The flywheel of a gasoline engine is required to give up 500 J of kinetic energy while its angular velocity decreases from 650 rev/min to 520 rev/min. What moment of inertia is required?

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Convert the angular velocities from revolutions per minute (rev/min) to radians per second (rad/s). Use the conversion factor: \( 1 \text{ rev} = 2\pi \text{ rad} \) and \( 1 \text{ min} = 60 \text{ s} \). For example, \( \omega = \text{(angular velocity in rev/min)} \times \frac{2\pi}{60} \).

Write the expression for the rotational kinetic energy of the flywheel: \( KE = \frac{1}{2} I \omega^2 \), where \( KE \) is the kinetic energy, \( I \) is the moment of inertia, and \( \omega \) is the angular velocity.

Calculate the change in kinetic energy, \( \Delta KE \), using the given values: \( \Delta KE = KE_{\text{initial}} - KE_{\text{final}} = 500 \text{ J} \).

Substitute the expressions for \( KE_{\text{initial}} \) and \( KE_{\text{final}} \) into \( \Delta KE \): \( \Delta KE = \frac{1}{2} I \omega_{\text{initial}}^2 - \frac{1}{2} I \omega_{\text{final}}^2 \).

Solve for the moment of inertia \( I \) by isolating it in the equation: \( I = \frac{2 \Delta KE}{\omega_{\text{initial}}^2 - \omega_{\text{final}}^2} \). Substitute the known values for \( \Delta KE \), \( \omega_{\text{initial}} \), and \( \omega_{\text{final}} \) to find \( I \).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Kinetic Energy of Rotation

The kinetic energy (KE) of a rotating object is given by the formula KE = 1/2 I ω², where I is the moment of inertia and ω is the angular velocity in radians per second. This concept is crucial for understanding how energy is stored in a rotating system and how it changes with variations in angular velocity.

Moment of Inertia

Moment of inertia (I) is a measure of an object's resistance to changes in its rotational motion, depending on the mass distribution relative to the axis of rotation. It plays a key role in determining how much torque is needed to change the angular velocity of an object, making it essential for solving problems involving rotational dynamics.

Angular Velocity Conversion

Angular velocity is often expressed in revolutions per minute (rev/min) but must be converted to radians per second (rad/s) for calculations involving kinetic energy. The conversion factor is 2π rad per revolution, and understanding this conversion is necessary to accurately apply the kinetic energy formula in the context of the problem.