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M58; Derivation of the Constants of the Motion for Central Forces MISN-0-58 DERIVATION OF THE CONSTANTS OF THE MOTION FOR CENTRAL FORCES DERIVATION OF THE CONSTANTS OF THE MOTION FOR CENTRAL FORCES ^q by ^r Peter Signell ` 1. Introduction . 1 r 2. Motion In a Plane . 1 q 3. Central Forces a. De¯nition . 2 origin b. Angular Component: Conserv. of Angular Momentum . .2 c. Radial Components: Conservation of Energy . .3 4. Conclusion . 4 Acknowledgments. .4 Project PHYSNET·Physics Bldg.·Michigan State University·East Lansing, MI 1 2 ID Sheet: MISN-0-58 THIS IS A DEVELOPMENTAL-STAGE PUBLICATION Title: Derivation of the Constants of the Motion for Central OF PROJECT PHYSNET Forces The goal of our project is to assist a network of educators and scientists in Author: Peter Signell, Michigan State University transferring physics from one person to another. We support manuscript Version: 2/1/2000 Evaluation: Stage 4 processing and distribution, along with communication and information systems. We also work with employers to identify basic scienti¯c skills Length: 1 hr; 12 pages as well as physics topics that are needed in science and technology. A Input Skills: number of our publications are aimed at assisting users in acquiring such skills. 1. State the Law of Conservation of Angular Momentum (MISN-0- 41). Our publications are designed: (i) to be updated quickly in response to 2. State the Law of Conservation of Energy (MISN-0-21). ¯eld tests and new scienti¯c developments; (ii) to be used in both class- room and professional settings; (iii) to show the prerequisite dependen- Output Skills (Knowledge): cies existing among the various chunks of physics knowledge and skill, K2. Separate the polar-coordinate variables in Newton's Second Law as a guide both to mental organization and to use of the materials; and and thereby derive the constants of motion, angular momentum (iv) to be adapted quickly to speci¯c user needs ranging from single-skill and energy, for central forces. instruction to complete custom textbooks. K3. Derive the time derivatives of the polar coordinate unit vectors. New authors, reviewers and ¯eld testers are welcome. Post-Options: PROJECT STAFF 1. \Derivation of Inverse Square Law Force Field Orbits" (MISN-0- 106). Andrew Schnepp Webmaster Eugene Kales Graphics Peter Signell Project Director ADVISORY COMMITTEE D. Alan Bromley Yale University E. Leonard Jossem The Ohio State University A. A. Strassenburg S. U. N. Y., Stony Brook Views expressed in a module are those of the module author(s) and are not necessarily those of other project participants. °c 2001, Peter Signell for Project PHYSNET, Physics-Astronomy Bldg., Mich. State Univ., E. Lansing, MI 48824; (517) 355-3784. For our liberal use policies see: http://www.physnet.org/home/modules/license.html. 3 4 MISN-0-58 1 MISN-0-58 2 DERIVATION OF THE That is: r^(t + dt) = r(t) + (dθ)s^. Then: CONSTANTS OF THE MOTION dr^ r^(t + dt) ¡ r^(t) dθ = = s:^ FOR CENTRAL FORCES dt dt dt by We will henceforth use the \dot" notation for time derivatives. Then the above equation becomes: Peter Signell r^_ = θ_s^: Similarly, you can easily derive that 1. Introduction s^_ = ¡θ_r^: In this module the constancy of angular momentum and energy for Then returning to ~r = rr^, we can easily ¯nd its derivative by taking the central forces is derived from Newton's Second Law in a straightforward derivative of the product: manner using vector algebra. Elsewhere the results of this module are ~v = ~r_ = r_r^ + rr^_ = r_r^ + rθ_s^: used to derive the equations for planetary orbits.1 Taking the derivative again, _ Ä _ _ Ä _2 _2 _ Ä 2. Motion In a Plane ~a = ~v = ~r = rÄr^ + r_θs^ + r_θs^ + rθs^ ¡ rθ r^ = (rÄ ¡ rθ )r^ + (2r_θ + rθ)s^: We will deal only with motion in a plane, which is the general case for motion of an object in a central force. The coordinates used will be polar, as shown in Fig. 1. Unit vectors will be denoted r^ and s^ as shown 3. Central Forces in Fig. 1. Thus ~r = rr^. For motion at ¯xed θ, which is radial motion, 3a. De¯nition. A central force is one that is completely radial: both r^ and s^ remain constant (unchanged) as time increases. However if s^ changes with time, then in time dt the unit vector r^ will change by an F~c = f(r; θ)r^: dθ s amount ( )^ as shown in Fig. 2. Then by Newton's Second Law: 1See \Derivation of Inverse Square Law Force Field Orbits" (MISN-0-106). 2 F~c = f(r; θ)r^ = m~a = m(rÄ ¡ rθ_ )r^ + m(2r_θ_ + rθÄ)s^: (1) 3b. Angular Component: Conserv. of Angular Momentum. q^ ^ qq^ ^ Taking angular components by taking the dot product of both sides of ^ r(t+dt) d = dr r Eq. (1) by s^: ` _ Ä r 0 = m(2r_θ + rθ) : r q d ^r(t) In order to form a complete di®erential, we multiply both sides by r: q origin d 0 = m(2rr_θ_ + r2θÄ) = (mr2θ_) : dt origin Thus the time derivative of mr2θ_ is zero, so Figure 1. Polar coordinate Figure 2. The radial unit vector at 2 symbols used here. two times. mr θ_ = constant; it is, in fact, the angular momentum about an axis normal (perpendicular) to the plane of motion and passing through the origin: L = mr2θ_ = I! = constant: (2) 5 6 MISN-0-58 3 MISN-0-58 4 3c. Radial Components: Conservation of Energy. Taking radial and Eq. (5) can be written: components by taking the dot product of both sides of Eq. (1) with r^: d 1 2 2 mv + Ep(r) = 0: f(r; θ) = m(rÄ ¡ rθ_ ) : dt ·2 ¸ We can eliminate θ_ by solving Eq. (2) for it. Then Then: 1 2 mv + Ep(r) = constant; L2 L2 2 f(r; θ) = m rÄ ¡ 2 3 = mrÄ ¡ 3 : µ m r ¶ mr where Ep(r) = ¡ f(r)dr is the potential energy. The equation is some- times written in theR \radial plus angular kinetic energy" form, obtained If the force function f(r; θ) is a function of radius alone, then by substituting Eq. (4) into Eq. (5): L2 2 1 2 L mrÄ ¡ 3 ¡ f(r) = 0: mr_ + + E (r) = constant: mr 2 2mr2 p In order to form a complete di®erential we multiply both sides by r_ to This, in turn, can be written in terms of radial momentum and moment get: 2 of inertia as: 2 2 L pr L mr_rÄ ¡ r_ ¡ r_f(r) = 0: (3) + + Ep(r) = constant: (6) mr3 2m 2I To see that this is a complete di®erential, form: Then Eq. (6) is the sum of radial kinetic, angular kinetic, and potential energy. L2 v2 = ~v ¢ ~v = r_2 + r2θ_2 = r_2 + ; (4) m2r2 4. Conclusion so that: 2 d 2 2L The appearance of Conservation of Angular Momentum from the v = 2r_rÄ ¡ 2 3 r_: dt m r angular equation and Conservation of Energy from the radial equation is Then Eq. (3) can be written: very interesting. A whole di®erent formulation, called the Hamiltonian formalism, uses such ideas as its basis. Identi¯cation of the constants d 1 2 of motion is thereby greatly simpli¯ed, especially in cases of complex mv ¡ r_f(r) = 0: (5) dt µ2 ¶ motions. The last term can also be converted to di®erential form by de¯ning: Acknowledgments Ep(r) = f(r) dr; Z Preparation of this module was supported in part by the National Science Foundation, Division of Science Education Development and Re- so that: search, through Grant #SED 74-20088 to Michigan State University. dE (r) f(r) = ¡ p : dr Then: dE (r) dE (r) p = p r_ = ¡f(r)r_; dt dr 7 8 MISN-0-58 ME-1 MODEL EXAM 1. See Output Skills K1-K3 in this module's ID Sheet. Credit may be received by taking a written exam (closed book, no notes) or by means of a lecture (closed book, one 3"£5" card of notes). For lecture credit, there must be no gaps, erasures, or pauses. Be sure to include diagrams in either exam mode. 9 10 11 12.
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