An installation consisting of an electric motor driving a water pump left 75 L of water per second to a height of 4.7 m . If the motor consumes a power of 5 k W , then efficiency of the installation is

Options

A. `39%`
B. `69%`
C. `93%`
D. `96%`

Answer

To find the efficiency of the installation consisting of an electric motor driving a water pump, we can follow these steps:

### Step 1: Calculate the Work Done by the Motor
The work done (W) by the motor in lifting the water can be calculated using the formula:
\[ W = mgh \]
where:
- \( m \) is the mass of the water,
- \( g \) is the acceleration due to gravity (approximately \( 9.8 \, \text{m/s}^2 \)),
- \( h \) is the height to which the water is lifted.

### Step 2: Determine the Mass of Water Lifted
Given that the pump lifts 75 liters of water per second, we convert this volume to mass. The density of water is \( 1 \, \text{kg/L} \), so:
\[ m = 75 \, \text{L} = 75 \, \text{kg} \]

### Step 3: Substitute Values into the Work Done Formula
Now, we can substitute the values into the work done formula:
\[ W = 75 \, \text{kg} \times 9.8 \, \text{m/s}^2 \times 4.7 \, \text{m} \]

### Step 4: Calculate the Work Done
Calculating the above expression:
\[ W = 75 \times 9.8 \times 4.7 \]
\[ W = 75 \times 46.06 \approx 3454.5 \, \text{J} \]

### Step 5: Calculate the Power Output of the Motor
Since the work is done in one second, the power output (P) of the motor is equal to the work done:
\[ P = 3454.5 \, \text{W} \]

### Step 6: Calculate the Efficiency
Efficiency (\( \eta \)) is defined as the ratio of useful power output to the total power input:
\[ \eta = \frac{P_{\text{output}}}{P_{\text{input}}} \]
Given that the motor consumes a power of \( 5 \, \text{kW} = 5000 \, \text{W} \):
\[ \eta = \frac{3454.5 \, \text{W}}{5000 \, \text{W}} \]

### Step 7: Calculate the Efficiency Value
Calculating the efficiency:
\[ \eta = 0.6909 \]

### Step 8: Convert Efficiency to Percentage
To express efficiency as a percentage, multiply by 100:
\[ \eta = 0.6909 \times 100 \approx 69.09\% \]

### Final Answer
The efficiency of the installation is approximately **69.09%**.

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