A circular hole is cut from a disc of radius 6 cm in such a way that the radius of the hole is 1 cm and the centre of 3 cm from the centre of the disc. The distance of the centre of mass of the remaining part from the centre of the original disc is

Options

A. 3/35 cm
B. 1/35 cm
C. 3/10 cm
D. 7/35 cm

Answer

To find the distance of the center of mass of the remaining part from the center of the original disc after cutting a circular hole, we can follow these steps:

### Step 1: Define the system
We have a large disc of radius \( R = 6 \, \text{cm} \) and a small circular hole of radius \( r = 1 \, \text{cm} \) cut from it. The center of the hole is located \( d = 3 \, \text{cm} \) from the center of the disc.

### Step 2: Calculate the mass of the large disc
The mass \( M \) of the large disc can be calculated using the formula:
\[ M = \rho \cdot V = \rho \cdot (\pi R^2 \cdot t) \]
where \( \rho \) is the density of the material and \( t \) is the thickness of the disc. Substituting \( R = 6 \, \text{cm} \):
\[ M = \rho \cdot (\pi \cdot (6^2) \cdot t) = 36 \pi \rho t \]

### Step 3: Calculate the mass of the hole
The mass \( m \) of the small hole can be calculated similarly:
\[ m = \rho \cdot V = \rho \cdot (\pi r^2 \cdot t) = \rho \cdot (\pi \cdot (1^2) \cdot t) = \pi \rho t \]

### Step 4: Relate the masses
From the expressions for \( M \) and \( m \):
\[ m = \frac{M}{36} \]

### Step 5: Calculate the center of mass of the system
To find the center of mass of the remaining part after the hole is cut, we use the formula for the center of mass \( x_{cm} \):
\[ x_{cm} = \frac{M \cdot 0 - m \cdot d}{M - m} \]
where:
- \( 0 \) is the position of the center of the large disc,
- \( d = 3 \, \text{cm} \) is the distance of the hole's center from the center of the disc.

Substituting the values:
\[ x_{cm} = \frac{M \cdot 0 - \frac{M}{36} \cdot 3}{M - \frac{M}{36}} \]

### Step 6: Simplify the equation
This simplifies to:
\[ x_{cm} = \frac{-\frac{3M}{36}}{M - \frac{M}{36}} = \frac{-\frac{3M}{36}}{\frac{36M - M}{36}} = \frac{-\frac{3M}{36}}{\frac{35M}{36}} = -\frac{3}{35} \]

### Step 7: Interpret the result
The negative sign indicates that the center of mass of the remaining part is located on the opposite side of the center of the original disc. Therefore, the distance of the center of mass of the remaining part from the center of the original disc is:
\[ \frac{3}{35} \, \text{cm} \]

### Final Answer
The distance of the center of mass of the remaining part from the center of the original disc is \( \frac{3}{35} \, \text{cm} \).