Question

Multivariable Calculus help with the magnitude of angular momentum: My questions is exercise 4 but I have attached exercise 1 and other notes that I was provided

4 Exercise 4. In any mechanics problem where the mass m is constant, the position vector F sweeps out equal areas in equal times the magnitude of the angular momentum ILI is conserved (Note: be sure to prove "if and only if") (Note: don't try to use Exercise 2 in the proof of Exercise 4.) EaUAL AREAS EN In particular we have shown that Kepler's Second Law is true for any central forcel The reasoning goes: for central forces L is conserved, so IL] is also conserved, so equal areas are swept out in equal times by Exercise 4. The special inverse square law form of the gravitational force is not needed here. The other two laws, however, definitely require more particular assumptions and are not true for every central force. Deriva tion of Kepler's Laws from Newtonian Mechanics and the Law of Gravitation. pldntrod bed out of the medieval worldview by distilling two quantitative patterns from a mountain uction to Kepler's Laws. Around 1609 Johannes Kepler (1571-1630) stum- of planetary observations. : Bach planet moves in an elliptical orbit with the sun at one focus. 2. The line segments joining the sun to a planet sweep out equal areas in equal times. Kepler later added a third law; the publication by Napier of logarithms in the intervening ten years had enabled Kepler to complete his incredible computational labors. 3. For any planet orbiting the sun, the square of the period bears the same proportion to the cube of the semimajor axis of its elliptical orbit. Newtonian Mechanics. To deduce Kepler's three empirical laws from general mechan- ical principles became the outstanding problem among natural philosophers. In this Issac Newton (1642-1725) succeeded beyond his contemporaries' wildest dreams. He provided a quantitative science of mechanics 1. NEWTON'S LAWS OF MOTION The net force Facting on an object is equal to the rate of change of its momentum ỹ with respect to time t. dp dt F(or F-mä i the mass m is constant.) 2. FORCE ADDS LIKE A VECTOR. If many forces, Fi, F2,..., F, act on an object the net force is their vector sum. Any problem where the above two rules hold is called a mechanics problem. Within this framework for Mechanics the task of a physicist is to investigate the forces F of Nature. This Newton did magnificently in the Law of Universal Gravitation. 3. LAW OF UNIVERSAL GRAVITATION. Two masses mi and m2 mutually attract one another with forces in direct proportion to their masses, in inverse proportion to the square of their distances apart, and along the line segment between them. Gm1m2 12 m, Here F12 is the force on m2 due to mi; f is the unit vector from mi to m2; r is the distance from mi to m2; G some constant of proportionality. In 1798 Cavendish measured C to be 6.67 x 10-11Nm2/kg2. The mechanics problem with n masses all subject to the gravitational force is called the "a-body problem." Two masses moving according to the gravitational force is called the Kepler Problem Prom these principles the laws of Kepler may be derived and that is the goal of this homework assignment In fact, Newton outstripped not only the capabilities of his contemporaries but, in two important ways, even their worldview force of gravity is not a mechanical law but is "action at a distance" Newton stepped beyond the dominant mechanical worldview of his age. ing to publish speculation on why gravity excists Newton contented himself with experimental verification of how gravity behaves. Universal Gravitation is then an axiom tested by experiment, to be fed into the Newtonian system of Mechanics. $1 Proof of Kepler's Second Law. In this section we consider central forces in any mechanics problem. Let F be a central force pointing toward the origin o in R3 m. Definition. (Central Force) Letf be the vector from to a particle of mass m. The force F ezerted on the particle is central F ll . Exercise 1. Show that the force of gravity is central. (Note: this problem cannot be done without using the definition F -Grmymaf of the force of gravity given on the previous page. ) Let a particle of mass m move under the influence of a force F emminating from Ö. Let: F(t) = the vector from 0 to the particle at time t zi(t)-ї"(t)-the velocity of the particle at tine t a(t)-F"t) the acceleration of the particle at time t Pt) mo(t) the momentum of the particle at time t. L(t) = r(t) × t-the angular momentum. In handwritten work the notation r -IF is common and convenient. Exercise 2. In any mechanics problem, the angular momentum L is conservedthe force F is central. Proof [Hint: Use Newton's Law F-p. ] (Note: be sure to prove "if and only if.") (Note: don't try to use Exercise 1 in the proof of Exercise 2. )

Answer 1

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