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Gravitational Acceleration and Potential Field of a Cylinder

Friday, November 22, 2019 - 20:14 | Author: wabis | Topics: Physics, Mathematics, Interactive, Simulation

This page provides a Numerical calculation

of the gravitational acceleration and gravitational potential field around and inside a cylinder with homogeneous density. It shows that the gravitational field has no center of gravity like a sphere, but in the limit of big distances it can be treated as a point mass or sphere with the same mass.

Introduction

It may surprise you that only a sphere can be treated as a point mass to calculate the gravitational field near the surface applying Newton's law of universal gravitation. Other shapes can only be treated as point sources far away from them.

Calculating the gravitational field of a cylinder involves complicated integrals for the general case, that I am not able to solve analytically. I have found analytical solutions for special cases, e.g. the field along the axis of the cylinder [1]. I could not find an analytical solution for any point around the cylinder. So I decided to implement a numerical solution.

Numerical Calculation

The following graphic shows the Numerical calculation of the gravitational acceleration or gravitational potential around a cylinder. It is computed by decomposing the cylinder into many small elements and calculating for each field cell location the gravitatinoal contribution of all the elements of the cylinder. Each individual cylinder element is treated as a point mass, where the mass and center of mass depends on the individual shape of an element, see Center of Mass of a Ring Section Element.

Legend ▾

Legend ▴

Mouse interactions: You can click in the graphic window (or enter a value in the x and y input fields) to choose a location for which the gravitational field is calculated and displayed in the yellow box. You can use the scroll wheel to zoom in and out.

While a long field computation is running, you can stop the calculation by clicking the Progress text on the upper right corner.

Point Mass g: In the lower right corner this is the acceleration magnitude at the current cursor position as for a shpere or point mass of the same mass as the cylinder.

Analytical g(0,y): The gravitational acceleration can be calculated analytically for the rotation axis of the cylinder. If the cursur is on the y-axis (x=0), then the yellow box value and the Analytical value should be about the same.

Field Type: You can show the acceleration field (magnitude of acceleration) or the potential field by choosing the corresponding Field Type and activating the Display Field checkbox. The color encoding is such, that it corresponds to the log₁₀ of the magnitude and cycles through the colors Field Resolution times per decimal place.

Field Resolution: Increase this value to get more color cycles per decimal place of the field values. If the value is 1, the colors cycle one time per decimal place, e.g. from 10⁻⁵ to 10⁻⁶.

Field Value in yellow box: The value in the yellow box is the magnitude of the acceleration or potential at the location (x,y) as calculated numerically by summing the influence over for all cylinder elements. The red arrow shows the direction of the acceleration. It's length is constant and not proportional to the magnitude.

Center of Gravity Line: The green line is an extension of the acceleration direction at the cursor position until it intersects the cylinder axis. It shows that the acceleration field does not have a single center of gravity at the center of the cylinder.

Element size: Change the number of cylinder elements used for the numerical approximation by entering the minimal size of the element in the Element Size input field. Note: the number of elements increases with the n³ power with the decrease in element size. So the computation time increases accordingly. The App should stay responsive all the time, so you can change any value anytime to cancel the current calculations. See Cylinder Decomposition for how the elements are computed for the cylinder. The few small rectangles in the cross section of the cylinder indicate the current element size.

Grid Size: Sets the size of a field grid cell. If the grid size is bigger than half the cylinder height, the grid elements are no longer quadratic but rectangunlar. Small grid sizes can result in so many grid elements, that the drawing and responsivness of the App becomes slow. You can change the size any time while the field is computed to cancel the current run and start a new calculation. Or you can stop the current calculation by clicking the Progress text in the upper right corner.

The acceleration magnitude, direction and the field potential are calculated at the same time. So you can swich fast between the Display Vectors, Field and Field Type Acceleration or Potential without starting a recalculation of the fields, even when the calculation of the field is still in progress.

Observations

The gravitational field near the cylinder surface does not point to the center of mass of the cylinder (black cross). In fact there is no such thing as the center of gravity for a cylinder.

Very near the surface the gravity vectors are almost vertical to the surface for some distance from the axis and the acceleration/potential are not changing much. In fact, for an infinite plane, the gravity vectors will point exactly perpendicular to the surface and the magnitude is the same everywhere, see Gravity on an infinite Flat Earth Plane for a derivation.

