Question

The following are the centroidal coordinates of semicircular plane of \(16\pi\) cm diameter and is symmetric about \(x\)-axis.

• A. \(\left(0, \frac{32}{3}\right)\)
• B. \(\left(\frac{32}{3}, 0\right)\)
• C. \(\left(0, \frac{16}{3}\right)\)
• D. \(\left(\frac{16}{3}, 0\right)\)

Hint:

Always check the axis of symmetry first to eliminate options: - Symmetric about \(x\)-axis \(\Rightarrow \bar{y} = 0\). This immediately eliminates choices (1) and (3). - Then, simply evaluate \(\frac{4R}{3\pi}\) using the correct radius value.

Answer 1

The Correct Option is B

Concept: The centroid defines the geometric center of a continuous planar shape. For a uniform semicircular lamina of radius \(R\), the centroid lies along its geometric axis of symmetry. The distance from the straight bounding base edge to the centroid along that axis of symmetry is given by the standard formula: \[ \bar{y}_{\text{local}} = \frac{4R}{3\pi} \]

Step 1: Extracting geometric parameters from the problem statement.
We are given that the total diameter of the semicircular plane is: \[ D = 16\pi \text{ cm} \] Since the radius \(R\) of a circle is half of its total diameter, we calculate: \[ R = \frac{D}{2} = \frac{16\pi}{2} = 8\pi \text{ cm} \]

Step 2: Implementing the symmetry condition to determine the coordinates.
The problem states that the semicircular plane is symmetric about the \(x\)-axis.
• If a geometric plane figure is perfectly symmetric about the \(x\)-axis, its centroid must lie directly on the \(x\)-axis.
• Any point that lies on the \(x\)-axis has a \(y\)-coordinate equal to zero: \[ \bar{y} = 0 \]
• This means the straight bounding baseline of this semicircle lies along the vertical \(y\)-axis, and the shape extends symmetrically above and below the \(x\)-axis into the right half-plane. The distance from the base (the \(y\)-axis) to the centroid along the symmetry axis (the \(x\)-axis) is given by the coordinate \(\bar{x}\).

Step 3: Calculating the \(\bar{x}\) coordinate value.
Using the standard centroid distance formula along the axis of symmetry: \[ \bar{x} = \frac{4R}{3\pi} \] Substitute the value of the radius \(R = 8\pi\) cm into this expression: \[ \bar{x} = \frac{4 \times (8\pi)}{3\pi} \] Simplifying the expression by canceling the factor of \(\pi\) from both the numerator and the denominator: \[ \bar{x} = \frac{32}{3} \text{ cm} \]

Step 4: Combining the coordinates into a final ordered pair.
Combining our calculated components into standard Cartesian coordinates \((\bar{x}, \bar{y})\): \[ (\bar{x}, \bar{y}) = \left(\frac{32}{3}, 0\right) \] This matches option (2).