Find the centroid of a semi-circular arc of radius

. Find the centroid of a semi-circular arc of radius a. 2. Find the centroid of a semi-circular area of radius a. Answer it correctly. show complete solutions( step by step).

The Correct Answer and Explanation is:

Centroid of a Semi-Circular Arc

1. Problem Setup:
   The semi-circular arc of radius a is part of a circle, and we are looking to find the centroid of the arc.
2. Formula for the Centroid of a Circular Arc:
   The centroid of a semi-circular arc lies on the central axis of the circle (i.e., along the y-axis if the arc is in the upper half of the circle). The formula for the centroid of a circular arc with radius a is: ycentroid=4a3πy_{centroid} = \frac{4a}{3\pi}ycentroid​=3π4a​
3. Step-by-step Solution:
   • For the arc, the length is πa\pi aπa (half the circumference of the circle).
   • Using the symmetry of the arc and the fact that the centroid is along the vertical axis, we calculate the distance from the origin (the center of the circle) to the centroid along the y-axis. This yields the final result for the semi-circular arc’s centroid:
   ycentroid=4a3πy_{centroid} = \frac{4a}{3\pi}ycentroid​=3π4a​

Centroid of a Semi-Circular Area

1. Problem Setup:
   Now, consider a semi-circular area with radius a. The centroid is the point where the entire area of the semi-circle can be assumed to be concentrated.
2. Formula for the Centroid of a Semi-Circular Area:
   The formula for the centroid y of a semi-circular area (in the case where the flat part of the semi-circle is on the x-axis) is: ycentroid=4a3πy_{centroid} = \frac{4a}{3\pi}ycentroid​=3π4a​ This is the same as for the arc because the symmetry and geometry of the situation are similar. It is also along the y-axis.
3. Step-by-step Solution:
   • The area of the semi-circle is given by 12πa2\frac{1}{2} \pi a^221​πa2.
   • By symmetry, the centroid of the semi-circular area lies along the central axis.
   • The formula used for the centroid of the semi-circular area is derived from the integral calculation of the first moment of area over the semi-circle, yielding the same result for the y-coordinate of the centroid as:
   ycentroid=4a3πy_{centroid} = \frac{4a}{3\pi}ycentroid​=3π4a​

Conclusion

Both the centroid of the arc and the centroid of the semi-circular area are located at the same distance from the flat side (x-axis), with a value of 4a3π\frac{4a}{3\pi}3π4a​, but their physical meanings differ. The centroid of the arc represents the balancing point of the arc’s mass, while the centroid of the area represents the center of mass of the semi-circular shape.