Find coordinates of mass center of a uniform semicircular plate of radius r placed symmetric to the y-axis of a Cartesian coordinate system, with centre at origin.

Options

A. `(4R)/(3pi)`
B. `(2R)/(3pi)`
C. `(4R)/(9pi)`
D. `(2R)/(pi)`

Answer

To find the coordinates of the mass center (centroid) of a uniform semicircular plate of radius \( r \) placed symmetrically about the y-axis with its center at the origin, we can follow these steps:

### Step 1: Understand the Geometry
The semicircular plate is symmetric about the y-axis. Therefore, the x-coordinate of the center of mass, \( x_{cm} \), will be 0.

**Hint:** For symmetric shapes about an axis, the center of mass along that axis will be at the origin (0).

### Step 2: Set Up the Coordinate System
We can place the semicircular plate in the Cartesian coordinate system such that the flat edge lies along the x-axis, and the curved edge is above the x-axis. The center of the semicircle is at the origin (0,0).

**Hint:** Visualize the semicircle and its placement in the coordinate system to understand the area distribution.

### Step 3: Calculate the y-coordinate of the Center of Mass
The y-coordinate of the center of mass \( y_{cm} \) for a semicircular plate can be calculated using the formula:
\[ y_{cm} = \frac{1}{A} \int y \, dA \]
where \( A \) is the area of the semicircle and \( dA \) is a differential area element.

### Step 4: Area of the Semicircle
The area \( A \) of a semicircle is given by:
\[ A = \frac{1}{2} \pi r^2 \]

**Hint:** Remember the formula for the area of a semicircle when calculating \( y_{cm} \).

### Step 5: Set Up the Integral for \( y_{cm} \)
To find \( y_{cm} \), we can use polar coordinates where:
- \( x = r \cos \theta \)
- \( y = r \sin \theta \)

The differential area element in polar coordinates is \( dA = \frac{1}{2} r^2 d\theta \).

The limits for \( \theta \) will be from \( 0 \) to \( \pi \) for the semicircle.

### Step 6: Calculate the Integral
Now, we can express \( y_{cm} \) as:
\[ y_{cm} = \frac{1}{A} \int_0^{\pi} (r \sin \theta) \left(\frac{1}{2} r^2 d\theta\right) \]
Substituting the area \( A \):
\[ y_{cm} = \frac{1}{\frac{1}{2} \pi r^2} \int_0^{\pi} \frac{1}{2} r^3 \sin \theta \, d\theta \]

### Step 7: Evaluate the Integral
Calculating the integral:
\[ \int_0^{\pi} \sin \theta \, d\theta = 2 \]
Thus,
\[ y_{cm} = \frac{1}{\frac{1}{2} \pi r^2} \cdot \frac{1}{2} r^3 \cdot 2 = \frac{r^2}{\pi r^2} = \frac{2r}{\pi} \]

### Step 8: Final Coordinates
The coordinates of the center of mass are:
\[ (x_{cm}, y_{cm}) = \left(0, \frac{2r}{\pi}\right) \]

### Summary
The coordinates of the mass center of the uniform semicircular plate are:
\[ \boxed{\left(0, \frac{2r}{\pi}\right)} \]