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 1.5 m diameter and a mass of 250 kg. Its maximum angular velocity is 1200 rpm. How much energy is stored in the flywheel?

Verified step by step guidance

1. Determine the moment of inertia of the flywheel. Since the flywheel is a solid disk, its moment of inertia is given by the formula: $I=\frac{1}{2}m{r}^2$, where $m$ is the mass of the flywheel and $r$ is its radius. The radius is half the diameter, so $r=\frac{1.5}{2}=0.75m$.
2. Convert the maximum angular velocity from rpm (revolutions per minute) to radians per second. Use the conversion factor: $1\backslash (\backslash text\{\; revolution\; \}\backslash )=2\pi \backslash (\backslash text\{\; radians\; \}\backslash )$ and $1\backslash (\backslash text\{\; minute\; \}\backslash )=60\backslash (\backslash text\{\; seconds\; \}\backslash )$. The angular velocity in radians per second is: $\omega =1200imes\frac{\mathrm{2\pi }}{60}$.
3. Calculate the rotational kinetic energy stored in the flywheel using the formula: $E=\frac{1}{2}I{\omega }^2$, where $I$ is the moment of inertia and $\omega$ is the angular velocity in radians per second.
4. Substitute the values of $I$ and $\omega$ into the formula for rotational kinetic energy. Ensure that all units are consistent (mass in kilograms, radius in meters, angular velocity in radians per second).
5. Simplify the expression to find the total energy stored in the flywheel. This will give the energy in joules (J), which is the standard unit of energy in the SI system.

Key Concepts

Rotational Kinetic Energy

Rotational kinetic energy is the energy possessed by an object due to its rotation. It is calculated using the formula KE_rot = 0.5 * I * ω², where I is the moment of inertia and ω is the angular velocity in radians per second. This concept is crucial for determining how much energy is stored in a flywheel as it spins.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion. For a solid disk, it is calculated using the formula I = 0.5 * m * r², where m is the mass and r is the radius. Understanding the moment of inertia is essential for calculating the rotational kinetic energy of the flywheel.

Angular Velocity

Angular velocity is a measure of how quickly an object rotates around an axis, expressed in radians per second. To convert from revolutions per minute (rpm) to radians per second, the formula ω = (rpm * 2π) / 60 is used. Knowing the angular velocity is necessary for calculating the energy stored in the flywheel.