DREXEL PHYS 113 - More on the Momentum Principle (40 pages)

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More on the Momentum Principle



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More on the Momentum Principle

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Pages:
40
School:
Drexel University
Course:
Phys 113 - Contemporary Physics I
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Chapter 2b More on the Momentum Principle 2b 1 Derivative form of the Momentum Principle 2b 1 1 An approximate result F ma 2b 2 Momentum not changing 2b 2 1 Static equilibrium 2b 2 2 Uniform motion 2b 2 3 Momentarily at rest vs static equilibrium 2b 2 4 Finding the rate of change of momentum 2b 2 5 Example Hanging block static equilibrium 2b 2 6 Preview of the Angular Momentum Principle 2b 3 Curving motion 2b 3 1 Example The Moon around the Earth curving motion 2b 3 2 Example Tarzan swings from a vine curving motion 2b 3 3 The Momentum Principle relates different things 2b 3 4 Example Sitting in an airplane curving motion 2b 3 5 Example A turning car curving motion 2b 4 A special case Circular motion at constant speed 2b 4 1 The initial conditions required for a circular orbit 2b 4 2 Nongravitational situations 2b 4 3 Period 2b 4 4 Example Circular pendulum curving motion 2b 4 5 Comment Forward reasoning 2b 4 6 Example An amusement park ride curving motion 2b 5 Systems consisting of several objects 2b 5 1 Collisions Negligible external forces 2b 5 2 Momentum flow within a system 2b 6 Conservation of momentum 2b 7 Predicting the future of complex gravitating systems 2b 7 1 The three body problem 2b 7 2 Sensitivity to initial conditions 2b 8 Determinism 2b 8 1 Practical limitations 2b 8 2 Chaos 2b 8 3 Breakdown of Newton s laws 2b 8 4 Probability and uncertainty 2b 9 Derivation of the multiparticle Momentum Principle 2b 10 Summary 2b 11 Review questions 2b 12 Problems 2b 13 Answers to exercises 114 115 116 116 116 117 118 119 120 120 121 122 123 123 125 126 127 128 129 129 131 131 132 133 134 134 136 136 137 137 138 138 139 139 140 142 143 144 151 114 Chapter 2 The Momentum Principle Chapter 2b More on the Momentum Principle In this chapter we continue to apply the Momentum Principle to a variety of systems The major topics in this chapter are The derivative form of the Momentum Principle with applications Applying the Momentum Principle to multiparticle systems Conservation of momentum 2b 1 Derivative form of the Momentum Principle The forms of the Momentum Principle given so far p F net t and p f p i F net t are particularly useful when we know the momentum at a particular time and want to predict what the momentum will be at a later time We use these forms in repetitive computer calculations to predict future motion Another important form is obtained by dividing by the time interval t p F net t Just as we did in finding the instantaneous velocity of an object see Section 1 7 6 we can find the instantaneous rate of change of momentum by letting the time interval t approach zero an infinitesimal time interval The ratio of the infinitesimal momentum change written as dp to the infinitesimal time interval written as dt is the instantaneous rate of change of the momentum As the time interval gets very small t 0 the ratio of the momentum change to the time interval approaches the ratio of the infinitesimal quantities dp p lim dt t 0 t and we obtain the following form THE MOMENTUM PRINCIPLE DERIVATIVE FORM dp F net dt In words the instantaneous time rate of change of the momentum of an object is equal to the net force acting on the object Or in calculus terms since the limit of a ratio of infinitesimal quantities is called a derivative we can say that the derivative of the momentum with respect to time is equal to the net force acting on the object This form of the Momentum Principle is useful when we know something about the rate of change of the momentum at a particular instant Knowing the rate of change of momentum we can use this form of the Momentum Principle to deduce the net force acting on the object which is numerically equal to the rate of change of momentum Knowing the net force we may be able to figure out particular contributions to the net force 2b 1 Derivative form of the Momentum Principle Ex 2b 1 At a certain instant the z component of the momentum of an object is changing at a rate of 4 kg m s per second At that instant what is the z component of the net force on the object Ex 2b 2 If an object is sitting motionless what is the rate of change of its momentum What is the net force acting on the object Ex 2b 3 If an object is moving with constant momentum 10 12 8 kg m s what is the rate of change of momentum dp dt What is the net force acting on the object 2b 1 1 An approximate result F ma In simple cases the derivative form of the Momentum Principle reduces to a form that may be familiar to you from a previous course If the mass is constant the usual situation and the speed is small enough compared to the speed of light that the momentum is well approximated by p mv we have this dp d mv dv m F net nonrelativistic form constant mass dt dt dt dv The quantity is called acceleration and is often given the symbol a dt This form of the Momentum Principle Newton s second law says that mass times acceleration time rate of change of velocity dv dt is equal to the net force or in simplified scalar form ma F or F ma The approximate form is not valid in situations where an object s mass isn t constant One example is a rocket with exhaust gases ejecting out the back as a result the rocket has decreasing mass In such cases the momentum based formula dp dt F net gives the correct results whereas the constant mass formula mdv dt F net cannot be used The approximate form is also not valid for objects moving at speeds close to the speed of light because in these circumstances the approximation p mv is not valid When the principle is written in terms of momentum it is valid even for objects whose mass changes or which are moving at speeds close to the speed of light as long as we use the relativistically correct definition of momentum So dp dt F net is the more general and more powerful form The extra symbols on dp dt F net compared to the simplified version F ma are important The Momentum Principle is a vector principle so the arrows over the symbols are extremely important they remind us that there are really three separate component equations for x y and z It is also important to remember that we have to add up all vector forces to give the net force so we write F net not just F Ex 2b 4 The velocity of a 80 gram ball changes from 5 0 3 m s to 5 02 0 …


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