Yeu-Sheng Paul Shiue, Ph.D 薛宇盛 Professor and Chair Mechanical Engineering Department Christian Brothers University 650 East Parkway South Memphis, TN

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2 Yeu-Sheng Paul Shiue, Ph.D 薛宇盛 Prfessr and Chair Mechanical Engineering Department Christian Brthers University 650 East Parkway Suth Memphis, TN Office: (901) Rm: N-110 Fax : (901) pshiue@cbu.edu Webpage:

3 Statics -- (dictinary) (a) relating t bdies r frces at rest. (b) nt mving. -- Deal with the actin f frces n bdies at rest.

4 Dynamics -- (dictinary) (a) relating t physical frce prducing mtin. Dynamics -- (1) Kinematics (2) Kinetics Kinematics -- The study f mtin withut reference t the frces which cause mtin. (MOTION) Kinetics -- Study the actin f frces n bdies t their resulting mtins. (FORCE, POWER, WORK)

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6 Ch Kinematics f a particle. Ch Kinetic f a Particle: Frce and Acceleratin. Ch Kinetic f a Particle: Wrk and Energy Ch Kinetic f a Particle: Impulse and Mmentum Ch Planar Kinematics f a Rigid Bdy Ch Planar Kinematics f a Rigid Bdy: Frce and Acceleratin Ch Planar Kinematics f a Rigid Bdy: Wrk and Energy Ch Planar Kinematics f a Rigid Bdy: Impulse and Mmentum Ch. 20, Ch21, Ch22 will be intrduced if time allwed.

7 (a)knwing Subjects (b) A lt f Practice (Prblem slving)

8 INTRODUCTION & RECTILINEAR KINEMATICS: CONTINUOUS MOTION Tday s Objectives: Students will be able t: 1. Find the kinematic quantities (psitin, displacement, velcity, and acceleratin) f a particle traveling alng a straight path.

9 READING QUIZ 1. In dynamics, a particle is assumed t have. A) bth translatin and rtatinal mtins B) nly a mass C) a mass but the size and shape cannt be neglected D) n mass r size r shape, it is just a pint 2. The average speed is defined as. A) Dr/Dt B) Ds/Dt C) s T /Dt D) Nne f the abve.

10 APPLICATIONS The mtin f large bjects, such as rckets, airplanes, r cars, can ften be analyzed as if they were particles. Why? If we measure the altitude f this rcket as a functin f time, hw can we determine its velcity and acceleratin?

11 APPLICATIONS (cntinued) A sprts car travels alng a straight rad. Can we treat the car as a particle? If the car accelerates at a cnstant rate, hw can we determine its psitin and velcity at sme instant?

12 An Overview f Mechanics Mechanics: The study f hw bdies react t frces acting n them. Statics: The study f bdies in equilibrium. Dynamics: 1. Kinematics cncerned with the gemetric aspects f mtin 2. Kinetics - cncerned with the frces causing the mtin

13 RECTILINEAR KINEMATICS: CONTINIOUS MOTION (Sectin 12.2) Vectr frm: D r = r - r A particle travels alng a straight-line path defined by the crdinate axis s. The psitin f the particle at any instant, relative t the rigin, O, is defined by the psitin vectr r, r the scalar s. Scalar s can be psitive r negative. Typical units fr r and s are meters (m) r feet (ft). The displacement f the particle is defined as its change in psitin. Scalar frm: D s = s - s The ttal distance traveled by the particle, s T, is a psitive scalar that represents the ttal length f the path ver which the particle travels.

14 VELOCITY Velcity is a measure f the rate f change in the psitin f a particle. It is a vectr quantity (it has bth magnitude and directin). The magnitude f the velcity is called speed, with units f m/s r ft/s. The average velcity f a particle during a time interval Dt is v avg = Dr / Dt The instantaneus velcity is the time-derivative f psitin. v = dr / dt Speed is the magnitude f velcity: v = ds / dt Average speed is the ttal distance traveled divided by elapsed time: (v sp ) avg = s T / Dt

15 ACCELERATION Acceleratin is the rate f change in the velcity f a particle. It is a vectr quantity. Typical units are m/s 2 r ft/s 2. The instantaneus acceleratin is the time derivative f velcity. Vectr frm: a = dv / dt Scalar frm: a = dv / dt = d 2 s / dt 2 Acceleratin can be psitive (speed increasing) r negative (speed decreasing). As the bk indicates, the derivative equatins fr velcity and acceleratin can be manipulated t get a ds = v dv

