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

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Answered: Consider a uniform gravitational field…

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

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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Concept explainers

Gravitational force

In nature, every object is attracted by every other object. This phenomenon is called gravity. The force associated with gravity is called gravitational force. The gravitational force is the weakest force that exists in nature. The gravitational force is always attractive.

Acceleration Due to Gravity

In fundamental physics, gravity or gravitational force is the universal attractive force acting between all the matters that exist or exhibit. It is the weakest known force. Therefore no internal changes in an object occurs due to this force. On the other hand, it has control over the trajectories of bodies in the solar system and in the universe due to its vast scope and universal action. The free fall of objects on Earth and the motions of celestial bodies, according to Newton, are both determined by the same force. It was Newton who put forward that the moon is held by a strong attractive force exerted by the Earth which makes it revolve in a straight line. He was sure that this force is similar to the downward force which Earth exerts on all the objects on it.

Question

Part A
Learning Goal:
To understand the relationship between the force and the potential energy changes
associated with that force and to be able to calculate the changes in potential energy as
definite integrals.
Consider a uniform gravitational field (a fair approximation near the surface of a planet). Find
Imagine that a conservative force field is defined in a certain region of space. Does this
sound too abstract? Well, think of a gravitational field (the one that makes apples fall down
and keeps the planets orbiting) or an electrostatic field existing around any electrically
charged object.
U(y:) – U(yo) = -
where
F =-mg j and ds = dy j.
If a particle is moving in such a field, its change in potential energy does not depend on the
particle'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 negative
derivative 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,
%3D
dz
?
Therefore, the change in potential energy can be found as the integral
AU = -
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 small
displacement of the particle along its path from 1 to 2.
Submit
Evaluating such an integral in a general case can be a tedious and lengthy task. However,
two circumstances make it easier:
Part B
1. Because the result is path-independent, it is always possible to consider the
most 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. Find
U(xf) – U(xo) = - S F, dš,
where
F, = -ka i, ds = dæ î
In this problem, you will practice calculating the change in potential energy for a particle
moving 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 unit
vector in the y direction, and f is the unit vector in the radial direction in case of a
spherically symmetrical force field.
• View Available Hint(s)
?
U(Tt) – U(x0) =

Part C
Finally, 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. Find
U(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

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