1 joule of energy is to be converted into new system of units in which length is measured in 10 metre, mass in 10 kg and time in 1 minute. The numerical value of 1 J in the new system is :-

Options

A. `36xx10^(-4)`
B. `36xx10^(-3)`
C. `36xx10^(-2)`
D. `36xx10^(-1)`

Answer

To solve the problem of converting 1 joule of energy into a new system of units where length is measured in 10 meters, mass in 10 kg, and time in 1 minute, we will follow these steps:

### Step-by-Step Solution:

1. **Understand the Definition of Joule**:
- The unit of energy, Joule (J), is defined as the work done when a force of one newton displaces an object by one meter in the direction of the force.
- Mathematically, \(1 \text{ J} = 1 \text{ N} \cdot 1 \text{ m}\).

2. **Express Newton in Fundamental Units**:
- A newton (N) is defined as \(1 \text{ N} = 1 \text{ kg} \cdot \text{m/s}^2\).
- Therefore, \(1 \text{ J} = 1 \text{ kg} \cdot \text{m/s}^2 \cdot 1 \text{ m} = 1 \text{ kg} \cdot \text{m}^2/\text{s}^2\).

3. **Identify the Dimensions of Energy**:
- The dimensional formula for energy (or work) is:
\[ [E] = [M][L^2][T^{-2}] \]
- Where \(M\) is mass, \(L\) is length, and \(T\) is time.

4. **Convert the Units in the New System**:
- In the new system:
- Length = 10 meters
- Mass = 10 kg
- Time = 1 minute = 60 seconds

5. **Substitute the New Units into the Dimensional Formula**:
- Substitute the new units into the formula for energy:
\[ [E] = (10 \text{ kg}) \cdot (10 \text{ m})^2 \cdot \left(\frac{1}{60 \text{ s}}\right)^2 \]

6. **Calculate Each Component**:
- Calculate the length component:
\[ (10 \text{ m})^2 = 100 \text{ m}^2 \]
- Calculate the time component:
\[ \left(\frac{1}{60 \text{ s}}\right)^2 = \frac{1}{3600 \text{ s}^2} \]

7. **Combine the Components**:
- Now substitute these values back into the energy equation:
\[ E = 10 \text{ kg} \cdot 100 \text{ m}^2 \cdot \frac{1}{3600 \text{ s}^2} \]
- This simplifies to:
\[ E = \frac{1000 \text{ kg m}^2}{3600 \text{ s}^2} \]

8. **Calculate the Numerical Value**:
- Simplifying \( \frac{1000}{3600} \):
\[ \frac{1000}{3600} = \frac{10}{36} = \frac{5}{18} \]
- Therefore, the new value of 1 joule in the new system is:
\[ E = \frac{5}{18} \text{ new joules} \]

### Final Answer:
The numerical value of 1 J in the new system is \( \frac{5}{18} \) new joules.

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