{"id":1130,"date":"2022-03-24T08:38:00","date_gmt":"2022-03-24T06:38:00","guid":{"rendered":"https:\/\/support.centreforelites.com\/centre-of-mass\/"},"modified":"2024-03-14T23:50:16","modified_gmt":"2024-03-14T21:50:16","slug":"centre-of-mass","status":"publish","type":"post","link":"https:\/\/support.centreforelites.com\/en\/centre-of-mass\/","title":{"rendered":"How to Find Centre of Mass"},"content":{"rendered":"

The centre of mass or mass centre is the mean<\/span><\/a> location of all the mass<\/span><\/a>in a system. <\/span><\/i><\/b>In the case of a rigid body<\/span><\/a>, the position of the centre of mass is fixed in relation to the body. In the case of a loose distribution of masses in free space<\/span><\/a>, such as shot<\/span><\/a>from a shotgun<\/span><\/a>or the planets<\/span><\/a>of the solar system<\/span><\/a>, the position of the centre of mass is a point in space<\/span><\/a> among them that may not correspond to the position of any individual mass.<\/span><\/p>\n

\"The<\/a><\/div>\n

The term centre of mass<\/b> is often used interchangeably with centre of gravity<\/b><\/span>, but they are physically different concepts. They happen to coincide in a uniform gravitational field<\/span>, but where gravity is not uniform, centre of gravity<\/b> refers to the mean location of the gravitational force <\/i>acting on a body.<\/span><\/p>\n

The centre of mass of a body does not generally coincide with its geometric centre, and this property can be exploited. Engineers try to design a sports car<\/span><\/a>‘s<\/i>centre of mass as low as possible to make the car handle<\/span><\/a><\/i>better. When high jumpers<\/span><\/a>perform a “Finsbury Flop<\/span><\/a><\/i><\/b>“, they bend their body in such a way that it is possible for the jumper to clear the bar while his or her centre of mass does not. <\/span><\/p>\n

\u00a0<\/span><\/span><\/span><\/b>CENTRE OF MASS OF A REGULAR OBJECT<\/span><\/b><\/b><\/p>\n

If an object has uniform density<\/span><\/a><\/i>then its centre of mass is the same as the centroid<\/span><\/a><\/i>of its shape.<\/span><\/p>\n

Examples:<\/span><\/b><\/p>\n

\u00b7\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/span><\/span><\/span>The centre of mass of a ring is at the centre of the ring (in the air).<\/span><\/i><\/p>\n

\u00b7\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/span><\/span><\/span>The centre of mass of a solid triangle lies on all three medians<\/span><\/a> and therefore at the centroid<\/span><\/a>, which is also the average of the three vertices.<\/span><\/i><\/p>\n

\u00b7\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/span><\/span><\/span>The centre of mass of a rectangle is at the intersection of the two diagonals.<\/span><\/i><\/p>\n

\u00b7\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/span><\/span><\/span>In a spherically symmetric body, the centre of mass is at the centre. This approximately applies to the Earth<\/span><\/a>: the density varies considerably, but it mainly depends on depth and less on the latitude<\/span><\/a>and longitude<\/span><\/a>coordinates.<\/span><\/i><\/p>\n

\u00b7\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/span><\/span><\/span>More generally, for any symmetry of a body, its centre of mass will be a fixed point of that symmetry.<\/span><\/i><\/p>\n

CENTRE OF MASS OF AN ARBITRARY 2D PHYSICAL SHAPE<\/span><\/b><\/p>\n

The Centre of Mass of an arbitrary 2D physical shape <\/b>can be found using the following plumbline method as illustrated in the following table:<\/span><\/p>\n\n\n\n\n
\n\n\"Centre<\/a><\/p>\n

\n<\/td>\n<\/tr>\n

\n

\n

Step 1:<\/span> An arbitrary 2D shape.<\/b><\/div>\n<\/td>\n
\n

Step 2:<\/span><\/b> Suspend the shape from a location near an edge. Drop a plumb line<\/span><\/a> and mark on the object.<\/span><\/p>\n<\/td>\n

\n

Step 3:<\/span><\/b> Suspend the shape from another location not too close to the first. Drop a plumb line again and mark. The intersection of the two lines is the centre of mass.<\/span><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Table 4.2: <\/span><\/i><\/b>Centre of Mass of an arbitrary 2D physical shape<\/span><\/i><\/b><\/p>\n

This method is useful when one wishes to find the centroid<\/span><\/a>of a complex planar shape with unknown dimensions. It relies on finding the centre of mass of a thin body of homogenous density<\/span><\/a>having the same shape as the complex planar shape.<\/span><\/p>\n

CENTRE OF MASS OF AN L-SHAPED OBJECT<\/span><\/b><\/b><\/p>\n

This is a method of determining the centre of mass of an L-shaped object.<\/span><\/p>\n

\n\n\n\n\n
\"Centre<\/a><\/td>\n<\/tr>\n
Centre of Mass of an L-shaped object<\/span><\/i><\/b><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n


\n<\/span>1.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/span><\/span>Divide the shape into two rectangles, as shown in fig 2. Find the centre of masses of these two rectangles by drawing the diagonals. Draw a line joining the centres of mass. The centre of mass of the shape must lie on this line AB.<\/span><\/p>\n

2.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/span><\/span><\/span>Divide the shape into two other rectangles, as shown in fig 3. Find the centres of mass of these two rectangles by drawing the diagonals. Draw a line joining the centres of mass. The centre of mass of the L-shape must lie on this line CD.<\/span><\/p>\n

3.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/span><\/span><\/span>As the centre of mass of the shape must lie along AB and also along CD, it is obvious that it is at the intersection of these two lines; at O. (The point O may or may not lie inside the L-shaped object.)<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"

The centre of mass or mass centre is the mean location of all the massin a system. In the case […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"set","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"wds_primary_category":148,"footnotes":""},"categories":[148],"tags":[298,299],"class_list":["post-1130","post","type-post","status-publish","format-standard","hentry","category-physical-science","tag-physics","tag-secondary-school-physics"],"aioseo_notices":[],"jetpack_featured_media_url":"","jetpack-related-posts":[{"id":771,"url":"https:\/\/support.centreforelites.com\/en\/3-things-about-the-centre-of-mass\/","url_meta":{"origin":1130,"position":0},"title":"3 Things about the Centre of Mass","author":"centreforelites","date":"April 18, 2022","format":false,"excerpt":"The centre of mass or mass centre is the\u00a0mean\u00a0location of all the\u00a0mass\u00a0in a system.\u00a0In the case of a\u00a0rigid body, the position of the centre of mass is fixed in relation to the body. 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