Find the center of mass for a shape consisting of three quarters of the unit circle, removing the quarter in the third quadrant
You know the area of the region is A = 3π/4
Also,
x̅ = ∫x dA
y̅ = ∫y dA
The region is easily expressed in polar coordinates, making these integrals
x̅ = 1/A ∫∫x r dr dθ = 1/A ∫[0..1] ∫[-π/2..π] r^2 cosθ dθ dr
= 1/A ∫[0..1] r^2 dr = 4/(3π)
y̅ = ∫∫y r dr dθ = ∫[0..1] ∫[-π/2..π] r^2 sinθ dθ dr
= 1/A ∫[0..1] r^2 dr = 4/(3π)
Note that this is the same as for the smaller area just in QI, since the parts in QII and QIV balance each other out.
To find the center of mass of a shape, we need to calculate the weighted average of the positions of all the individual elements in the shape. In this case, the shape consists of three quarters of a unit circle with one quarter removed from the third quadrant.
To find the center of mass, we can split the shape into smaller parts that are easier to calculate and then find the weighted average of their centers. Let's break down the shape into two parts: the first quarter-circle and the remaining half-circle.
1. First Quarter-Circle:
- The first quarter-circle has an area of π/4.
- The center of mass of a quarter-circle lies at a distance of 4r/3π from the center, where r is the radius.
- For our unit circle, the center of mass of the first quarter-circle is (0.5, 0.5).
2. Half-Circle:
- The half-circle (excluding the first quarter) has an area of (π/2) - (π/4) = π/4.
- The centroid of a semicircle coincides with the centroid of its base, which is the straight line segment joining the two endpoints of the arc. Therefore, the centroid of a half-circle is located at the midpoint of the base.
- Since the base of the half-circle lies along the x-axis, the centroid lies at (r, 0) where r is the radius.
- For our unit circle, the center of mass of the half-circle is (1, 0).
Now, we can calculate the weighted average of the centers of these two parts to find the overall center of mass:
Center of mass = ( (Area1 * Center1) + (Area2 * Center2) ) / (Total Area)
Total Area = π/4 + π/4 = π/2
Center of mass = ( (π/4) * (0.5, 0.5) + (π/4) * (1, 0) ) / (π/2)
= ( (π/8, π/8) + (π/4, 0) ) / (π/2)
= ( π/8 + π/4, π/8 ) / (π/2)
= ( 3π/8, π/8 )
Therefore, the center of mass for the given shape is approximately (3π/8, π/8).