The analytically calculated acceleration along the axis of the cylinder is in good concordance with the numerically calculated value. Only very near the surface and inside the cylinder are some differences, depending on the choosen size of the cylinder elements. The smaller the element size, the better the concordance.

Width increasing distance from the cylinder, the gravity field approaches the field of a point mass with the center of gravity at the center of mass of the cylinder and the same mass as the cylinder. So in this case the cylinder can be regarded as a sphere or point mass of the same mass.

For a distance from the surface of about the element size or more the numerical approximation is very accurate. This can be confirmed by changing the element size and calculating the field for the same point.

Algorithm

The cylinder is decomposed into many small elements, see Cylinder Decomposition. For the computed field, a cross section plane is choosen and subdivided into a grid of cells. Because the cylinder is axial symmetric, this plane shows the field values for all points around the cylinder.

The volume, mass and center of mass of each cylinder element is computed according to the shape of the element. Then for each grid cell the gravitational acceleration and gravitational potential due to each cylinder element is calculated according to Newton's law of gravitation. So in this calculations each cylinder element is approximated by a point mass of the mass of the cylinder element.

The contributions of all cylinder elements are summed over by vector addition to get the final field value and direction of acceleration for one grid cell. The process is repeated for each grid cell of the cross section plane. The grid cell is colored according to the field strength and the direction of acceleration is indicated by a short line.

To cross check the validity of the computation, the analytical solution along the cylinder axis is computed as well and displayed. Additionally the field for a point mass or sphere of the same mass as the cylinder is computed.

The main challenge was to implement all this in JavaScript in such a way, that the browser keeps responding to user input during the whole computation, which can last for many minutes. The whole Code can be inspected on the page JavaScript: Gravitational Acceleration and Potential Field of a Cylinder.

Cylinder Decomposition

Depending on the Radius and Height the cylinder is decomposed into small elements with an approximate size as entered in Element Size. The following graph shows how this decomposition is executed. Each element is calculated such, that the height, the radial size and the outer border size of each element is between Element Size and 2 times Element Size.

Center of Mass of a Ring Section Element

To calculate the gravitational influence of a single element, we have to know its center of mass. To calculate the center of mass of an element, we have to calculate the plane that divides the ring section element into 2 equal areas. The center of mass is then the intersection of this plane and the other perpendicular symmetry planes.

The blue and yellow areas are equal in size, see Calculation of Center of Mass Line. The red line mark shows the position of the center of mass for a triangle with 2 sides r. If r = dr and the angle φ is small, the divider line and the mark coinside.

Calculation of Center of Mass Line

The center of mass of any cylinder element in the radial direction of the cylinder is calculated with the following equation.

(1)

$x = \sqrt{ { \left[ { r }^2 + { ( r - \Delta r ) }^2 \right] \cdot \varphi \over 4 \cdot \tan\left( \varphi / 2 \right) } }$

where^'

$x$ ^'	=^'	^'location of the center of mass line from the center of the arc
$r$ ^'	=^'	^'radius of the arc
$\Delta r$ ^'	=^'	^'dr = thickness of the arc ring
$\varphi$ ^'	=^'	^'phi = pie section angle

This equation is only valid for a certain maximal value of $\varphi$ . To ensure that this only valid values are used, the minimum number of cylinder elements in the center of the cylinder is set to 8.

Derivation

The blue and yellow area have to be the same. We have to figure out at which distance x from the center of the circle this is the case. The blue area A is the pie area P_A with radius r minus the triangle area T with height x and angle φ. The yellow area B is the triangle area T minus the pie area P_B with radius r − Δr. We need to find the value x for which the areas A and B are equal:

(2)	$A = B \qquad \Rightarrow \qquad P_\mathrm{A} - T = T - P_\mathrm{B} \qquad \Rightarrow \qquad T = { P_\mathrm{A} + P_\mathrm{B} \over 2 }$

The area of the triangle T depends on its height x and half the base length s, which depends on the height x and the angle φ:

(3)	$T = s \cdot x \qquad \qquad s = x \cdot \tan\left( \varphi / 2 \right) \qquad \Rightarrow \qquad T = { x }^2 \cdot \tan\left( \varphi / 2 \right)$

The areas of the pie sections of the circle are:

(4)	$P_\mathrm{A} = { 1 \over 2 } \cdot r^2 \cdot \varphi \qquad\qquad P_\mathrm{B} = { 1 \over 2 } \cdot (r - \Delta r)^2 \cdot \varphi$

