Center of Gravity questions - Center of Area

In summary, the problem is to find the centroid of a T-shaped object, using equations for X and Y COM, with the total mass being represented by M.
  • #1
physicx_1
14
0

Homework Statement


Find the problem of the centroid of each of the shapes

http://img29.imageshack.us/img29/7563/64279412.jpg


this makeshift diagram is not as accurate as it should be, but it is a plain T shape that is perfectly symmetrical.

Homework Equations



Not sure

The Attempt at a Solution



I know I should work out the distance to centroid from yy(x), as in the vertical length. and the distance to centroid from xx(y) that is the horizontal length. also I should find out about the moment of the area or something?

someone please help
 
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  • #2
Hi physicx_1 :smile:
Welcome to PF !

From symmetry you can tell that COM will be at the central point for X COM

For YCOM use eqn:

[tex]Y_{COM} = \frac{\sum{m_ix_i}}{\sum{m_i}}[/tex]

i.e.

[tex]Y_{COM} = \frac{\sum{m_ix_i}}{M}[/tex]
 
  • #3
cupid.callin said:
Hi physicx_1 :smile:
Welcome to PF !

From symmetry you can tell that COM will be at the central point for X COM

For YCOM use eqn:

[tex]Y_{COM} = \frac{\sum{m_ix_i}}{\sum{m_i}}[/tex]

i.e.

[tex]Y_{COM} = \frac{\sum{m_ix_i}}{M}[/tex]

Thanks. Looking forward to my stay here :)

So what does M stand for? Moment?
 
  • #4
The total mass
 
  • #5


I would approach this problem by first defining the problem of finding the centroid of each shape. The centroid is the point at which the shape would balance if it were suspended from that point. In other words, it is the center of gravity of the shape. To find the centroid, we need to calculate the centroid coordinates, which are the average of the coordinates of all the points in the shape.

To find the centroid of the T shape, we can divide it into two rectangles and a triangle. The centroid of a rectangle is at the midpoint of its base and the centroid of a triangle is at one-third of the distance from its base to its apex. Using this information, we can calculate the centroid coordinates for each shape.

For the rectangle on the left, the centroid coordinates would be (0, 2). For the rectangle on the right, the centroid coordinates would be (4, 2). And for the triangle, the centroid coordinates would be (2, 0.67). To find the overall centroid of the T shape, we can take the average of these coordinates, giving us a centroid of (2, 1.56).

In terms of equations, we can use the formula for the centroid of a composite shape, which is the sum of the individual centroids multiplied by their respective areas, divided by the total area. So for the T shape, the centroid coordinates would be:

x̄ = (A1x1 + A2x2 + A3x3) / (A1 + A2 + A3)

ȳ = (A1y1 + A2y2 + A3y3) / (A1 + A2 + A3)

Where A1, A2, and A3 are the areas of the three shapes and x1, x2, x3 and y1, y2, y3 are the respective centroid coordinates for each shape.

In summary, to find the centroid of a shape, we need to divide it into simpler shapes and calculate the centroid coordinates for each shape. Then, we can use the formula for the centroid of a composite shape to find the overall centroid coordinates. This approach can be applied to any shape, not just the T shape shown in the problem.
 

FAQ: Center of Gravity questions - Center of Area

1. What is the difference between center of gravity and center of area?

The center of gravity is the point at which the entire weight of an object can be considered to act, while the center of area is the geometric center of an object's shape.

2. How is the center of gravity of an irregularly shaped object determined?

The center of gravity of an irregularly shaped object can be determined by suspending the object from different points and finding the point at which it balances perfectly. This point will be the center of gravity.

3. Does the center of gravity of an object change when it is in motion?

Yes, the center of gravity of an object changes when it is in motion. It shifts towards the direction of the movement and can affect the stability and balance of the object.

4. How does the center of gravity affect the stability of an object?

The lower the center of gravity of an object, the more stable it is. This is because the weight of the object is concentrated closer to the ground, making it harder to tip over. Additionally, a wider base of support also increases stability by spreading the weight over a larger area.

5. Can the center of gravity be outside of an object?

No, the center of gravity is always located within the boundaries of the object. However, for objects with irregular shapes, the center of gravity may appear to be outside of the physical boundaries due to the distribution of weight and shape of the object.

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