If the unit of power is million erg/minute, the unit of force is 1000 dyne and the unit of length is 5/3 cm then the unit of time is (in second).

Options

A. `10`
B. `1`
C. `1//10`
D. `1//100`

Answer

To solve the problem, we need to find the unit of time in seconds given the units of power, force, and length. Let's break down the solution step by step.

### Step-by-Step Solution:

1. **Identify the Given Units**:
- Power (P) = 1 million erg/minute = \(10^6\) erg/minute
- Force (F) = 1000 dyne
- Length (L) = \( \frac{5}{3} \) cm

2. **Convert Units to Standard SI Units**:
- 1 erg = \(10^{-7}\) joules
- 1 dyne = \(10^{-5}\) newtons
- 1 cm = \(10^{-2}\) meters
- Convert power to joules per second:
\[ P = 10^6 \text{ erg/minute} = 10^6 \times 10^{-7} \text{ J/minute} = 10^{-1} \text{ J/minute} \]
Since 1 minute = 60 seconds:
\[ P = \frac{10^{-1}}{60} \text{ J/s} = \frac{10^{-1}}{60} \text{ W} \]

3. **Calculate the Energy in Terms of Force and Length**:
- The formula for power is:
\[ P = \frac{F \cdot L}{t} \]
- Rearranging for time (t):
\[ t = \frac{F \cdot L}{P} \]

4. **Substitute the Values**:
- Convert force and length:
\[ F = 1000 \text{ dyne} = 1000 \times 10^{-5} \text{ N} = 10^{-2} \text{ N} \]
\[ L = \frac{5}{3} \text{ cm} = \frac{5}{3} \times 10^{-2} \text{ m} \]
- Substitute into the equation for time:
\[ t = \frac{(10^{-2} \text{ N}) \cdot \left(\frac{5}{3} \times 10^{-2} \text{ m}\right)}{\frac{10^{-1}}{60} \text{ W}} \]
\[ = \frac{10^{-2} \cdot \frac{5}{3} \cdot 10^{-2}}{\frac{10^{-1}}{60}} \]

5. **Simplify the Expression**:
- The numerator becomes:
\[ 10^{-2} \cdot \frac{5}{3} \cdot 10^{-2} = \frac{5 \cdot 10^{-4}}{3} \]
- The denominator becomes:
\[ \frac{10^{-1}}{60} = \frac{60 \cdot 10^{-1}}{1} \]
- Therefore:
\[ t = \frac{\frac{5 \cdot 10^{-4}}{3}}{\frac{10^{-1}}{60}} = \frac{5 \cdot 10^{-4} \cdot 60}{3 \cdot 10^{-1}} = \frac{5 \cdot 60}{3} \cdot 10^{-3} = 100 \cdot 10^{-3} = 10^{-1} \text{ seconds} \]

6. **Final Result**:
- Hence, the unit of time is:
\[ t = \frac{1}{10} \text{ seconds} \]

### Conclusion:
The unit of time is \( \frac{1}{10} \) seconds.