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#redirect [[Center of mass#Center of gravity]] {{R to section}} |
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| date = October 2016 |
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[[File:Bird toy showing center of gravity.jpg|thumb|right|This toy uses the principles of center of gravity to keep balance on a finger.]] |
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The '''center of gravity''' of a body is that point through which the [[resultant]] of the system of parallel forces formed by the weights of all the particles constituting the body passes for all positions of the body. It is denoted as "'''C.G'''" or "'''G'''". In a uniform gravitational field the center of gravity is identical to the [[center of mass]].<ref>Strength Of Materials - By S.Ramamrutham and R.Narayan , ISBN 81-87433-54-X</ref> |
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==Introduction== |
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Every body is attracted by gravity towards the center of the earth. This force of attraction is proportional to the mass of the body, perpendicular to the surface of the earth, and directed towards the center of the earth. This is known as the [[weight]] of the body. For bodies that are small relative to the earth, the constituting parts of the body can be assumed to be at equal distances from the center of the earth and therefore can also be assumed that the forces formed by those parts are also parallel to each other (in fact, those parts further from the center of the earth experience less gravitational pull than those closer to it— the gravitational pull of the earth is greater at sea level than it is on Mount Everest; also, the directions of pull are in fact always slightly angular).<br/> |
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The [[resultant]] of all these parallel forces is the total ''weight'' of the body. This resultant force passes through a single point for all positions of the body. That point is called the ''center of gravity''. |
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==Centroid== |
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{{main article|Centroid}} |
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Different geometrical shapes such as the circle, triangle, and rectangle are plane figures having only 2-dimensions. They have area but no [[mass]]. The center of gravity of these plane figures is called the centroid or geometrical center. The method of finding out the centroid of a plane figure is the same as that of finding out the center of gravity of a body. If the object is assumed to have uniform mass per unit area, then the centroid is also the center of gravity in a uniform gravitational field. |
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By Moyukh Chakrabarti. |
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==Methods to calculate center of gravity== |
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# By geometrical consideration |
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# By [[Moment (physics)|moments]] |
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# By graphical method |
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The first two methods are generally used to find out the center of gravity or centroid, as the third method can become tedious. |
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===Center of gravity by geometrical consideration=== |
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# The center of gravity of a circle is its center. |
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# The center of gravity of a square, rectangle or a [[parallelogram]] is at the points where its diagonals meet each other. It is also the middle point of the length as well as the width. |
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# The center of gravity of a triangle is at the point where the [[median (geometry)|medians]] of the triangle meet. |
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# The center of gravity of a right circular [[Cone]] is at a distance of <math> \frac{h}{4}\!</math> from its base. |
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# The center of gravity of a [[Sphere|hemisphere]] is at a distance of <math> \frac{3r}{8}\!</math> from its base. |
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# The center of gravity of a segment of a [[sphere]] of [[radius]] '''h''' is at a [[perpendicular]] distance of <math> \frac{3}{4} \frac{(2r-h)^2}{(3r-h)}\!</math> from the center of the sphere. |
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# The center of gravity of a semicircle is at a [[perpendicular]] distance of <math> \frac{4r}{3\pi} \!</math> from its center. |
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# The center of gravity of a [[Trapezoid|trapezium]] with parallel side '''a''' and '''b''' is at a distance of <math> \frac{2}{3} \frac{(b+2a)}{(b+a)}\!</math> measured from the base '''b'''. |
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# The center of gravity of a cube of side '''L''' is at a distance of <math> \frac{L}{2}\!</math> from every face. |
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# The center of gravity of a Sphere of diameter '''d''' is at a distance of <math> \frac{d}{2}\!</math> from every point. |
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<gallery style="margin:auto;"> |
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image:Circular section.jpg|thumb|Circle |
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image:Rectangle.jpg|thumb|Square and Rectangle |
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image:Right circular Cone.jpg|thumb|Circular Cone |
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image:Hemisphere.jpg|thumb|Hemisphere |
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image:Segment of a Sphere.jpg|thumb|Segment Of a Sphere |
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image:Semicircle.jpg|thumb|Semicircle |
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image:Trapizium.jpg|thumb|Trapezium |
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</gallery> |
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# |
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===Center of gravity by Moments=== |
