Problem 45 - University Physics with Modern Physics

Question

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

Step-by-Step Solution

Answer

The required moment of inertia is approximately 0.60 kg·m².

Step 1: Understand the given problem

The problem involves finding the moment of inertia of a flywheel. We know that it must lose 500 J of kinetic energy as its angular velocity decreases from 650 revolutions per minute (RPM) to 520 RPM.

Step 2: Convert angular velocities to rad/s

First, we need to convert the angular velocities from RPM to radians per second (rad/s). The conversion factor is \(\frac{2\pi}{60}\). - Initial angular velocity: \[ \omega_i = 650 \, \text{RPM} \times \frac{2\pi}{60} = \frac{1300\pi}{60} \, \text{rad/s} \approx 68.1 \, \text{rad/s} \]- Final angular velocity:\[ \omega_f = 520 \, \text{RPM} \times \frac{2\pi}{60} = \frac{1040\pi}{60} \, \text{rad/s} \approx 54.5 \, \text{rad/s} \]

Step 3: Use the kinetic energy formula

The kinetic energy (KE) of a rotating object is given by:\[ KE = \frac{1}{2}I\omega^2 \]where \(I\) is the moment of inertia and \(\omega\) is the angular velocity.

Step 4: Apply energy conservation

The change in kinetic energy can be described by:\[ \Delta KE = KE_{initial} - KE_{final} = \frac{1}{2}I\omega_i^2 - \frac{1}{2}I\omega_f^2 = 500 \, \text{J} \]

Step 5: Rearrange the equation for moment of inertia

Rearrange the equation to solve for \(I\):\[ 500 = \frac{1}{2}I(\omega_i^2 - \omega_f^2) \]\[ I = \frac{1000}{\omega_i^2 - \omega_f^2} \]

Step 6: Calculate \(\omega_i^2\) and \(\omega_f^2\)

Calculate the squared values using converted angular velocities:\[ \omega_i^2 = (68.1)^2 = 4637.61 \, \text{rad}^2/ ext{s}^2 \]\[ \omega_f^2 = (54.5)^2 = 2970.25 \, \text{rad}^2/ ext{s}^2 \]

Step 7: Solve for moment of inertia \(I\)

Substitute the squared values into the equation from Step 5:\[ I = \frac{1000}{4637.61 - 2970.25} \approx \frac{1000}{1667.36} \approx 0.60 \, \text{kg} \cdot \text{m}^2 \]

Key Concepts

Flywheel

A flywheel is an important mechanical device used to store or release energy in a system. It plays a vital role in engines and various machines by smoothing out the fluctuations in angular velocity and ensuring consistent energy supply.

Flywheels achieve this through their rotational motion, which means they can store energy when spinning and release that energy when slowing down. This balance is especially useful in situations like vehicles or energy-generating machines, where the power output needs to be stable.

• In the context of a gasoline engine, the flywheel helps balance the energy cycles, making the engine run more smoothly.
• The efficiency of a flywheel depends on its size and the moment of inertia, which allows it to store more energy during its motion.

By understanding flywheels, we can appreciate how they play a key role in managing and storing kinetic energy across different systems.

Kinetic Energy

Kinetic energy is the energy of motion. For rotating objects, such as a flywheel, we often talk specifically about rotational kinetic energy.

The formula for the kinetic energy (KE) of a rotating object is \[ KE = \frac{1}{2} I \omega^2 \] where \(I\) is the moment of inertia—an object's resistance to changes in its rotation—and \(\omega\) is the angular velocity, or how quickly something is spinning.

• Rotational kinetic energy is greater in objects that have a larger moment of inertia and higher angular velocity.
• When a flywheel accelerates, its kinetic energy increases, storing energy within the system.

In practical terms, managing kinetic energy is about controlling how much energy is stored or released from the flywheel during operation. This principle is why flywheels are used in engines, aiding in stabilizing the energy they emit or absorb.

Angular Velocity

Angular velocity is the rate at which an object rotates or spins. It's a critical aspect when studying the dynamics of rotating systems like flywheels.

When we talk about angular velocity, we refer to how fast something rotates, usually measured in radians per second (rad/s).

• For a flywheel, higher angular velocity means more kinetic energy is stored, while a lower angular velocity means energy is being released.
• Conversions from revolutions per minute (RPM) to radians per second are often needed to calculate angular velocity accurately.

In systems where precision and timing are essential, controlling angular velocity ensures consistent performance. Understanding this concept helps us in recalibrating and optimizing systems, whether in vehicles or other mechanical settings.