按一下以編輯母片標題樣式,按一下以編輯母片,第二層,第三層,第四層,第五層,*,*,*,19.1,線動量與角動量,19.2,衝量動量原理,作業十,19.3,動量守恆,19.4,偏心碰撞,作業十,CH19,剛體之平面運動力學,:,衝量與動量法,1,本章學習目標,1.,了解定義並計算動量,角動量,衝量,角衝量,2.,了解並應用動量衝量原理,3.,了解並應用動量守恆原理,4.,了解並計算非彈性碰撞問題,衝量動量法對動力學某一類型的問題特別有效,衝量動量法可以用來驗證、補充力與加速度法,2,基本定理,:,其中,:,線動量,:,外力,:,對,P,點之角動量,:,對,P,點之扭力,角動量必須相對於一,P,點選取,19.1,線動量與角動量,定義,3,證明,:,0,請參考課本,Ch17,4,P:,剛體上任一點,G:,剛體上的質心,5,特例,1.,若,P,為固定點,(,靜止點,),且剛體在平移,2.,若,P,為固定點且為剛體之旋轉中心,(O),3.,若,P,為質心,(,運動中,),定義,6,證明,3:,證明,2:,7,例題,19-1(P.470)fig 19-3,At a given instant the 10-kg disk and 5-kg bar have the motions shown in,fig.19-3a.Determine their angular,momenta,about point G and about point B for the disk and about G and about the IC for the bar at this instant,.,目標,:,公式運用,1.,座標,B,2.,待求,3.,已知,8,4.,解,Disk:,Rod:,1.,使用 求,3.,質心,2.,9,19-2,衝量、動量原理,衝量,=,動量變化,角衝量,=,角動量變化,對多剛體系統,線衝量,角衝量,10,若,P,為定點且剛體作平移運動,若,P,為定點且為旋轉中心,若,P,點為質心,(,動點,),11,證明,:,12,中許多內力會互相抵消,或使許多單剛體之外力相對系統成為成對的內力而彼此相抵消,或由於,P,點之選擇會使許多外力對,P,點之扭力 為零,因此可簡化計算,使用共同,P,點,(,只有一個,P,點,),例,:EX19-4,13,例題,:19-2(p.474)fig 19-5,The 100-N(10-kg)disk shown in Fig.19-5a is assumed to be uniform and is pin-supported at its center.If it is acted upon by a constant couple moment of,6 N m and a force of 50 N which is applied to a cord wrapped around its periphery,determine the angular velocity of the disk two seconds after starting from rest.Also,what are the force components of reaction at the pin?,0.,目標：公式運用,1.,座標：,3.,已知,2.,待求,(,建議使用,),14,4.,解,:FBD,四個力：,W,、,F,、,、,一個力偶,M,(A,點為固定點,),外力也會造成扭矩,15,例題,:19-3(p.475)fig 19-6,The 100-kg spool shown in Fig.19-6a has a radius of gyration k,G,=0.35 m.A cable is wrapped around the central hub of the spool,and a horizontal force having a variable magnitude of P=(t+10)N is applied,where t is in seconds.If the spool is initially at rest,determine its angular velocity in 5 s.Assume that the spool rolls without slipping at A.,0.,目標,:,題型辦認,1.,座標,:,;at rest,(,拘束條件,:at A,No slipping),3.,已知,2.,待求,(,建議使用,),16,4.,解,:FBD,拘束條件,(at A no slipping),1,3,2,17,例題,19-4(P.476),參考,fig 19-7,The block shown in Fig.19-7a has a mass of 6 kg.It is attached to a cord which is wrapped around the periphery of a 20-kg disk that has a moment of inertia I,A,=0.40 .If the block is initially moving downward with a speed of,2,m/s,determine its speed in 3 s.Neglect the mass of the cord in the calculation.,0.,目標,:,公式運用,1.,座標,r,A,=0.2 m I,A,=0.4 kg-m,m,A,=20 kg,m,B,=6 kg,拘束條件,No Slipping,2.,待求,3.,已知,(,建議使用,),18,4.,解,:FBD,Disk:,四力,:A,x,A,y,W,A,T,Block:,二力,:T,W,B,Disk(,轉動,),Block:,拘束條件,:,解之,:,A,聯立,rad/s,m/s,19,例題,19-5(P.478)fig 