Find the value of 100 J on a system which has 20 cm, 250g and half minute as fundamental units of length, mass and time.

Explanation

To find the value of 100 J in a system with fundamental units of length, mass, and time as 20 cm, 250 g, and half a minute respectively, follow these steps:

### Step-by-Step Solution:

1. **Identify the given quantities in the SI system:**
- Energy (E) = 100 J
- Unit mass (M1) = 1 kg
- Unit length (L1) = 1 m
- Unit time (T1) = 1 s

2. **Identify the given quantities in the new system:**
- Unit mass (M2) = 250 g = 0.25 kg
- Unit length (L2) = 20 cm = 0.2 m
- Unit time (T2) = 30 s (since half a minute = 30 s)

3. **Write the dimensional formula for energy:**
- Energy (E) = \(\frac{1}{2}mv^2\)
- Dimensional formula: \([M^1L^2T^{-2}]\)

4. **Use the conversion formula:**
\[ N2 = N1 \left(\frac{M1}{M2}\right)^a \left(\frac{L1}{L2}\right)^b \left(\frac{T1}{T2}\right)^c \]
where \(a\), \(b\), and \(c\) are the powers of mass, length, and time in the dimensional formula of energy.

5. **Substitute the values:**
- \(N1 = 100\)
- \(M1 = 1\) kg, \(M2 = 0.25\) kg
- \(L1 = 1\) m, \(L2 = 0.2\) m
- \(T1 = 1\) s, \(T2 = 30\) s
- \(a = 1\), \(b = 2\), \(c = -2\)

\[ N2 = 100 \left(\frac{1}{0.25}\right)^1 \left(\frac{1}{0.2}\right)^2 \left(\frac{1}{30}\right)^{-2} \]

6. **Simplify the expression:**
\[ N2 = 100 \left(4\right) \left(25\right) \left(30^2\right) \]

7. **Calculate the powers:**
\[ 4 \times 25 = 100 \]
\[ 30^2 = 900 \]

8. **Combine all the factors:**
\[ N2 = 100 \times 100 \times 900 \]
\[ N2 = 100 \times 90000 \]
\[ N2 = 9000000 \]

9. **Final result:**
\[ N2 = 9 \times 10^6 \text{ (new units)} \]