ft B) 6.0 ft C) 18.0 ft D) 9.0 ft