19-8,The,Charpy,impact test is used in materials testing to determine the energy absorption characteristics of a material during impact.The test is performed using the pendulum shown in fig.19-8a,which has a mass,m,mass center at G,and a radius of gyration about G.Determine the distance from the pin at A to the point,P,where the impact with the specimen S should occur so that the horizontal force at the pin is essentially zero during the impact.For the calculation,assume the specimen absorbs all the pendulum,s kinetic energy gained during the time it falls and thereby stops the pendulum from swinging when,0.,目標：基本公式運用,1.,座標：,m,3.,已知,2.,待求,(,建議使用,),20,4.,解：,FBD,三個力,(,注意,:,碰撞前瞬間為狀態,1,，碰撞後為,2),21,作業十,合作習題,:,19-17 (P.481),19-18 (P.482),19-20 (P.482),19-22 (P.483),19-,24,(P.48,3,),19-25,(P.483),22,第十作業題型分類,題型 例題 作業,習題,軸心不為形心,ex1,5 24,25 1,2,3,4,11,16,30,有滑動,17,18 21,28,軸心為形心,ex2,4 22 5,6,7,8,9,10,12,15,無滑動,ex3 20 13,14,19,29,定 軸 旋 轉,滾 動 問 題,23,26,27,31,32,23,19-3,動量守恆,(P.486),若系統所受線性衝量之和為零，則,：,線動量守恆,不同的兩時間點,若系統所受角衝量之和為零，則,O,：,表示相對一固定點或質心,動量守恆常可用以驗證力與加速度法的答案是否正確,.,：角動量守恆,24,例題,19-6(P.488)ref fig 19-9,The 10-kg wheel shown in Fig 19-9a has a moment of inertia,Assuming that the wheel does not slip or rebound,determine the minimum velocity it must have to just roll over the obstruction at A.,目標,：,角動量守恆,座標,：,Can roll over,待求：,已知：,前,後,3.,爬上,A,點之上,25,解,FBD,：,注意 由於碰撞時間極短,與 相比可視為無作用,!,G,0,再選,A,為角衝量計算,則,故角衝量為零,(,記得,),角動量,(,相對,A,點,),守恆,故線衝量中,線動量不守恆,When roll over=0,沒有接觸,四個力：,重力,碰撞力,摩差力,正向力,26,因此,:,以,A,為參考點,(,之作用點,),力臂為零。使得 形成之,角衝量為零。,線動量不守恆但角動量守恆,No slipping,代入上式得,27,能量守恆,T,2,+V,2,=T,3,+V,3,抬升到最高點之最小能量,.,動能為零,基準線定在碰撞後，未抬升之前,28,例題,19-7(P.489)fig 19-10,The 5-kg slender rod shown in Fig.19-20a is pinned at,O,and is initially at rest.If a 4-g bullet is fired into the rod with a velocity of 400,m/s,as shown in the figure,determine the angular velocity of the rod just after the bullet becomes embedded in it.,目標,：角動量守恆,座標,：,O,待求：,2.,1.,已知：,29,解,：,FBD,將子彈,+,桿看成一個系統,則只有六個力,：,梢的反作用力,桿與子彈之重力,子彈與桿之間之作用力，反作用力,(,注意,F,與,-F,大小相等，方向相反，是系統之內力。,形成之衝量及角衝量會彼此相消,),所以有角動量守恆,以,O,點為參考點,，,以消去,O,x,，,未知力的影響,(,因為力臂長為零,，,所以力偶為零。,),30,桿靜止,碰撞前,:,碰撞後,:,又,:,31,19.4,偏心碰撞,(P.490),解碰撞的問題要使用,1.,角動量守恆公式,(,能量不守恆,),2.,恢復係數公式,若,完全彈性碰撞，則,完全非彈性碰撞,(,塑性碰撞,),，則,32,例題,19-8(P.493)fig19-2,The 5-kg slender rod is suspended from the pin at A,Fig 19-12a.If a 1-kg ball B is thrown at the rod and strikes its center with a horizontal velocity of 9m/s,determine the angular velocity of the rod just