The ratio of SI to cgs units of force is

Options

A. 10
B. `10^(2)`
C. `10^(3)`
D. `10^(5)`

Explanation

To find the ratio of SI to CGS units of force, we will follow these steps:

### Step 1: Understand the formula for force
The formula for force is given by:
\[ F = m \cdot a \]
where \( F \) is force, \( m \) is mass, and \( a \) is acceleration.

### Step 2: Identify the SI unit of force
In the SI system:
- The unit of mass (m) is kilograms (kg).
- The unit of acceleration (a) is meters per second squared (m/s²).

Thus, the SI unit of force is:
\[ \text{SI unit of force} = \text{kg} \cdot \text{m/s}^2 = \text{kg} \cdot \text{m} \cdot \text{s}^{-2} \]

### Step 3: Identify the CGS unit of force
In the CGS system:
- The unit of mass (m) is grams (g).
- The unit of acceleration (a) is centimeters per second squared (cm/s²).

Thus, the CGS unit of force is:
\[ \text{CGS unit of force} = \text{g} \cdot \text{cm/s}^2 = \text{g} \cdot \text{cm} \cdot \text{s}^{-2} \]

### Step 4: Write down the ratio of SI unit to CGS unit
Now, we can write the ratio of the SI unit of force to the CGS unit of force:
\[ \text{Ratio} = \frac{\text{SI unit of force}}{\text{CGS unit of force}} = \frac{\text{kg} \cdot \text{m/s}^2}{\text{g} \cdot \text{cm/s}^2} \]
The units of seconds squared (s²) will cancel out:
\[ \text{Ratio} = \frac{\text{kg} \cdot \text{m}}{\text{g} \cdot \text{cm}} \]

### Step 5: Convert units to the same system
We need to convert kg to grams and meters to centimeters:
- \( 1 \text{ kg} = 1000 \text{ g} = 10^3 \text{ g} \)
- \( 1 \text{ m} = 100 \text{ cm} = 10^2 \text{ cm} \)

Substituting these conversions into the ratio:
\[ \text{Ratio} = \frac{10^3 \text{ g} \cdot 10^2 \text{ cm}}{\text{g} \cdot \text{cm}} = \frac{10^5 \text{ g} \cdot \text{cm}}{\text{g} \cdot \text{cm}} \]

### Step 6: Simplify the ratio
Now, we can cancel out grams (g) and centimeters (cm):
\[ \text{Ratio} = 10^5 \]

### Final Answer
The ratio of SI to CGS units of force is:
\[ \text{Ratio} = 10^5 \]

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