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17Calculus Physics - Moments and Center of Mass of Discrete Objects

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Center of Mass of Discrete Objects and Moments About a Line

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

example

\(m_1=6, m_2=3, m_3=2, m_4=9\)

The moment about the y-axis is \(\displaystyle{M_y = \sum_{i=1}^{n}{x_im_i}}\) where each \(x_i\) is the directed distance to the y-axis.
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 x-direction is \(\bar{x}=M_y/m=44/20=2.2\).

We can do the same calculations about the x-axis.
The moment about the x-axis is \(\displaystyle{M_x=\sum_{i=1}^{n}{y_im_i}}\) where \(y_i\) is the directed distance to the x-axis.   For our example, \(M_x=(-2)6+0{3}+3(2)+2(9)=12\).
The center of mass in the y-direction 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.

How to Ace the Rest of Calculus: The Streetwise Guide, Including MultiVariable Calculus

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

The Organic Chemistry Tutor - 3599 video solution

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An 8kg mass is placed at y=3m.   Where should a 10kg mass be placed along the y-axis 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 y-axis so that the center of mass will be located at y=4.5m?

Solution

The Organic Chemistry Tutor - 3600 video solution

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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 x-axis?
(b) What is the total moment with respect to the y-axis?
(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 x-axis?
(b) What is the total moment with respect to the y-axis?
(c) What is the center of mass of the system?

Solution

patrickJMTPhysics - 3598 video solution

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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

The Organic Chemistry Tutor - 3601 video solution

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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

The Organic Chemistry Tutor - 3602 video solution

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