Find the value 15 joule in a system which has 10 cm, 100 gm and 20sec as a fundamental units?

Explanation

To find the value of 15 joules in a system with fundamental units of 10 cm, 100 gm, and 20 sec, we will follow these steps:

### Step 1: Understand the dimensions of energy
The dimension of energy (Joule) is given by:
\[ \text{Energy} = \text{ML}^2\text{T}^{-2} \]
where:
- M = mass
- L = length
- T = time

### Step 2: Identify the units in the new system
In the new system, we have:
- Length (L) = 10 cm = \(10^{-1}\) m
- Mass (M) = 100 gm = \(10^{-1}\) kg
- Time (T) = 20 sec

### Step 3: Convert the given energy from joules to the new system
Using the relation:
\[ n_1 u_1 = n_2 u_2 \]
where:
- \( n_1 = 15 \) joules (the energy we want to convert)
- \( u_1 = 1 \text{ J} = 1 \text{ kg m}^2/\text{s}^2 \) (energy in SI units)
- \( n_2 \) is the energy in the new system
- \( u_2 \) is the energy in the new system expressed in terms of the new units.

### Step 4: Express \( u_2 \) in terms of the new units
Using the dimensions:
\[ u_2 = M^1 L^2 T^{-2} \]
Substituting the new units:
- \( M = 100 \text{ gm} = 0.1 \text{ kg} \)
- \( L = 10 \text{ cm} = 0.1 \text{ m} \)
- \( T = 20 \text{ sec} \)

Now substituting these values into the expression for \( u_2 \):
\[ u_2 = (0.1 \text{ kg}) \times (0.1 \text{ m})^2 \times (20 \text{ sec})^{-2} \]
\[ u_2 = 0.1 \times 0.01 \times \frac{1}{400} \]
\[ u_2 = \frac{0.001}{400} = 2.5 \times 10^{-6} \text{ J} \]

### Step 5: Calculate \( n_2 \)
Now, substituting back into the relation:
\[ n_2 = \frac{u_1}{u_2} \times n_1 \]
\[ n_2 = \frac{1 \text{ J}}{2.5 \times 10^{-6} \text{ J}} \times 15 \]
\[ n_2 = \frac{15}{2.5 \times 10^{-6}} \]
\[ n_2 = 6 \times 10^{6} \]

### Conclusion
Thus, the value of 15 joules in the new system is:
\[ n_2 = 6 \times 10^{6} \text{ new units of energy} \]

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