Flywheels are large massive wheels (disks) used to store energy. They can be spun up slowly, and then the wheel’s energy can be released quickly to accomplish a task that demands high power. A large flywheel has a 2.4 m diameter and a mass of 300 kg. Its maximum angular velocity is 1500 rpm.
a. The flywheel is spun up at a constant torque of 70 Nm. How long does it take the flywheel to reach top speed?
b. How much energy is stored in the flywheel?
c. After reaching top speed, the flywheel is connected to a machine to which it will deliver energy. Half of the energy stored in the flywheel is delivered in 2.5 sec. What is the average power delivered to the machine?
d. How much torque does the flywheel exert on the machine?
To solve these questions, we need to use the concept of rotational motion and the properties of flywheels. Let's tackle each question step by step:
a. To find the time it takes for the flywheel to reach top speed, we can use the equation:
Torque = Moment of Inertia * Angular acceleration
The moment of inertia of a disc is given by:
Moment of Inertia = (1/2) * Mass * Radius^2
where the radius is half the diameter.
First, convert the angular velocity from rpm to rad/s:
Angular velocity = 1500 rpm * (2π rad/1 min) * (1 min/60 s)
Next, calculate the moment of inertia:
Radius = 2.4 m / 2 = 1.2 m
Moment of Inertia = (1/2) * 300 kg * (1.2 m)^2
Then, rearrange the torque equation to solve for angular acceleration:
Angular acceleration = Torque / Moment of Inertia
Finally, substitute the given torque value to find the angular acceleration. Since the torque is constant, it will not depend on time.
Now you can solve for the time it takes for the flywheel to reach top speed using the equation:
Angular velocity = Initial angular velocity + Angular acceleration * Time
Since the flywheel starts from rest (initial angular velocity is zero), you can rearrange the equation to solve for time.
b. The energy stored in the flywheel is equal to its rotational kinetic energy. The formula for rotational kinetic energy is:
Rotational Kinetic Energy = (1/2) * Moment of Inertia * Angular velocity^2
Substitute the given values for moment of inertia and angular velocity, and calculate the energy stored in the flywheel.
c. We are given that half of the energy stored in the flywheel is delivered in 2.5 seconds. Average power can be calculated using the formula:
Average Power = Energy / Time
Substitute the given energy and time values to find the average power delivered to the machine.
d. To find the torque exerted by the flywheel on the machine, we can use the equation:
Torque = Moment of Inertia * Angular acceleration
Since we already know the moment of inertia and angular acceleration from part a, we can substitute those values into the equation to find the torque.
By following these steps and performing the calculations, you can find the answers to all the given questions.