Question

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?

Short answer

The required moment of inertia is approximately 0.599 kg·m².

Step-by-step solution

Convert Angular Velocities to Radians per Second

First, we need to convert the initial and final angular velocities from revolutions per minute (rev/min) to radians per second (rad/s).Using the conversion factor, \( 1 ext{ rev/min} = \frac{2\pi ext{ rad}}{60 ext{ s}} \).So, the initial angular velocity \( \omega_i \) is:\[ \omega_i = 650 \times \frac{2\pi}{60} = \frac{650 \times 3.1416}{60} = 68.068 \text{ rad/s} \]And the final angular velocity \( \omega_f \) is:\[ \omega_f = 520 \times \frac{2\pi}{60} = \frac{520 \times 3.1416}{60} = 54.453 \text{ rad/s} \]

Set up the Kinetic Energy Change Formula

The change in kinetic energy \( \Delta KE \) is given by:\[ \Delta KE = \frac{1}{2} I (\omega_i^2 - \omega_f^2) \]where \( I \) is the moment of inertia, \( \omega_i \) is the initial angular velocity, and \( \omega_f \) is the final angular velocity.In this problem, \( \Delta KE = 500 \, \text{Joules} \).

Calculate Moment of Inertia

Rearrange the kinetic energy formula to solve for \( I \):\[ I = \frac{2 \cdot \Delta KE}{\omega_i^2 - \omega_f^2} \]Substitute \( \Delta KE = 500 \, \text{J} \), \( \omega_i = 68.068 \, \text{rad/s} \), and \( \omega_f = 54.453 \, \text{rad/s} \):\[I = \frac{2 \times 500}{68.068^2 - 54.453^2} = \frac{1000}{4633.2246 - 2964.537} \= \frac{1000}{1668.6876} = 0.599 \text{ kg·m}^2\]

Verify and State the Required Moment of Inertia

The calculations show that the required moment of inertia is approximately \( 0.599 \, \text{kg·m}^2 \). It is verified based on the known values and formula used.

Key concepts

Angular Velocity

Angular velocity is a measure of how fast an object rotates or spins. It tells us the angle through which a point or a line has been rotated in a specified time.
Imagine the hands of a clock moving around its center; that motion can be described using angular velocity. It is usually represented by the symbol \( \omega \).
Angular velocity can be expressed in different units such as revolutions per minute (rev/min) or radians per second (rad/s). When solving physics problems, rad/s is the preferred unit because it is part of the International System (SI) of units.- **Example Usage:** In circular motion, a car moving around a racetrack uses angular velocity to describe its motion.- **Equation:** Angular velocity can be calculated using the formula \( \omega = \frac{\theta}{t} \), where \( \theta \) is the angle moved through, and \( t \) is time.

Kinetic Energy

Kinetic energy is the energy possessed by an object due to its motion. Anything that moves - from a car rolling down a hill to a flywheel spinning in an engine - has kinetic energy.
In rotational motion, the kinetic energy has a specific formula that involves the moment of inertia and angular velocity:- **Formula:** \( KE = \frac{1}{2} I \omega^2 \)- \( KE \) stands for kinetic energy.- \( I \) is the moment of inertia, which measures how much an object resists rotational motion.- \( \omega \) is the angular velocity.
Key takeaway: The faster an object spins, or the bigger its moment of inertia, the more kinetic energy it has.

Conversion to Radians per Second

Many angular velocity problems begin with values given in revolutions per minute (rev/min), which are practical for everyday contexts but need conversion into radians per second (rad/s) for use in scientific calculations.
The conversion factor is crucial here:- **Conversion Factor:** \( 1 \text{ rev/min} = \frac{2\pi \text{ rad}}{60 \text{ s}} \)- **Conversion Steps:** 1. Multiply the number of revolutions by \( 2\pi \) to change it to radians. 2. Divide by 60 to convert from per minute to per second.
**Example:** To convert 650 rev/min: - Multiply by \( \frac{2\pi}{60} \): \[ 650 \times \frac{2\pi}{60} = 68.068 \text{ rad/s} \]
This conversion simplifies using angular velocity in other formulas.

Change in Kinetic Energy Formula

The change in kinetic energy formula is a key concept in rotational dynamics. It helps calculate how much energy is gained or lost as an object speeds up or slows down.
The formula is:- \( \Delta KE = \frac{1}{2} I (\omega_i^2 - \omega_f^2) \)- \( \Delta KE \) represents the change in kinetic energy.- \( I \) is the moment of inertia, indicating object resistance to changes in its rotation.- \( \omega_i \) and \( \omega_f \) are the initial and final angular velocities, respectively.
In the given exercise, this formula helps determine how much energy is lost when the angular velocity of the flywheel decreases. It's essential for solving many physics problems related to rotational motion.