Flywheels are large, massive wheels used to store energy. They can be spun up slowly, then the wheel's energy can be released quickly to accomplish a task that demands high power. An industrial flywheel has a 1.7 m diameter and a mass of 270 kg . Its maximum angular velocity is 1400 rpm .
The flywheel is disconnected from the motor and connected to a machine to which it will deliver energy. Half the energy stored in the flywheel is delivered in 2.1 s . What is the average power delivered to the machine?
Energy stored = (1/2) I omega^2
7587
To find the average power delivered to the machine, we need to calculate the energy stored in the flywheel and then divide it by the time it takes to deliver half of that energy.
First, let's find the energy stored in the flywheel. The energy stored in a rotating object, such as a flywheel, can be calculated using the formula:
E = (1/2) * I * ω²
where E is the energy, I is the moment of inertia, and ω is the angular velocity.
The moment of inertia of a solid cylinder (flywheel) can be calculated using the formula:
I = (1/2) * m * r²
where m is the mass of the flywheel and r is its radius.
Given:
Mass (m) = 270 kg
Diameter = 1.7 m
Radius (r) = diameter / 2 = 1.7 m / 2 = 0.85 m
Angular velocity (ω) = 1400 rpm
First, let's convert the angular velocity from rpm to radians per second:
1 revolution = 2π radians
1 minute = 60 seconds
Angular velocity (ω) = 1400 rpm * (2π radians/1 revolution) * (1 minute/60 seconds)
ω ≈ 146.67 radians/second
Now, let's calculate the moment of inertia (I):
I = (1/2) * m * r²
I = (1/2) * 270 kg * (0.85 m)² ≈ 995.18 kg m²
Then, let's calculate the energy (E) stored in the flywheel:
E = (1/2) * I * ω²
E = (1/2) * 995.18 kg m² * (146.67 radians/second)²
E ≈ 16382305.8 Joules
Now, let's calculate the time taken to deliver half the energy:
Given:
Time (t) = 2.1 seconds
The energy delivered in t seconds can be calculated using the formula:
E_delivered = average power * t
Since we want to find the average power, we need to rearrange the formula:
Average power = E_delivered / t
We are given that half of the energy is delivered, so:
E_delivered = (1/2) * E = (1/2) * 16382305.8 Joules = 8191152.9 Joules
Now, let's substitute the values and calculate the average power:
Average power = 8191152.9 Joules / 2.1 seconds ≈ 3895820.9 Watts
Therefore, the average power delivered to the machine is approximately 3.8958 MW (megawatts).