A circular hole of 50 mm diameter is cut out from a circular disc of 100 mm diameter as shown in the figure below. Find the centre of gravity of the section from A.

To find the center of gravity (centroid) of the shaded area from point A, we will treat the shaded area as a composite of a large circular disc and a smaller circular hole. We'll use point A as our origin $(0,0)$.

First, let's define the properties of the two shapes:

• Large Circular Disc (LC):

  • Diameter $D_{LC} = 100 \text{ mm}$
  • Radius $R_{LC} = \frac{100}{2} = 50 \text{ mm}$
  • Area $A_{LC} = \pi R_{LC}^2 = \pi (50)^2 = 2500\pi \text{ mm}^2$
  • Centroid $x_{LC}$: Since A is the leftmost point of the large circle and the diameter is $100 \text{ mm}$, the center of the large circle is at $x_{LC} = 50 \text{ mm}$ from A.
  • Centroid $y_{LC}$: The line A-B is the axis of symmetry, so $y_{LC} = 0 \text{ mm}$.

• Circular Hole (H):

  • Diameter $D_H = 50 \text{ mm}$
  • Radius $R_H = \frac{50}{2} = 25 \text{ mm}$
  • Area $A_H = \pi R_H^2 = \pi (25)^2 = 625\pi \text{ mm}^2$
  • Centroid $y_H$: The hole is also symmetric about the line A-B, so $y_H = 0 \text{ mm}$.
  • Centroid $x_H$: The diagram's dimensioning for the hole's position is ambiguous. However, the visual representation clearly shows the hole is offset to the right, not concentric with the large circle. A common interpretation for such offset holes, consistent with the visual, is that the right edge of the hole is tangent to the right edge of the large circle.
    • The right edge of the large circle is at $x = 100 \text{ mm}$ from A.
    • If the right edge of the hole is tangent to this, then $x_H + R_H = 100 \text{ mm}$.
    • $x_H + 25 = 100 \implies x_H = 75 \text{ mm}$.
    • Alternatively, if the $50 \text{ mm}$ dimension from A to the center of the hole is interpreted as the distance from A to the left edge of the hole, then $x_H - R_H = 50 \text{ mm}$, which gives $x_H - 25 = 50 \text{ mm}$, so $x_H = 75 \text{ mm}$. This interpretation is consistent with the visual and leads to a centroid shift.