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[[File:Center of gravity by Moments.jpg|thumb|Center of gravity By [[Moment (physics)|moments]]]] |
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Consider a body of mass "M" whose center of gravity is required to be found out. Let "g" be the [[Gravitational acceleration|acceleration due to gravity]]. Then the weight of the body is "Mg". Divide the body into small particles having equal masses, whose center of gravity is known as shown in the figure. Let their weights be m<sub>1</sub>g<sub>1</sub>, m<sub>2</sub>g<sub>2</sub>, m<sub>3</sub>g<sub>3</sub>............,etc., and (X<sub>1</sub>, Y<sub>1</sub>), (X<sub>2</sub>, Y<sub>2</sub>), (X<sub>3</sub>,Y<sub>3</sub>)..............,etc., be the coordinates of their center of gravity from a fixed point <big>"o"</big>. |
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Let "G" be the center of gravity of the whole body then, <math>\bar X</math> and <math>\bar Y</math> are the coordinates of "G" from "o".<br /> |
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From the [[Torque#Principle of moments|principle of moments]], we know that,<br /> |
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<big><big><big>Mg</big></big></big> <math>\bar X</math> = m<sub>1</sub>gX<sub>1</sub> + m<sub>2</sub>gX<sub>2</sub> + m<sub>3</sub>gX<sub>3</sub>............, <br /> |
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<big><big><big>Mg</big></big></big> <math>\bar X</math> = <big><big>g</big></big>( m<sub>1</sub>X<sub>1</sub> + m<sub>2</sub>X<sub>2</sub> + m<sub>3</sub>X<sub>3</sub>............,) |
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<big><big><big>M</big></big></big> <math>\bar X</math> = m<sub>1</sub>X<sub>1</sub> + m<sub>2</sub>X<sub>2</sub> + m<sub>3</sub>X<sub>3</sub>............, |
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<big><big><big>M</big></big></big> <math>\bar X</math> = <math>\sum mx</math> |
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but, <big><big>M</big></big> = m<sub>1</sub> + m<sub>2</sub> + m<sub>3</sub>............ = <math>\sum m</math> |
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therefore, <math>\bar X </math> = <math> \frac{\sum mx}{\sum m}\!</math> |
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Similarly, <math>\bar Y </math> = <math> \frac{\sum my}{\sum m}\!</math> |
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==[[Axis (mathematics)|Axis]] of reference== |
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The center of gravity of a body is always calculated with reference to some assumed axis known as the axis of reference. The axis of reference for plane figures (laminas) is usually taken as the lowest line touching the lamina which is parallel to the horizontal X axis, for calculating <math>\bar Y </math> the vertical distance of the center of gravity from this axis. Similarly, the line touching the left outermost edge which is parallel to the vertical Y axis is usually used for calculating <math>\bar X </math>, the horizontal distance of the center of gravity from this axis. |
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==Centroid of plane figures<ref>A Text Book Of Engineering Mechanics - By R.S.Khurmi. ISBN 81-219-0651-2</ref>== |
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Plane geometrical figures such as T-sections, I-sections, L-sections etc., have only [[areas]] but no [[mass]]. The centroid (center of area) of those figures is found out in the same way as that of solid bodies. The centroid will also be the center of gravity if the lamina has uniform mass per unit area. |
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Consider a lamina as the above figure,let its area be '''"A"''' .Divide the lamina into elemental areas '''a<sub>1</sub>, a<sub>2</sub>, a<sub>3</sub>............''',etc.,. And '''(X<sub>1</sub>, Y<sub>1</sub>), (X<sub>2</sub>, Y<sub>2</sub>), (X<sub>3</sub>,Y<sub>3</sub>)'''..............,etc., are the [[coordinates]] of their center of areas from the reference axis '''"Y-Y"''' . |
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Let '''"G"''' centroid of the whole lamina, and <math>\bar X</math> and <math>\bar Y</math> are the coordinates of '''"G"'''(Centroidal distances) from the reference axes '''"Y-Y"''' and '''"X-"'''. <br /> |
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From the principle of moments, we know that,<br /> |
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<big><big><big>'''A'''</big></big></big> <math>\bar X</math> = '''a<sub>1</sub>X<sub>1</sub> + a<sub>2</sub>X<sub>2</sub> + a<sub>3</sub>X<sub>3</sub>............,''' |
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<big><big><big>'''A'''</big></big></big> <math>\bar X</math> = <math>\sum ax</math> |
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but, <big><big>'''A'''</big></big> = '''a<sub>1</sub> + a<sub>2</sub> + a<sub>3</sub>............''' = <math>\sum a</math> |
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therefore, '''<math>\bar X </math> = <math> \frac{\sum ax}{\sum a}\!</math><br />''' |
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Similarly, '''<math>\bar Y </math> = <math> \frac{\sum ay}{\sum a}\!</math>''' |
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==See also== |
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# [https://www.grc.nasa.gov/www/k-12/airplane/cg.html Center Of Gravity-NASA] |
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# [http://www.britannica.com/science/centre-of-gravity Center Of Gravity-Encyclopaedia Britannica] |
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# http://www.explainthatstuff.com/center-of-gravity.html |
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==References== |
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{{Reflist}} |
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[[Category:Mechanical engineering]] |
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[[Category:Civil engineering]] |
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[[Category:Physics]] |
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[[Category:Mechanics]] |
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[[Category:Engineering]] |
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[[Category:Science]] |
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Latest revision as of 08:34, 15 January 2017
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