Trying to derive equation for acceleration in spherical coordinate system

In summary, the conversation discusses the process of deriving acceleration in spherical coordinates and the need to find the projection of spherical unit vectors onto Cartesian unit vectors. The person is unsure about how to calculate the projections for e(θ) and e(φ) and has attached their work so far for reference. It is noted that the unit vectors in spherical coordinates are different from those in Cartesian coordinates, and there may be a mistake in the drawing of the unit vector e(θ).
  • #1
vweltin
2
0

Homework Statement


I was trying to figure out how to derive acceleration in spherical coordinates, and I realized that I need to find the projection of each spherical unit vector [ e(r), e(θ), and e(φ)] onto each Cartesian unit vector [î, j, and k], but I'm not quite sure as to how to do that. It's pretty intuitive for me to calculate the projections of e(r), but the other components confuse me a bit.

Homework Equations


see attached image

The Attempt at a Solution


I've attached the work I've done so far as an image (I didn't mean for that ' near the î on the bottom to be there..)
ntsOv.png
 
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  • #2
http://en.wikipedia.org/wiki/Unit_vectors#Spherical_coordinates" are the unit vectors in the spherical coordinate system from wikipedia. Your [itex]\hat{e_r}[/itex] is given as [itex]\hat{r}[/itex], [itex]\hat{e_\theta}[/itex] is given as [itex]\hat{\theta}[/itex] and [itex]\hat{e_\phi}[/itex] is given as [itex]\hat{\psi}[/itex].

It looks to me that your [itex]\hat{e_\theta}[/itex] is drawn incorrectly in the diagram. Is it drawn to go straight into the page?
 
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FAQ: Trying to derive equation for acceleration in spherical coordinate system

What is the spherical coordinate system?

The spherical coordinate system is a three-dimensional coordinate system used to locate points in space. It consists of three coordinates: radius (r), inclination (θ), and azimuth (φ). The origin of the system is located at the center of a sphere, and the coordinates are measured in terms of distance, angle, and rotation from this origin point.

Why is it important to derive an equation for acceleration in spherical coordinates?

Deriving an equation for acceleration in spherical coordinates is important in many fields of science, such as astronomy, physics, and engineering. It allows us to better understand the motion and forces acting on objects in three-dimensional space, and can be used to solve complex problems involving acceleration in curved or rotating systems.

How is the equation for acceleration in spherical coordinates derived?

The equation for acceleration in spherical coordinates is derived using vector calculus and the chain rule. It involves taking the second derivative of the position vector with respect to time, and then converting the resulting expression into spherical coordinates using trigonometric identities and the definitions of the coordinates.

What are the components of the equation for acceleration in spherical coordinates?

The equation for acceleration in spherical coordinates has three components: radial acceleration (ar), tangential acceleration (aθ), and normal acceleration (aφ). These components represent the acceleration in the radial, tangential, and normal directions, respectively, and their magnitudes can be calculated using the derived equation.

Can the equation for acceleration in spherical coordinates be used in any situation?

The equation for acceleration in spherical coordinates is a general equation that can be applied in any situation where an object is moving in three-dimensional space. However, it is most commonly used in situations involving curved or rotating systems, as it provides a more accurate representation of the acceleration compared to equations in other coordinate systems.

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