Moment and Impulse

Moment and Impulse - 8 Momentum and impulse 8.1 Linear...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
8 Momentum and impulse 8.1 Linear momentum LO = c R, (m, oR, ) We have shown in Chapter 3 that, for any system of particles or rigid body, (3.15) Integrating this equation with respect to time = wxrn,R,2 Since c rn, R: is Io, the moment of inertia of the body about the z-axis, the total moment of momentum for this case is F = IMifc Lo = low (8-2) gives [f Fdl= [2Mi;;dt = ~i~~-~i~~ (8.1) The integral J:Fdt is known as the impulse and In Chapter 6 we showed that, for a rigid body in (6.11) 1 general plane motion, MG = IGh is a vector quantity. Because c m,i, = MiG = the linear momentum gives we can write [ r MGdt = [ f IG hdt = IGO~ - 6J1 (8.3) impulse = change in linear momentum or, symbolically, that is, I= AG moment of impulse = change in moment of momentum 8.2 Moment of momentum From Fig. 8.1 we see that the moment momentum about the z-axis of a particle which is KG = ALG If rotation is taking place about a fixed axis then equation 6.13 applies which, when inte- grated, leads to M,dt = Iowdt = Io%- lowl (8.4) or Ko=ALo 8.3 Conservation of momentum If we now consider a collection of particles or effects from bodies outside the system, then moving On a circu1ar path, radius R1, about the rigid bodies interacting without any appreciable z-axis is = R, (m, OR, 1 Crn,Fl = o (8.5) For a rigid body rotating about the z-axis with angular velocity w, the total moment of momen- tum is so that c~,~, = constant i.e. linear momentum is conserved.
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
112 Momentum and impulse Extending equation 6.12a for a system of bodies, c ZG h + TGMaGe = 0 (8.6) c o + 2 rG MvGe = constant (8-7) Integrating with respect to time gives which is an expression of the conservation of moment of momentum. The term ‘angular momentum’ is often used in this context but is not used in this book since the term suggests that only the moments of momentum due to rotation are being considered whereas, for example, a particle moving along a straight line will have a moment of momentum about a point not on its path. 8.4 Impact of rigid bodies We can make use of these conservation principles very effectively in problems involving impact. In many cases of collision between solid objects the time of contact is very small and hence only small changes in geometry take place during the contact period, although finite changes in velocity occur. As an example, consider the impact of a small sphere with a rod as shown in Fig. 8.2. The rod is initially at rest prior to the impact, so that u1 = 0 w1 = 0; u2, v2 y are the velocities after impact. Figure 8.2 Conservation of linear momentum gives mvl = Mu2 + mv2 (8.8) and conservation of moment of momentum about an axis through G gives mvla = IG(r4?+mv2a (8.9) So far we have two equations, but there are three unknowns. To provide the third equation we shall make some alternative assumptions: i) the rigid-body kinetic energies are con- served, or ii) the two objects coalesce and continue as a single rigid body.
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 15

Moment and Impulse - 8 Momentum and impulse 8.1 Linear...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online