Center of Gravity questions - Center of Area

[physicx_1]

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

 

[cupid.callin]

Hi physicx_1
Welcome to PF !

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

For Y_COM use eqn:

$$Y_{COM} = \frac{\sum{m_ix_i}}{\sum{m_i}}$$

i.e.

$$Y_{COM} = \frac{\sum{m_ix_i}}{M}$$

 

[physicx_1]

Hi physicx_1
Welcome to PF !

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

For Y_COM use eqn:

$$Y_{COM} = \frac{\sum{m_ix_i}}{\sum{m_i}}$$

i.e.

$$Y_{COM} = \frac{\sum{m_ix_i}}{M}$$

Thanks. Looking forward to my stay here :)

So what does M stand for? Moment?

 

[cupid.callin]

The total mass

 

[rwisz]

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.