Water falls from a 40 m high dam at the rate of \(9 \times 10^4\) kg per hour. Fifty percentage of gravitational potential energy can be converted into electrical energy. Using this hydroelectric energy number of 100 W lamps, that can be lit, is: (Take g=10ms-2)
To solve this problem, we need to determine how much electrical energy can be produced from the gravitational potential energy of water falling from a dam and then calculate how many 100 W lamps can be lit using this energy.
Step-by-Step Solution:
Calculate the gravitational potential energy (GPE) of the water falling each hour.
Formula for gravitational potential energy is:
GPE = mgh,
where:
m = 9 \times 10^4 kg (mass of water falling per hour)
g = 10 \, \text{m/s}^2 (acceleration due to gravity)
h = 40 \, \text{m} (height of the dam)
Substituting the values, we get:
GPE = 9 \times 10^4 \times 10 \times 40 = 3.6 \times 10^7 \, \text{Joules}
Calculate the electrical energy converted from the gravitational potential energy.
It is given that 50% of GPE can be converted into electrical energy.
Electrical energy produced = 50% of GPE = 0.5 \times 3.6 \times 10^7 = 1.8 \times 10^7 \, \text{Joules}
Determine the number of 100 W lamps that can be lit with this energy.
Power of one lamp = 100 W = 100 J/s.
Since energy is in Joules and power is in watts (Joules per second), we need to calculate how many Joules are used by one lamp in one hour.
Energy used by one lamp in one hour = 100 \times 3600 = 3.6 \times 10^5 \, \text{Joules}
Number of lamps = Total electrical energy / Energy used by one lamp = \frac{1.8 \times 10^7}{3.6 \times 10^5} = 50
Therefore, 50 lamps of 100 W each can be lit using the available hydroelectric energy.
