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?

Verified step by step guidance

1. Convert the angular velocities from revolutions per minute (rev/min) to radians per second (rad/s). Use the conversion factor where $1 \text{ rev} = 2\pi \text{ rad}$ and $1 \text{ min} = 60 \text{ s}$.
2. Calculate the initial angular velocity, $\omega_i$, and the final angular velocity, $\omega_f$, using the formula $\omega = 2\pi \times \text{(revolutions per minute)}/60$.
3. Use the formula for the change in kinetic energy of a rotating object, which is given by $\Delta K = \frac{1}{2} I (\omega_f^2 - \omega_i^2)$, where $I$ is the moment of inertia, $\omega_i$ is the initial angular velocity, and $\omega_f$ is the final angular velocity.
4. Rearrange the kinetic energy formula to solve for the moment of inertia $I$. The formula becomes $I = \frac{2 \Delta K}{\omega_f^2 - \omega_i^2}$.
5. Substitute the values of $\Delta K$, $\omega_i$, and $\omega_f$ into the rearranged formula to find the moment of inertia $I$.

Key Concepts

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 an object's angular velocity, 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.