A circular hole of radius ( a/2 ) is cut out of a circular disc of radius 'a' as shown in figure. The centroid of the remaining circular portion with respect to point 'O' will be :

Solution

Physics sir · 3 steps

Treat the hole as a disc of negative mass, and the remaining centroid falls out of one line.

1. Set up positions. Take O as the origin on the X-axis. The big disc has its centre at x = a. The hole touches the right edge at x = 2a, so its centre is at x = 2a - a/2 = 3a/2.
2. Mass goes with area. For a uniform disc, mass is proportional to area. Take the full disc as π a² and the hole as π (a/2)² = (π a²)/4. The hole enters with a minus sign.

   x_(cm) = (π a² (a) - (π a²)/4(3a/2))/(π a² - (π a²)/4) = (a - 3a/8)/(3/4) = 5a/8 × 4/3

3. So. x_(cm) = 5a/6 from O. It sits a little left of the big disc's centre, away from the hole, as expected. The answer is (c).

The idea to keep. For a body with a piece cut out, write the full body minus the removed piece, using area as mass. The centroid always shifts away from the hole.

Answer from the NTA final answer key. Solution by PrepWiser.