A Verdant Power water turbine (a "windmill" in water) turns in the East River near New York City. Its propeller is 2.5 m in radius and spins at 32 rpm when in water that is moving at 2.0 m/s. The rotational inertia of the propeller is approximately 3.0 kg⋅m2.Determine the electric energy in joules that it could provide in 1 day if it is 100% efficient at converting its kinetic energy into electric energy. Assume that the energy delivered per revolution is equal to the rotational kinetic energy of the turbine.
To determine the electric energy that the Verdant Power water turbine could provide in 1 day, we need to calculate the kinetic energy of the turbine and then multiply it by the number of revolutions it makes in a day.
Step 1: Calculate the kinetic energy of the turbine
The kinetic energy (KE) of a rotating object is given by the formula:
KE = (1/2) * I * ω^2
Where:
KE is the kinetic energy
I is the moment of inertia of the propeller (given as 3.0 kg⋅m^2)
ω is the angular velocity in radians per second. We need to convert the rpm (revolutions per minute) to radians per second.
To convert rpm to radians per second, we use the formula:
ω = (2π * n) / 60
Where:
n is the number of revolutions per minute.
Substituting the given values:
ω = (2π * 32) / 60
Step 2: Calculate the number of revolutions in a day
Since we are interested in the energy generated in 1 day, we need to convert the number of minutes in a day to revolutions. There are 24 hours in a day, and 60 minutes in an hour.
Number of revolutions in a day = (24 * 60)/ n
Substituting the given value of n = 32.
Step 3: Calculate the electric energy generated in 1 day
The energy delivered per revolution is equal to the rotational kinetic energy of the turbine. Therefore, we multiply the kinetic energy (KE) by the number of revolutions (rev) in a day.
Energy in joules that the turbine could provide in 1 day = KE * rev
Calculating each step:
Step 1:
ω = (2π * 32) / 60 = 3.36 radians/second
Step 2:
Number of revolutions in a day = (24 * 60)/ 32 = 45 revolutions/day (rounded to the nearest whole number)
Step 3:
Energy in joules that the turbine could provide in 1 day = KE * rev
Plugging in the values:
Energy = (1/2) * 3.0 kg⋅m^2 * (3.36 radians/second)^2 * 45 revolutions/day
Calculating this expression will give us the final answer.