Question

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

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

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.

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- \[ F = F_{x} \hat{i} + F_{y} \hat{j} + F_{z} \hat{k} \]

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.

Key concepts

Understanding Vector Force

In physics, force is represented as a vector, which means that it has both magnitude and direction. This is crucial for understanding how objects move or change their motion. A vector force is typically described using its components along various axes. In the context of a three-dimensional space, these are the x, y, and z axes.

For this particular exercise, we derived the force components from the given potential energy function. Potential energy, represented as \( U(x, y, z) \), relates to force through the negative gradient. Therefore, the force in the x-direction, \( F_{x} \), is given as \( -\frac{dU}{dx} \), and similarly for \( F_{y} \) and \( F_{z} \).

These partial derivatives tell us how the potential energy changes in each direction, which directly translates to the force exerted along these directions.

Exploring Cartesian Coordinates

Cartesian coordinates are a standard way of describing a point in a space using three values \( (x, y, z) \). Each value represents a point on one of the three perpendicular axes. In our exercise, these coordinates are essential to finding the vector force components.

When we express the force vector \( F \) in terms of its components, it looks like this: \( F = F_{x} \hat{i} + F_{y} \hat{j} + F_{z} \hat{k} \). Here, \( \hat{i}, \hat{j}, \hat{k} \) represent the unit vectors along the x, y, and z axes, respectively.

This is not just an abstract representation; it helps in visualizing how forces break down in different directions, which is fundamental when predicting how an object will move. Using cartesian coordinates provides a clear and straightforward method to perform calculations involving forces in three dimensions.

Transitioning to Spherical Polar Coordinates

Spherical polar coordinates provide a different way of representing points in three-dimensional space, especially appropriate when dealing with spherical symmetry. They use three parameters: radial distance \( r \), polar angle \( \theta \), and azimuthal angle \( \phi \).

For this exercise, after determining the force vector in cartesian coordinates, converting to spherical polar coordinates simplifies the expression when the problem involves radial symmetry. This transformation is common in fields like electromagnetism and gravitational physics, where such symmetry frequently occurs.

The conversion helps to find the radial component of force \( F_{r} \) efficiently, which aligns with the direction of the radius. The use of spherical coordinates makes calculations more intuitive and manageable in cases where the system's symmetry is radial.