Center of Mass of Discrete Objects and Moments About a Line
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how to solve word problems basics of moments and center of mass 
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The moment about a line is very similar to a moment about a point. The \(x_i\)'s are just the perpendicular distance from the mass to the line. Here is an example.
Example
\(m_1=6, m_2=3, m_3=2, m_4=9\) 

The moment about the yaxis is \(\displaystyle{M_y = \sum_{i=1}^{n}{x_im_i}}\) where each \(x_i\) is the directed distance to the yaxis.
In this example, \(M_y = 3(6)+0(3)+(5)(2)+4(9) = 44\).
Just like we did for center of mass about a point, the center of mass about a line is the moment divided by the total mass.
In this example, total mass is \(m=6+3+2+9=20\) and the center of mass in the xdirection is \(\bar{x}=M_y/m=44/20=2.2\).
We can do the same calculations about the xaxis.
The moment about the xaxis is \(\displaystyle{M_x=\sum_{i=1}^{n}{y_im_i}}\) where \(y_i\) is the directed distance to the xaxis.
For our example, \(M_x=(2)6+0{3}+3(2)+2(9)=12\).
The center of mass in the ydirection is \(\bar{y}=M_x/m=12/20=0.6\)
So the center of mass in the plane is \((\bar{x},\bar{y})=(2.2,0.6)\).
Okay, let's work some practice problems before we go on.
Practice
Unless otherwise instructed, solve these problems giving your answers in exact, simplified form.
A 5kg mass is placed at the origin and a 9kg mass is placed at \(x=2\)m. What is the center to mass?
Problem Statement
A 5kg mass is placed at the origin and a 9kg mass is placed at \(x=2\)m. What is the center to mass?
Solution
video by The Organic Chemistry Tutor 

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An 8kg mass is placed at y=3m. Where should a 10kg mass be placed along the yaxis so that the center of mass will be located at y=4.5m?
Problem Statement
An 8kg mass is placed at y=3m. Where should a 10kg mass be placed along the yaxis so that the center of mass will be located at y=4.5m?
Solution
video by The Organic Chemistry Tutor 

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We have three masses at various points in the plane as follows
2lbs at \( (3,2) \), 6lbs at \( (1,4) \) and 1lb at \( (0,6) \)
(a) What is the total moment with respect to the xaxis?
(b) What is the total moment with respect to the yaxis?
(c) What is the center of mass of the system?
Problem Statement
We have three masses at various points in the plane as follows
2lbs at \( (3,2) \), 6lbs at \( (1,4) \) and 1lb at \( (0,6) \)
(a) What is the total moment with respect to the xaxis?
(b) What is the total moment with respect to the yaxis?
(c) What is the center of mass of the system?
Solution
video by patrickJMTPhysics 

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We have four masses at various points in the plane as follows.
4kg at \((0,0)\), 6kg at \((4,0)\), 7kg at \((0.5)\) and 10kg at \((5,6)\). Determine the center of mass of this system.
Problem Statement
We have four masses at various points in the plane as follows.
4kg at \((0,0)\), 6kg at \((4,0)\), 7kg at \((0.5)\) and 10kg at \((5,6)\). Determine the center of mass of this system.
Solution
video by The Organic Chemistry Tutor 

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A 10m long 4000kg truck has its center of mass 4m behind the front of the truck. Where should a 1400kg load be placed so that the center of mass will be right in the middle of the truck?
Problem Statement
A 10m long 4000kg truck has its center of mass 4m behind the front of the truck. Where should a 1400kg load be placed so that the center of mass will be right in the middle of the truck?
Solution
video by The Organic Chemistry Tutor 

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