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Chapter 12: Problem 32

The potential energy of a three-dimensional force is given by \(U(x, y, z)=-k / \sqrt{x^{2}+y^{2}+z^{2}}\). (a) Derive \(F_{x}, F_{y}\), and \(F_{z}\) and then describe the vector force at each point in terms of its coordinates \(x, y\), and \(z .(b)\) Convert to spherical polar coordinates and find \(F_{r}\).

Short Answer

Expert verified
The force components in cartesian coordinates can be derived from potential energy function and then the force vector can be described in terms of \( x \), \( y \), and \( z \). After converting this to spherical polar coordinates, the radial force component \( F_{r} \) can be determined.

Step by step solution

01

Derive \(F_{x}\), \(F_{y}\) and \(F_{z}\)

From the definition of force in terms of potential energy, we have- \[ F_{x}=-\frac{dU}{dx} \]- \[ F_{y}=-\frac{dU}{dy} \]- \[ F_{z}=-\frac{dU}{dz} \].Plug in the given function \( U(x, y, z) = -k/\sqrt{x^{2}+y^{2}+z^{2}} \) into these equations and then execute the derivatives to get the desired force components.
02

Describe the vector force in terms of coordinates

Using the results from Step 1, the total force vector \( F \) can be expressed in terms of its components, \( F_{x} \), \( F_{y} \), and \( F_{z} \). The formula for the force vector is- \[ F = F_{x} \hat{i} + F_{y} \hat{j} + F_{z} \hat{k} \]
03

Convert to spherical polar coordinates

After expressing the force vector in cartesian coordinates, it is then converted into spherical polar coordinates by using the transformation formulas for spherical polar coordinates in three dimensions. The force component \( F_{r} \) can then be found.

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Most popular questions from this chapter

An object falls from rest from a height \(h\). Determine the kinetic energy and the potential energy of the object as a function \((a)\) of time and \((b)\) of height. Graph the expressions and show that their sum - the total mechanical energy - is constant in each case.

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