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Chapter 6: Problem 34

A particle is moving along the \(x\) -axis subject to the potential energy function \(U(x)=1 / x+x^{2}+x-1\) a) Express the force felt by the particle as a function of \(x\). b) Plot this force and the potential energy function. c) Determine the net force on the particle at the coordinate \(x=2.00 \mathrm{~m}\)

Short Answer

Expert verified
Question: Determine the force acting on a particle at x=2.00 m, given that its potential energy function is U(x) = 1/x + x^2 + x - 1. Answer: The net force acting on the particle at x = 2.00 m is 4.25 N.

Step by step solution

01

(Part a: Derive force function F(x))

We know that the force acting on a particle due to potential energy is given by: F(x) = -dU(x)/dx Given, U(x) = 1/x + x^2 + x - 1 We will differentiate U(x) with respect to x to find the force function, F(x). F(x) = - d(1/x + x^2 + x - 1)/dx
02

(Differentiate U(x) with respect to x)

Differentiating each term of U(x) with respect to x, we get: -d(1/x)/dx = - (-1/x^2) = 1/x^2 -d(x^2)/dx = -2x -d(x)/dx = -1 -d(-1)/dx = 0 So, F(x) = - (1/x^2 - 2x - 1)
03

(Part b: Plot F(x) and U(x))

To plot F(x) = - (1/x^2 - 2x - 1) and U(x) = 1/x + x^2 + x - 1, you can use graphing software or online tools such as Desmos, Wolfram Alpha, or GeoGebra. Enter the given functions as mentioned and adjust the range of x values as needed to see the behavior of the functions.
04

(Part c: Find net force at x=2.00 m)

Using the derived force function F(x) = - (1/x^2 - 2x - 1), we can determine the net force on the particle at x=2.00 m: F(2) = - (1/(2^2) - 2(2) - 1) F(2) = - (1/4 - 4 - 1) F(2) = - (-4.25) F(2) = 4.25 N The net force acting on the particle at x = 2.00 m is 4.25 N.

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