A dam is used to block the passage of a river and to generate electricity. Approximately 58.4 x 103 kg of water falls each second through a height of 20.1 m. If one half of the gravitational potential energy of the water were converted to electrical energy, how much power (in MW) would be generated?
Can some one please help me with this problem? Thank you
To find the power generated by the dam, we need to calculate the amount of gravitational potential energy being converted to electrical energy per second.
The formula for gravitational potential energy is given by:
Potential Energy = mass * gravity * height
In this case, the mass is given as 58.4 x 10^3 kg, the gravity is 9.8 m/s^2, and the height is 20.1 m.
Potential Energy = 58.4 x 10^3 kg * 9.8 m/s^2 * 20.1 m
Potential Energy = 1.138872 x 10^7 J
Since the problem states that only half of the potential energy is being converted to electrical energy, we can calculate the power using the formula:
Power = (1/2) * Potential Energy / time
The time is 1 second, so:
Power = (1/2) * 1.138872 x 10^7 J / 1 s
Power = 5.69436 x 10^6 J/s
Finally, to convert the power from joules per second to megawatts, we divide by 10^6:
Power = 5.69436 x 10^6 J/s / 10^6
Power = 5.69436 MW
Therefore, approximately 5.69436 MW of power would be generated by the dam.
Of course, I can help you with this problem! To find the power generated by the falling water, we need to determine the amount of gravitational potential energy converted to electrical energy per second.
The formula for gravitational potential energy is given by: PE = mgh, where m is the mass of the object (in this case, the falling water), g is the acceleration due to gravity (approximately 9.8 m/s^2 on Earth), and h is the height of the fall.
Given:
Mass of water, m = 58.4 x 10^3 kg
Height, h = 20.1 m
First, let's calculate the gravitational potential energy of the water falling per second using the formula. Then, we will find half of that energy, since only half is converted to electrical energy.
PE = mgh
PE = (58.4 x 10^3 kg) x (9.8 m/s^2) x (20.1 m)
PE = 113,842,400 J
Now, we need to convert joules to megawatts. Remember that 1 megawatt (MW) is equal to 1 million watts (W), and 1 watt is equivalent to 1 joule per second.
To convert joules to megawatts:
Power (MW) = Energy (J) / Time (s) x 1 million
Since we have the amount of energy per second, the time value is 1 second.
Power (MW) = 113,842,400 J / 1 s x (1/1,000,000) MW
Power (MW) = 0.1138424 MW
Therefore, approximately 0.1138424 MW of power would be generated by the falling water if one half of the gravitational potential energy is converted to electrical energy.
power= PE rate*efficiency
= 58.4kg/s *9.8m/s^2*20.1m*1/2