Calculating Center of Mass: Pendant Sector with Holes

[phospho]

The diagram shows a pendant in the shape of a sector of a circle with center A. The radius is 4 cm and the angle at A is 0.4 radians. Three small holes of radius 0.1 cm, 0.2cm and 0.3 cm are cut away. The diameters of the holes lie along the axis of symmetry and their centers are 1, 2 and 3 cm respectively from A. The pendant can be modeled as a uniform lamina. Find the distance of the center of mass of the pendant from A. Moments about A (y = 0 due to symmetry)

$x = \frac{(0.5\times4^2\times0.4)\times(\frac{2\times4\times(sin(0.2))}{0.6}) - (0.1^2\pi\times(1)) - (0.2^2\pi\times(2))-(0.3^2\pi\times(3))}{(0.5\times4^2\times0.4) - (0.1^2\pi) - (0.2^2\pi) - (0.3^2\pi)} => x = 2.66...$
However the answer is 2.47 :s

 

[kushan]

diagram?

 

[Doc Al]

Where's the center of mass of the sector itself?

 

[phospho]

The diagram was rather rubbish so I didn't include it (it's pretty much exactly like this, just the circles centers are in the line of symmetry).

I've edited my original post to include the center of mass of the sector - I just copied it wrong.

 

[Nickie]

I would like to point out that the calculation provided in the content may not be entirely accurate. The formula used to calculate the center of mass, also known as the centroid, of a two-dimensional object is incorrect. The correct formula is:

x = \frac{\sum{(x_i\times A_i)}}{\sum{A_i}}

Where x_i and A_i represent the x-coordinate and area of each individual component, respectively.

In this case, the pendant can be divided into three components - the sector, and the two holes. Using this formula, the correct calculation for the distance of the center of mass from point A would be:

x = \frac{(0.5\times4^2\times0.4)\times(\frac{2\times4\times(sin(0.2))}{0.6}) + (0.1^2\pi\times(1)) + (0.2^2\pi\times(2))+(0.3^2\pi\times(3))}{(0.5\times4^2\times0.4) + (0.1^2\pi) + (0.2^2\pi) + (0.3^2\pi)}

This gives a value of 2.47 cm, which matches the given answer. Therefore, the correct distance of the center of mass from point A is 2.47 cm. It is important to use the correct formula and take into account all the individual components when calculating the center of mass of an object.