In a system of units, the units of mass, length and time are 1 quintal, 1 km and 1h, respectively. In this system 1 N force will be equal to

Options

A. 1 new unit
B. 129.6 new unit
C. 427.6 new unit
D. 60 new unit

Explanation

To find out how much 1 Newton (N) of force is in the given system of units where the unit of mass is 1 quintal, the unit of length is 1 kilometer, and the unit of time is 1 hour, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Definition of Force**:
The dimensional formula for force is given by:
\[ F = MLT^{-2} \]
where \( M \) is mass, \( L \) is length, and \( T \) is time.

2. **Convert Newton to Base Units**:
We know that:
\[ 1 \, \text{N} = 1 \, \text{kg} \cdot \text{m} \cdot \text{s}^{-2} \]

3. **Convert Units**:
We need to express kg, m, and s in terms of the new units:
- 1 quintal = 100 kg, therefore:
\[ 1 \, \text{kg} = \frac{1}{100} \, \text{quintals} \]
- 1 kilometer = 1000 meters, therefore:
\[ 1 \, \text{m} = \frac{1}{1000} \, \text{km} \]
- 1 hour = 3600 seconds, therefore:
\[ 1 \, \text{s} = \frac{1}{3600} \, \text{hours} \]

4. **Substituting the Values**:
Now substituting these conversions into the equation for Newton:
\[ 1 \, \text{N} = \left(\frac{1}{100} \, \text{quintals}\right) \cdot \left(\frac{1}{1000} \, \text{km}\right) \cdot \left(\frac{1}{3600} \, \text{h}\right)^{-2} \]

5. **Simplifying the Expression**:
This can be simplified as follows:
\[ 1 \, \text{N} = \frac{1}{100} \cdot \frac{1}{1000} \cdot \left(\frac{1}{3600^2}\right) \, \text{quintals} \cdot \text{km} \cdot \text{h}^{-2} \]
\[ = \frac{1}{100 \cdot 1000} \cdot \frac{1}{3600^2} \, \text{quintals} \cdot \text{km} \cdot \text{h}^{-2} \]
\[ = \frac{1}{10^5} \cdot \frac{1}{3600^2} \, \text{quintals} \cdot \text{km} \cdot \text{h}^{-2} \]

6. **Calculating the Numerical Value**:
Now we calculate \( 3600^2 \):
\[ 3600^2 = 12960000 \]
Thus, we have:
\[ 1 \, \text{N} = \frac{1}{10^5} \cdot \frac{1}{12960000} \, \text{quintals} \cdot \text{km} \cdot \text{h}^{-2} \]
\[ = \frac{1}{1296000} \, \text{quintals} \cdot \text{km} \cdot \text{h}^{-2} \]

7. **Final Result**:
Therefore, we can conclude that:
\[ 1 \, \text{N} \approx 129.6 \, \text{quintals} \cdot \text{km} \cdot \text{h}^{-2} \]