Part ALearning Goal:To understand the relationship between the force and the potential energy changesassociated with that force and to be able to calculate the changes in potential energy asdefinite integrals.Consider a uniform gravitational field (a fair approximation near the surface of a planet). FindImagine that a conservative force field is defined in a certain region of space. Does thissound too abstract? Well, think of a gravitational field (the one that makes apples fall downand keeps the planets orbiting) or an electrostatic field existing around any electricallycharged object.U(y:) – U(yo) = -whereF =-mg j and ds = dy j.If a particle is moving in such a field, its change in potential energy does not depend on theparticle's path and is determined only by the particle's initial and final positions. Recall that,in general, the component of the net force acting on a particle equals the negativederivative of the potential energy function along the corresponding axis:Express your answer in terms of m, g, yo, and yf.• View Available Hint(s)dU(r)F,%3Ddz?Therefore, the change in potential energy can be found as the integralAU = -U(yt) – U(yo) =where AU is the change in potential energy for a particle moving from point 1 to point 2,F is the net force acting on the particle at a given point of its path, and ds is a smalldisplacement of the particle along its path from 1 to 2.SubmitEvaluating such an integral in a general case can be a tedious and lengthy task. However,two circumstances make it easier:Part B1. Because the result is path-independent, it is always possible to consider themost straightforward way to reach point 2 from point 1.2. The most common real-world fields are rather simply defined.Consider the force exerted by a spring that obeys Hooke's law. FindU(xf) – U(xo) = - S F, dš,whereF, = -ka i, ds = dæ îIn this problem, you will practice calculating the change in potential energy for a particlemoving in three common force fields.and the spring constant k is positive.Express your answer in terms of k, xo, and æf.Note that, in the equations for the forces, i is the unit vector in the x direction, j is the unitvector in the y direction, and f is the unit vector in the radial direction in case of aspherically symmetrical force field.• View Available Hint(s)?U(Tt) – U(x0) = Part CFinally, consider the gravitational force generated by a spherically symmetrical massive object. The magnitude and direction of such a force are given by Newton's law of gravity:Gmim2 î.where ds = drî; G, mj, and m, are constants; and r>0. FindU(r¡) – U(ro) = -F • dš.Express your answer in terms of G, m1, m2, ro, and rf.• View Available Hint(s)Πν ΑΣφ?U(r:) – U(ro) =Submit

Answered: Image /qna-images/answer/6b023201-8358-4575-8648-1ed1ee28997c.jpg

Answered: Consider a uniform gravitational field…

Science

Physics

Consider a uniform gravitational field (a fair approximation near the surface of a planet). Find

Principles of Physics: A Calculus-Based Text

5th Edition

ISBN:9781133104261

Author:Raymond A. Serway, John W. Jewett

Publisher:Raymond A. Serway, John W. Jewett

Chapter6: Energy Of A System

Section: Chapter Questions

Problem 52P

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