Review 2 Summary of principles Newton s second law F ma Technique Force Diagrams Circular motion Energy principle Technique Identify states with energy diagrams Circular motion Momentum principle Technique Draw momentum diagrams Two objects slide without friction on a circular ring of radius R oriented in a vertical plane The heavier object of mass 3m is attached to a spring with an unstretched length of zero admittedly an unphysical assumption and spring constant k The xed end of the spring is attached to a point a horizontal distance 2R from the center of the circle The lighter object of mass m is initially at rest at the bottom of the ring The heavier object is released from rest at the top of the ring then collides with and sticks to the lighter object Find the value for m that will allow the combined object of mass 4m to just reach the point A on the ring but go no higher Express your answer in terms of some or all of the quantities k R and the acceleration of gravity g Energy principle State 1 Initial state State 2 When 3m is at the bottom U U Momentum principle State 2 When 3m is at the bottom p 3mv State 3 When 3m collides with m Assume t 0 p 4mv Energy principle State 3 When 3m collides with m State 4 4m is at point A U U State 1 The spring is stretched a distance x State 2 The spring is at equilibrium State one Before collision State 2 The spring is maximally compressed U U U p U p Without friction State 1 Initial state State 2 Final state With friction State 1 Initial state E E E E State 2 Right before the collision Escape velocity means E 0 Velocity of a setellite Jet engine Problem Rocket leaving a planet has a nal kinetic energy of 0 U We want a rule for Technique momentum diagrams at time t and t The important thing is to identify the small changes such as m v etc What mass elements are changing momentum In the case of a jet engine these mass elements are air and fuel At time t At time t t Conservation of momentum
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