In this problem, you will practice calculating the change inpotential energy for

Question

In this problem, you will practice calculating the change inpotential energy for a particle moving in three common forcefields.

Where is the change in potential energy for a particle movingfrom point 1 to point 2, is the net force acting on the particle at a given pointof its path, and is a small displacement of the particle along itspath from 1 to 2. Evaluating such an integral in a general case can be a tedious andlengthy task. However, two circumstances make it easier: Because the result is path-independent, it is alwayspossible to consider the most straightforward way to reach point 2from point 1. The most common real-world fields are rather simplydefined. In this problem, you will practice calculating the change inpotential energy for a particle moving in three common forcefields. Note that, in the equations for the forces, is the unit vector in the x direction, is the unit vector in the y direction, and is the unit vector in the radial direction in case of aspherically symmetrical force field. Consider a uniform gravitationalfield(a fair approximation near the surface of a planet).Find Express your answer in terms of Consider the force exerted by a spring thatobeys Hooke's law. Find and the spring constant Express your answer in terms of Finally, consider the gravitational forcegenerated by a spherically symmetrical massive object. Themagnitude and direction of such a force are given by Newton's lawof gravity: Express your answer in terms of

Explanation / Answer

In all the cases the force acts in the direction of motion i.e. the dot product between the force vector and displacement vector is a scalar denoted by Fds. In the second case: direction of motion is opposite to force and so, dot product is negative. delU = change in potential energy

A) Integrate: mg dy from y = y0 to yf => delU = mg(yf - y0)

B) Integrate: kx dx from x = x0 to xf => delU = 0.5 k (xf^2 - x0^2)

C) Integrate: G m1 m2 / r^2 dr from r = r0 to rf => delU = -G m1 m2 (1/rf - 1/r0)

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