Putting this together we get:

(5)	$T = { P_\mathrm{A} + P_\mathrm{B} \over 2 } \qquad \Rightarrow \qquad { x }^2 \cdot \tan\left( \varphi / 2 \right) = { 1 \over 2 } \cdot \left[ { 1 \over 2 } \cdot r^2 \cdot \varphi + { 1 \over 2 } \cdot (r - \Delta r)^2 \cdot \varphi \right]$
(6)	$x^2 \cdot \tan( \varphi / 2 ) = { \left[ { r }^2 + { ( r - \Delta r ) }^2 \right] \cdot \varphi \over 4 }$

Now we can solve for x:

(7)

$x = \sqrt{ { \left[ { r }^2 + { ( r - \Delta r ) }^2 \right] \cdot \Delta \varphi \over 4 \cdot \tan\left( \Delta \varphi / 2 \right) } }$

Analytical Solutions

There is a relative simple analytical equation for the gravitational acceleration of a cylinder along its symmetry axes. The result of the analytical equation is displayed in the App at the lower right corner. It should be close to the value displayed at the cursor position, if the cursor is on the vertical y-axis. Near the surface and inside the cylinder the value from the numerical calculation and the analytical value can differ slightly due to the limited resolution of the numerical solution.

Analytical solutions also exist for the potential of a cylinder but are not included here.

The gravitational field of a cylinder along its axis is: [1]

(8)

$g = 2 \cdot \pi \cdot G \cdot \rho \cdot \int_{ a }^{ b } \left[ 1 - { t \over \sqrt{ { t }^2 + { r }^2 } } \right] \ \mathrm{d} t$

where^'

$g$ ^'	=^'	^'gravitational acceleration on the cylinder axis at position y from the surface of the cylinder
$G$ ^'	=^'	^'6.674 × 10⁻¹¹ m²·kg⁻¹·s⁻² = gravitational constant
$\rho$ ^'	=^'	^'cylinder density
$r$ ^'	=^'	^'cylinder radius
$h$ ^'	=^'	^'cylinder height
$y$ ^'	=^'	^'distance on the cylinder axis from the surface of the cylinder
$a$ ^'	=^'	^'lower bound of integral. Inside and Outside of the cylinder: $a = y$
$b$ ^'	=^'	^'upper bound of integral. Outside: $b = h + y$ , Inside: $b = h - y$
$t$ ^'	=^'	^'integration variable

Depending on whether the point on the y-axis is outside or inside the cylinder we have to provide different bounds b for the integral. The y values are measured from the surface and always positive.

If we do the integral for both cases we get:

(9)

$g = 2 \cdot \pi \cdot G \cdot \rho \cdot \left[ h - \sqrt{ { ( h + y ) }^2 + { r }^2 } + \sqrt{ { y }^2 + { r }^2 } \right]$

Outside

(10)

$g = 2 \cdot \pi \cdot G \cdot \rho \cdot \left[ h - 2 \cdot y - \sqrt{ { ( h - y ) }^2 + { r }^2 } + \sqrt{ { y }^2 + { r }^2 } \right]$

Inside

where^'

$g$ ^'	=^'	^'gravitational acceleration on the cylinder axis at position y from the surface of the cylinder, always pointing towards the cylinder center
$G$ ^'	=^'	^'6.674 × 10⁻¹¹ m²·kg⁻¹·s⁻² = gravitational constant
$\rho$ ^'	=^'	^'cylinder density
$y$ ^'	=^'	^'distance on the cylinder axis from the surface of the cylinder
$r$ ^'	=^'	^'cylinder radius
$h$ ^'	=^'	^'cylinder height

References

[1][a][b]

GRAVITATIONAL FIELD AND POTENTIAL

; Page 12; Dr. J. B. Tatum
http://astrowww.phys.uvic.ca/~tatum/celmechs/celm5.pdf

About	Walter Bislin (wabis)
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Created	Friday, November 22, 2019

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walter.bislins.ch

Walter Bislin's Blog-En

Gravitational Acceleration and Potential Field of a Cylinder

Introduction

Numerical Calculation

Legend ▾

Legend ▴

Observations

Algorithm

Cylinder Decomposition

Center of Mass of a Ring Section Element

Calculation of Center of Mass Line

Derivation

Analytical Solutions

References

Blog-Functions

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