From a waterfall, water is falling down at the rate of 100kg / s on the blades of turbine. If the height of the fall is 100 m , then the power delivered to the turbine is approximately equal to

Options

A. 100 kW
B. 0.1 kW
C. 10 kW
D. 1 kW

Answer

To solve the problem of calculating the power delivered to the turbine by the falling water, we can follow these steps:

### Step 1: Understand the Concept of Power
Power is defined as the rate of doing work or the rate of energy transfer. In this case, the work done by the falling water can be related to the change in potential energy.

### Step 2: Identify the Given Values
- Rate of water flow (mass per second), \( \frac{dm}{dt} = 100 \, \text{kg/s} \)
- Height of the fall, \( h = 100 \, \text{m} \)
- Acceleration due to gravity, \( g = 10 \, \text{m/s}^2 \)

### Step 3: Calculate the Potential Energy Change
The potential energy (PE) change when water falls from a height \( h \) is given by the formula:
\[ PE = mgh \]
Where:
- \( m \) is the mass of water falling,
- \( g \) is the acceleration due to gravity,
- \( h \) is the height.

### Step 4: Calculate the Work Done per Second (Power)
Since we are interested in the power, we need to calculate the work done per unit time. The power \( P \) can be expressed as:
\[ P = \frac{dW}{dt} = \frac{d(PE)}{dt} = \frac{dm}{dt} \cdot g \cdot h \]
Substituting the known values:
\[ P = (100 \, \text{kg/s}) \cdot (10 \, \text{m/s}^2) \cdot (100 \, \text{m}) \]

### Step 5: Perform the Calculation
Now, we can calculate the power:
\[ P = 100 \cdot 10 \cdot 100 = 100000 \, \text{W} \]

### Step 6: Convert to Kilowatts
Since power is often expressed in kilowatts (kW), we convert watts to kilowatts:
\[ P = \frac{100000 \, \text{W}}{1000} = 100 \, \text{kW} \]

### Final Answer
The power delivered to the turbine is approximately equal to **100 kW**.

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