16 SUMMARY OF KINEMATIC RELATIONS: RECTILINEAR MOTION Differentiate psitin t get velcity and acceleratin. v = ds/dt ; a = dv/dt r a = v dv/ds Integrate acceleratin fr velcity and psitin. v t dv = v Velcity: a dt r v v dv = v s s a ds ds = Nte that s and v represent the initial psitin and velcity f the particle at t = 0. s s Psitin: t v dt

17 CONSTANT ACCELERATION The three kinematic equatins can be integrated fr the special case when acceleratin is cnstant (a = a c ) t btain very useful equatins. A cmmn example f cnstant acceleratin is gravity; i.e., a bdy freely falling tward earth. In this case, a c = g = 9.81 m/s 2 = 32.2 ft/s 2 dwnward. These equatins are: v v dv t = a c dt yields v = v + a t c s s ds t = v dt yields s = s + v t + (1/2) a c t 2 v v v dv s = s a c ds yields v 2 2 = (v ) + 2a (s - s ) c

18 EXAMPLE Given: A particle travels alng a straight line t the right with a velcity f v = ( 4 t 3 t 2 ) m/s where t is in secnds. Als, s = 0 when t = 0. Find: The psitin and acceleratin f the particle when t = 4 s. Plan: Establish the psitive crdinate, s, in the directin the particle is traveling. Since the velcity is given as a functin f time, take a derivative f it t calculate the acceleratin. Cnversely, integrate the velcity functin t calculate the psitin.

19 Slutin: EXAMPLE (cntinued) 1) Take a derivative f the velcity t determine the acceleratin. a = dv / dt = d(4 t 3 t 2 ) / dt =4 6 t => a = 20 m/s 2 (r in the directin) when t = 4 s 2) Calculate the distance traveled in 4s by integrating the velcity using s = 0: = v = ds / dt => ds = v dt => ds (4 t 3 t 2 ) dt => s s = 2 t 2 t 3 s => s 0 = 2(4) 2 (4) 3 => s = 32 m ( r ) s t

20 CONCEPT QUIZ 3 m/s 5 m/s t = 2 s t = 7 s 1. A particle mves alng a hrizntal path with its velcity varying with time as shwn. The average acceleratin f the particle is. A) 0.4 m/s 2 B) 0.4 m/s 2 C) 1.6 m/s 2 D) 1.6 m/s 2 2. A particle has an initial velcity f 30 ft/s t the left. If it then passes thrugh the same lcatin 5 secnds later with a velcity f 50 ft/s t the right, the average velcity f the particle during the 5 s time interval is. A) 10 ft/s B) 40 ft/s C) 16 m/s D) 0 ft/s

21 GROUP PROBLEM SOLVING Given: Ball A is released frm rest at a height f 40 ft at the same time that ball B is thrwn upward, 5 ft frm the grund. The balls pass ne anther at a height f 20 ft. Find: The speed at which ball B was thrwn upward. Plan: Bth balls experience a cnstant dwnward acceleratin f 32.2 ft/s 2 due t gravity. Apply the frmulas fr cnstant acceleratin, with a c = ft/s 2.

22 Slutin: GROUP PROBLEM SOLVING (cntinued) 1) First cnsider ball A. With the rigin defined at the grund, ball A is released frm rest ((v A ) = 0) at a height f 40 ft ((s A ) = 40 ft). Calculate the time required fr ball A t drp t 20 ft (s A = 20 ft) using a psitin equatin. s A = (s A ) + (v A ) t + (1/2) a c t 2 S, 20 ft = 40 ft + (0)(t) + (1/2)(-32.2)(t 2 ) => t = s

23 Slutin: GROUP PROBLEM SOLVING (cntinued) 2) Nw cnsider ball B. It is thrw upward frm a height f 5 ft ((s B ) = 5 ft). It must reach a height f 20 ft (s B = 20 ft) at the same time ball A reaches this height (t = s). Apply the psitin equatin again t ball B using t = 1.115s. s B = (s B ) + (v B ) t + (1/2) a c t 2 S, 20 ft = 5 + (v B ) (1.115) + (1/2)(-32.2)(1.115) 2 => (v B ) = 31.4 ft/s

24 ATTENTION QUIZ 1. A particle has an initial velcity f 3 ft/s t the left at s 0 = 0 ft. Determine its psitin when t = 3 s if the acceleratin is 2 ft/s 2 t the right. A) 0.0 ft B) 6.0 ft C) 18.0 ft D) 9.0 ft 2. A particle is mving with an initial velcity f v = 12 ft/s and cnstant acceleratin f 3.78 ft/s 2 in the same directin as the velcity. Determine the distance the particle has traveled when the velcity reaches 30 ft/s. A) 50 ft B) 100 ft C) 150 ft D) 200 ft

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