Question

Consider a piece of wire with uniform density. It is the quarter of a circle in the first quadrant. The circle is centered at the origin and has a radius of 6. Find the centroid of the wire. Recall that the mass of the wire is (M), and the moments about the (x)- and (y)-axes are (M_x) and (M_y), where (M) is the density.

a) ((3, 3))

b) ((4, 4))

c) ((6, 6))

d) ((8, 8))

Answer 1

The centroid of a quarter circle can be found using the formula for the y-coordinate: y = (4R)/(3π). For the given wire, the centroid is located at the point (0, 8/3) or approximately (0, 2.67).

Step-by-step explanation:

The centroid of a quarter circle with uniform density can be found by considering the geometry of the shape. The centroid is the point (x, y) that represents the average position of the mass distribution. For a quarter circle, the centroid is located along the y-axis, and its y-coordinate can be calculated using the formula: y = (4R)/(3π), where R is the radius of the circle.

Therefore, the centroid of the wire, which is the quarter of a circle with a radius of 6, is (0, 8/3) or approximately (0, 2.67).