(A) The total mass of the system is $M = 6m + m + m + m + m = 10m$.The $y$-coordinates of the centers of mass of the individual parts are:Outer circle

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(A) The total mass of the system is $M = 6m + m + m + m + m = 10m$.The $y$-coordinates of the centers of mass of the individual parts are:Outer circle: $y_1 = 0$,mass $m_1 = 6m$Left inner circle: $y_2 = a$,mass $m_2 = m$Right inner circle: $y_3 = a$,mass $m_3 = m$Horizontal line segment: $y_4 = -a$,mass $m_4 = m$The $y$-coordinate of the center of mass is given by:$Y_{cm} = \frac{m_1 y_1 + m_2 y_2 + m_3 y_3 + m_4 y_4}{M}$$Y_{cm} = \frac{(6m)(0) + (m)(a) + (m)(a) + (m)(-a)}{10m}$$Y_{cm} = \frac{0 + ma + ma - ma}{10m} = \frac{ma}{10m} = \frac{a}{10}$

Class 11 Physics System of Particles and Rotational Motion Centre of mass (Point Mass)

Medium

Look at the figure drawn with ink of uniform linear density. $A$ mass $m$ of ink is used to draw each of the two inner circles and each of the two line segments. $A$ mass $6m$ of ink is used to draw the outer circle. The coordinates of the centers of the different parts are: outer circle $(0, 0)$,left inner circle $(-a, a)$,right inner circle $(a, a)$,and the horizontal line segment $(0, -a)$. Find the $y$-coordinate of the center of mass of the ink in the figure.

A
$\frac{a}{10}$
B
$\frac{a}{8}$
C
$\frac{a}{12}$
D
$\frac{a}{3}$

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System of Particles and Rotational Motion

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Centre of mass (Point Mass)

The centre of mass of a triangle shown in the figure has coordinates

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Particles of masses $m, 2m, 3m, \ldots, nm$ grams are placed on the same line at distances $l, 2l, 3l, \ldots, nl$ cm from a fixed point. The distance of the centre of mass of the particles from the fixed point in centimetres is:

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$A$ uniform square plate $abcd$ has a mass of $1 \, kg$. If two point masses,each of $20 \, g$,are placed at the corners $b$ and $c$ as shown,then the centre of mass shifts on the line

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The velocities of three particles of masses $20 \ g$,$30 \ g$ and $50 \ g$ are $10 \hat{i}$,$10 \hat{j}$ and $10 \hat{k}$,respectively. The velocity of the centre of mass of the three particles is

MediumAP EAMCET 2001

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The centre of mass of two thin uniform rods of the same length $L$ but made of different materials and kept as shown in the figure can be,if the meeting point is the origin of coordinates:

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The centre of mass of a triangle shown in the figure has coordinates

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