Water falls from a 40 m high dam at the rate of $9\times {10}^4\mathrm{kg}$ per hour. Fifty percentage of gravitational potential energy can be converted into electrical energy. Using this hydro electric energy, number of 100 W lamps, that can be lit, is : (Take g = 10 ms^-2)

To solve this problem, we need to determine how many 100 W lamps can be powered using the hydroelectric energy generated by water falling from a dam. Below is a step-by-step breakdown of the solution:

Step 1: Calculate the Mass of Water Falling per Second

The given flow rate of water is 9 × 10⁴ kg per hour. To convert this to kg per second, we use the following calculation:

Mass flow rate = (9 × 10⁴ kg) / (3600 s) ≈ 25 kg/s    

Therefore, the mass of water falling per second is approximately 25 kg/s.

Step 2: Calculate the Gravitational Potential Energy (GPE)

The gravitational potential energy (GPE) of the falling water is calculated using the formula:

GPE = m × g × h    

Where:

• m is the mass of water falling per second (25 kg/s)
• g is the acceleration due to gravity (10 m/s²)
• h is the height from which the water falls (40 m)

Substituting the values into the formula:

GPE = 25 kg/s × 10 m/s² × 40 m = 10,000 J/s = 10,000 W    

The gravitational potential energy is therefore 10,000 J/s, or 10,000 watts (W). This is the power available from the falling water.

Step 3: Calculate the Electrical Energy Available

Since only 50% of the gravitational potential energy can be converted into electrical energy, the electrical energy available is:

Electrical energy = 0.5 × GPE = 0.5 × 10,000 W = 5,000 W    

Thus, the electrical energy available for powering lamps is 5,000 W (or 5 kW).

Step 4: Calculate the Number of 100 W Lamps That Can Be Lit

To calculate the number of 100 W lamps that can be powered by the available electrical energy, we divide the total electrical energy by the power of each lamp:

Number of lamps = Electrical energy / Power of each lamp    Number of lamps = 5,000 W / 100 W = 50 lamps    

Therefore, the number of 100 W lamps that can be lit using the hydroelectric energy is 50 lamps.

Final Answer

The number of 100 W lamps that can be lit is 50.