Lab 6 : Momentum and Center of Mass

6 Momentum and Center of MassIntroductionCollisionsBeforeAftervA1vA2vB2m ABv B1m mm ABFigure 1: A “bouncy” collision in one dimension.Figure 1 shows a diagram of a “bouncy” (elastic) collision in one dimension betweentwo objects of masses m A and m B initially moving with velocities v A1 and v B1 . The law ofconservation of translational momentum for this collision is expressedm A v A1 +m B v B1 = m A v A2 +m B v B2 (1)where v A2 and v B2 are the velocities of the objects after the collision.Beforev B1 v 2Afterv A1m A m A+m Bm BFigure 2: A “sticky” collision in one dimension.Figure 2 shows a “sticky” (perfectly inelastic) collision in one dimension in which twoobjects collide and stick together. In this case, Eq. 1 simplifies tom A v A1 +m B v B1 = (m A +m B )v 2 (2)You will investigate qualitatively one dimensional collisions involving two carts of the kindused in Lab 3.

Center of MassThe center of mass of a system of particles is given by⃗R com = 1M sys(m A ⃗r A +m B ⃗r B +m C ⃗r C +...) (3)where⃗r A ,⃗r B , ⃗r C , ... are the positions of all of the particles in the system. In two dimensions,Eq. 3 corresponds toX com =Y com =1(m A x A +m B x B +m C x C +...)M sys1(m A y A +m B y B +m C y C +...) (4)M sysEqs. 3 and 4 also work for systems of extended objects, where the labels A, B, C, ...refer to objects instead of particles, and the positions ⃗r A , ⃗r B , ⃗r C , ... are the positions of thecenters of masses of the objects. Here, you will be considering two systems of objects – amobile and a person walking on a cart with “frictionless” wheels. In both of these cases, youcan ignore the vertical component of equation 4.ExperimentsCollisions in One Dimension1. Investigate “sticky” and “bouncy” collisions with carts of equal mass (no additionalmass added).(a) Under what circumstances is one of the objects stationary after the collision?(b) Under what circumstances are both of the objects stationary after the collision?2. Repeat the above process with carts of unequal mass. (Add both masses to the cart.)Mobile ConstructionConstruct a mobile following the general design shown in Figure 3 from two meter sticks,three mass hangers carrying 50 g, 100 g, and 200 g, and several lengths of string. Usethe concept of center of mass to explain quantitatively the locations of the pivot points ofthe meter sticks in your completed mobile. Show your mobile to your instructor/TA beforemoving on to your next task.Walking the PlankAn 8 foot plank is mounted on two carts in the middle of the lab. Explain what you observewhen one of the members of your lab group walks at a brisk pace from one end of the plankto the other. Please be careful – a spotter on either side of the plank should keep pace withthe walker just in case.

QuestionsFigure 3: Mobile design. (Locations of your pivot points may vary.)1. (a) Under what circumstances in one dimensional collisions between two objects is oneof the objects stationary after the collision? (b) How about both objects? (c) How doyour answers change if the objects have different masses?2. Where were the pivot points of the meter sticks in your mobile? Give your reasoning.3. What happens when a person walks the plank-on-wheels? Explain.