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. Kinetic energy=17
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 can provide in 1 day, we need to calculate the rotational kinetic energy of the turbine and then convert it to electric energy.
First, let's calculate the rotational kinetic energy of the turbine. The formula for rotational kinetic energy is:
Rotational Kinetic Energy = (1/2) * Moment of Inertia * Angular Velocity^2
Given:
- Radius of the propeller (r) = 2.5 m
- Angular Velocity (ω) = 32 rpm = (32/60) * 2π rad/s (since 1 rpm = 2π/60 rad/s)
- Moment of Inertia (I) = 3.0 kg⋅m^2
Plugging in these values into the formula, we get:
Rotational Kinetic Energy = (1/2) * 3.0 kg⋅m^2 * ((32/60) * 2π rad/s)^2
Rotational Kinetic Energy = (1/2) * 3.0 * ((32/60) * 2π)^2 Joules
Now, we can calculate the electric energy that the turbine can provide in 1 day. Since the energy delivered per revolution is equal to the rotational kinetic energy of the turbine, we need to find the number of revolutions in 1 day.
Given:
- Speed of water (v) = 2.0 m/s
The distance covered by the propeller in one revolution is the circumference of the circle it traces, which is 2π * radius = 2π * 2.5 m.
The time taken for one revolution is the distance divided by the speed of the water, which is (2π * 2.5 m) / (2.0 m/s).
To find the number of revolutions in 1 day, we need to multiply the number of revolutions per second (angular velocity) by the number of seconds in a day.
Number of Revolutions in 1 Day = (32/60) * 2π rad/s * (86400 s) (since there are 86400 seconds in a day)
Finally, to determine the electric energy provided in 1 day, we multiply the rotational kinetic energy by the number of revolutions in 1 day:
Electric Energy in 1 Day = Rotational Kinetic Energy * Number of Revolutions in 1 Day
Now, you can plug in the values and calculate the result.