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

##### 17Calculus

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

$$m_1=6, m_2=3, m_3=2, m_4=9$$

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.

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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