Systems of Particles: Angular Momentum and Work Energy Principle

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1 J/1.053J Dyamics ad Cotrol I, Sprig 2007 Professor Thomas Peacock 2/20/2007 Lecture 4 Systems of Particles: Agular Mometum ad Work Eergy Priciple Systems of Particles Agular Mometum (cotiued) τ ext : Total Exteral Torque B H B : Total Agular Mometum P : Total Liear Mometum From ow o, τ ext = τ B. B d τ ext B = H B + v B P If τ B = 0 ad v B = 0 or if B is the ceter of mass or if v B v C the H B = costat (Coservatio of Agular Mometum). You may be familiar with τ B = d H B (oly valid if v B = 0 or v B P ). Agular mometum H B of a collectio of particles about poit B is give by: where h Bi = r i m i v i. N H B = h Bi If (H B ) is the sum of the agular mometa of the idividual particles about poit B, Cite as: Thomas Peacock ad Nicolas Hadjicostatiou, course materials for 2.003J/1.053J Dyamics ad Cotrol I, Sprig MIT OpeCourseWare ( Massachusetts Istitute of Techology. Dowloaded o [DD Moth YYYY].

2 2 Figure 1: Agular mometum about B for a system of particles. Each particle has mass m i positios r i with respect to the origi ad r i with respect to B. The ceter of mass C has positios r c with respect to B ad ρ i with respect to each poit mass m i. Figure by MIT OCW. Therefore, we write: H B = r i m i v i = r i m i r i = (r c + ρ ) m i v i i = (r c m i v i ) + ρ i m i v i = r c Mv c + ρ i m i v i where we have used m i v i = Mv c H B = H C + r c P Notice that v B does ot appear i this equatio. The agular mometum about B is the agular mometum about the ceter of mass (C) plus the momet of the system liear mometum (Mv C = P ) about B. We will use these equatios for rigid bodies. With rigid bodies will eed to use momets of iertia. Cite as: Thomas Peacock ad Nicolas Hadjicostatiou, course materials for 2.003J/1.053J Dyamics ad Cotrol I, Sprig MIT OpeCourseWare ( Massachusetts Istitute of Techology. Dowloaded o [DD Moth YYYY].

3 3 Work Eergy Priciple r [ ] 2i t 2 t d 2 d 1 W 12 = F i dr i = p v i = m i (v i v i ) r 1i t 1 i t 1 2 where: = T 2 T 1 r2i F it = W it r i dr i 12 r 1i 2i F ext dr i = W ext 1 T = m i (v i v i ) 2 r2i W 12 = (W 12 ) i = F i dr i r i 12 1i r 1i r 2i r 2i = F it i dr i + F ext i dr i r 1i r 1i This is o-zero i geeral. t 2 W 12 it = F i it v i t 1 t 2 = f v i t 1 j=1 j 1 t 2 = (f v i + f ji v j ) t 1 j>1 t 2 W 12 it = f (v i v j ) t 1 j>1 Cite as: Thomas Peacock ad Nicolas Hadjicostatiou, course materials for 2.003J/1.053J Dyamics ad Cotrol I, Sprig MIT OpeCourseWare ( Massachusetts Istitute of Techology. Dowloaded o [DD Moth YYYY].

4 4 Figure 2: Relative velocity probably has a compoet i the directio of f. The figure shows two radom poits with radomly chose velocities. Uless the differece betwee the velocities of the two poits is zero or perpedicular to the directio of force f, f (v i v j ) will ot be zero; there would be some compoet i the directio of f. Figure by MIT OCW. No reaso that differece betwee velocities should ot have a compoet i the directio of f. If particles are parts of a rigid body system, the there is o relative motio i the directio of f (e.g.) Figure 3: Two poit masses coected by a rod. This is a example of a rigid body where due to the rod, there is o relative motio of the two poit masses at each ed whe the rigid body moves. Figure by MIT OCW. d ri r j 2 = 0 [ ] d (r i r j ) (r i r j ) = 0 2(r i r j ) (v i v j ) = 0 Iteral forces f are alog the directio (r i r j ). f (v i v j ) = 0. Therefore, for a rigid body system we have proved: Cite as: Thomas Peacock ad Nicolas Hadjicostatiou, course materials for 2.003J/1.053J Dyamics ad Cotrol I, Sprig MIT OpeCourseWare ( Massachusetts Istitute of Techology. Dowloaded o [DD Moth YYYY].

5 Examples 5 f (v i v j ) = 0 Therefore, W it must be 0 (or if you show that iteral forces do o work). 12 Thus, More geerally: W ext = T 2 T 1 12 W ext + W it = T 2 T If all exteral forces are potetial forces or the oes who are exteral do o work ad W it = 0, 12 W = W ext = V ext V ext V = potetial work where V ext = V i ext ext. V i is the exteral force potetial of particle i. Examples Example 1 T 1 + V ext = T 2 + V ext 1 2 How does l affect the motio? How does θ affect the motio? No rotatios ivolved. Probably will ot eed agular mometum. Kiematics Describe the motio (kiematics) without forces Kowig the locatio of A is equivalet to kowig the locatio of the ceter of mass of M. v C = v A M m r A = xî r B = xî + sê s = xî + s cos θî + s si θĵ ṙ A = ẋî ṙ B = ẋî + ṡ cos θî + ṡ si θĵ r A = ẍî r B = ẍî + s cos θî + s si θĵ Note: Geeralized coordiates. î ad ê s are ot. Importat to defie coordiates. Cite as: Thomas Peacock ad Nicolas Hadjicostatiou, course materials for 2.003J/1.053J Dyamics ad Cotrol I, Sprig MIT OpeCourseWare ( Massachusetts Istitute of Techology. Dowloaded o [DD Moth YYYY].

6 Examples 6 Figure 4: Block o frictioless surface that moves o frictioless road. The mass (m) ca slide dow the iclie i a frictioless maer. Mass (M) is free to move horizotally without frictio. If mass (m) is released from rest at the l positio, fid the velocity of mass (M) at the momet (m) reaches the bottom of the iclie. Figure by MIT OCW. Figure 5: Diagram of kiematics of block o ramp. Need two sets of coordiates. M oly moves i the x-directio. m oly moves i the ê s directio. Figure by MIT OCW. Cite as: Thomas Peacock ad Nicolas Hadjicostatiou, course materials for 2.003J/1.053J Dyamics ad Cotrol I, Sprig MIT OpeCourseWare ( Massachusetts Istitute of Techology. Dowloaded o [DD Moth YYYY].

Problem 1. Problem Engineering Dynamics Problem Set 9--Solution. Find the equation of motion for the system shown with respect to:

Problem 1. Problem Engineering Dynamics Problem Set 9--Solution. Find the equation of motion for the system shown with respect to: 2.003 Egieerig Dyamics Problem Set 9--Solutio Problem 1 Fid the equatio of motio for the system show with respect to: a) Zero sprig force positio. Draw the appropriate free body diagram. b) Static equilibrium