Unit 5: Conservation of Momentum

1. Unit 5: Conservation of Momentum • Force and Momentum and Conservation of Momentum (9-1,9-2, 9-3) • Collisions (9-4, 9-5, 9-6) • Collisions, Center of Mass (9-7, 9-8) • Center of Mass, Rocket Propulsion, and Quiz Three Review (9-9, 9-10) Physics 253

2. Center of Mass and Translational Motion • We can prove that the translational motion of a system can be described by considering the effect of the net force on the center of mass. • In that sense the system is no different than a single force incident on a single particle. Physics 253

3. Since all the internal forces cancel by the 3rd Law, the summation involves only the external forces: • This is Newton’s 2nd Law applied to a system: the sum of all the forces acting on a system is equal to the total mass of the system times the acceleration of its CM. • In other words, the CM behaves as if all the mass of a system or object resides at the CM and as if all the external forces act directly upon it! • This makes treatment of the translation motion of a system quite straightforward. Physics 253

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5. Problem A rocket shot into space reaches it’s highest point and a horizontal distance d from it’s origin separates into two equal mass segments. Imagine the first part just stopped and dropped immediately to the earth. Where does part II land? Answer Because the explosion is “internal” the CM will continue on it’s path. The explosion happened at the top of the trajectory and if the rocket had maintained it’s trajectory the CM would have landed 2d from the origin. Thus the average position of the two equal mass fragments at impact must be 2d. This can only occur if the second stage travels a total horizontal distance of 3d Notice how we didn’t need to resort to any fancy equations of motion and just applied the 2nd Law for systems. Example: A Two-Stage Rocket Physics 253

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7. Systems of Variable Mass • Up until now we’ve limited ourselves to systems of constant mass such that • But what if the mass of the system varies? Then derivative will have two terms: one proportional to dv/dt and the other to dm/dt. Physics 253

8. Two Examples where dm/dt is Nonzero • Rockets where dm/dt < 0. • Conveyer belts where dm/dt>0. • Let’s set up a formalism for such cases. Physics 253

9. Time t Time t+dt Physics 253

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12. The left side of the equation is just the usual mass times acceleration term. • The summation • refers to all external forces such as gravity, friction, etc… • it does not include the force on M due to dM. • The term which depends on the change of mass with respect to time corresponds to the force exerted on M due to the addition or ejection of mass. It is also the change of momentum due to the addition of mass and often called thrust. • This equation can be used to describe a system subject to changes of mass. Physics 253

13. Example 1: A Conveyer Belt • A hopper drops gravel at a rate of 75.0kg/s onto a conveyer belt moving at a constant speed of v=2.20m/s. • Determine • The force needed to keep the belt moving. • The power output of the motor drive. Physics 253

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15. Example 2: A Rocket! • A vertically fired rocket of total mass 21,000kg carries 15,000kg fuel. The fuel is burned at a rate of 190kg/s and leaves the engine at a rate of 2800m/s relative to the rocket. • Calculate • The rocket thrust • The net force on the rocket at blastoff and at burnout • The velocity as a function of time • Final velocity at burnout. • Assume no friction and g. +X Physics 253

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19. Quiz Number Three • Same format as first two quizzes. • Total of 50 points • 4 multiple choice @ 5 pts each • 3 problems @ 5, 10, 15 points. • Bring a calculator • Bring a 3x5” note card crib sheet. Physics 253

20. Definition of Work • F is force • d is displacement • q is the angle between the force and displacement Physics 253

21. V1 V2 F d Definition of Kinetic Energy and the Work-Energy Theorem • Consider an object of mass m moving initially with speed v1. • Now accelerate it uniformly to speed v2 by applying a constant force F over a distance d. Physics 253

22. Gravitational Potential Energy • At the surface of the earth • U=mgh • At any radius • U= GMEm/r Physics 253

23. Potential Energy of a Spring • U = (1/2)kx2 • k = spring constant • x= displacement Physics 253

24. Conservation of Mechanical Energy • Let’s just consider a conservative system (earth and object, spring and mass…) • One in which the work does not depend on the path taken • Or equivalently one for which the work around a closed path is zero • One in which energy can be transformed from kinetic energy to potential and back again. • By the work-energy principle we know that Physics 253

25. The Classic Example: The Falling Rock • Anywhere along the path the total mechanical energy is E= (1/2)mv2+mgh. • When dropped, the rock has no velocity so K =(1/2)mv2=0 but it has potential energy U=mgh • As it falls, K increases and the potential energy decreases so that the total energy is constant. • At the surface, K is at a maximum and the potential at a minimum, U=mgh=0. Physics 253

26. What’s it good for? If h=3.0m, calculate the speed when the rock is 1.0 m above the ground. Physics 253

27. The Law of Conservation of Momentum: The total momentum of an isolated system of bodies remains constant. Physics 253

28. Example: The Kick of a Rifle • What is the recoil velocity of a 5.0-kg rifle that shoots at 0.050-kg bullet at a muzzle velocity of 120m/s? • What would be the initial velocity of a 75-kg rifleperson? Physics 253

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30. Center of Mass Physics 253