A vibrating system consists of mass 12.5 kg, a spring of spring constant 1000 ` (N ) /( m ) ` performing damped oscillation with damping coefficient of 15 kg `s^(-1)`. The value of critical damping coefficient of the system is

Options

A. ` 223.6 kg s ^( - 1 ) `
B. `71.4 kg s ^( - 1 ) `
C. `100 kg s ^( - 1 ) `
D. `110 kg s ^( - 1 ) `

Explanation

To find the critical damping coefficient (\(B_C\)) of the given vibrating system, we can follow these steps:

### Step 1: Identify the given values
- Mass (\(m\)) = 12.5 kg
- Spring constant (\(k\)) = 1000 N/m
- Damping coefficient (\(C\)) = 15 kg/s (not needed for finding critical damping)

### Step 2: Recall the formula for critical damping coefficient
The critical damping coefficient (\(B_C\)) is given by the formula:
\[ B_C = 2 \sqrt{k \cdot m} \]

### Step 3: Substitute the known values into the formula
Now, we will substitute the values of \(k\) and \(m\) into the formula:
\[ B_C = 2 \sqrt{1000 \, \text{N/m} \cdot 12.5 \, \text{kg}} \]

### Step 4: Calculate the product inside the square root
First, calculate the product:
\[ 1000 \cdot 12.5 = 12500 \]

### Step 5: Calculate the square root
Now, find the square root of 12500:
\[ \sqrt{12500} = 111.8 \, \text{(approximately)} \]

### Step 6: Multiply by 2
Now, multiply the square root by 2 to find \(B_C\):
\[ B_C = 2 \cdot 111.8 \approx 223.6 \, \text{kg/s} \]

### Final Answer
Thus, the critical damping coefficient of the system is:
\[ B_C \approx 223.6 \, \text{kg/s} \]