A flywheel is a solid disk that rotates about an axis that...

A flywheel is a solid disk that rotates about an axis that is perpendicular to the disk at its center. Rotating flywheels provide a means for storing energy in the form of rotational kinetic energy and are being considered as a possible alternative to batteries in electric cars. The gasoline burned in a 300 -mile trip in a typical midsize car produces about $1.2 \times 10^{9} \mathrm{J}$ of energy. How fast would a $13-\mathrm{kg}$ flywheel with a radius of $0.30 \mathrm{m}$ have to rotate to store this much energy? Give your answer in rev/min.

Key Concepts

Rotational Kinetic Energy

This concept involves the energy stored in a rotating object. It is given by the formula (1/2)I?², where I is the moment of inertia and ? is the angular velocity. This energy form is crucial when considering systems that store or use rotational motion, as it quantifies the capacity of a rotating body to do work.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotation. For different shapes, the moment of inertia depends on the mass distribution relative to the axis of rotation. For a solid disk rotating about an axis perpendicular to its face through its center, the moment of inertia is calculated using the formula I = (1/2)mR², which captures how mass and geometry influence the dynamics.

Angular Velocity

Angular velocity describes the rate of rotation of an object, usually measured in radians per second. It is central to calculating rotational kinetic energy because, in the energy formula (1/2)I?², energy increases with the square of the angular velocity. Determining the required angular velocity for a given energy storage helps in designing and evaluating the performance of rotating devices.

Unit Conversion Between Angular Measures

In many problems, it is necessary to convert angular velocity from one set of units to another, for instance, from radians per second (used in formulas) to revolutions per minute (a more practical unit of measure for mechanical applications). Understanding these conversions is essential to relate theoretical calculations to real-world system specifications.

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