91Ó°ÊÓ

The moment of inertia is a fundamental concept in rotational dynamics. It represents the distribution of mass in a rotating object and affects how easily that object can change its rotational speed. Just like mass affects linear motion, moment of inertia affects rotational motion.
The formula for the moment of inertia of a solid cylinder rotating about its central axis is given by:

• \[ I = \frac{1}{2} m r^2 \]
• Where,
  • \(m\) is the mass of the cylinder, and
  • \(r\) is the radius.

For our flywheel in the exercise, the mass \( m \) is 1000 kg and the radius \( r \) is half of the diameter, 0.90 m. Substituting these values into the formula gives the moment of inertia as 405 kg·m².
A larger moment of inertia means the flywheel is more resistant to changes in its rotational speed, important for storing kinetic energy efficiently. The moment of inertia is crucial in determining how much energy the flywheel can store and release, a key aspect of its energy efficiency in powering the bus.

Rotational kinetic energy is the energy possessed by an object due to its rotation. This type of energy is analogous to the linear kinetic energy of an object moving in a straight line. It's crucial in applications like our flywheel, which depends on rotational kinetic energy to perform its function as an energy storage device.
The formula for calculating rotational kinetic energy is:

• \[ KE = \frac{1}{2} I \omega^2 \]
• Where,
  • \(I\) is the moment of inertia, and
  • \(\omega\) is the angular speed in radians per second.

In our example, the flywheel's kinetic energy at its angular speed of 3000 rev/min (converted to 300π rad/s) is calculated using its moment of inertia of 405 kg·m². The result is an energy of 1.80 x 10^7 Joules.
This energy is what powers the bus, illustrating the efficiency and capacity of flywheel energy storage for sustained power supply.

Angular speed indicates how fast something is rotating, typically given in revolutions per minute (rev/min) but often needs to be converted to radians per second (rad/s) for calculations in physics. Understanding this conversion is essential for correctly determining the energies involved in rotational motion.
The conversion factor is rooted in the relationship between revolutions and radians:

• 1 revolution = \(2\pi\) radians

When converting from rev/min to rad/s, use the formula:

• \[\text{Angular speed in rad/s} = \text{Angular speed in rev/min} \times \frac{2 \pi \text{ rad}}{1 \text{ rev}} \times \frac{1 \text{ min}}{60 \text{ s}}\]

This calculation was used to convert the flywheel's angular speed of 3000 rev/min to 300Ï€ rad/s in the exercise, which is necessary to compute the rotational kinetic energy. Such conversions are vital in any task involving rotational dynamics.

Power and energy are core components of physics that describe how much work an object can do over time, and how fast that energy is expended, respectively. For a system like our flywheel-powered bus, understanding both concepts is key to determining its operational efficiency.
Energy is measured in Joules (J) and refers to the total amount of work that can be performed. Power, measured in Watts (W), is the rate at which energy is used or transferred.
To calculate how long the bus can operate using the flywheel, the formula used is:

• \[\text{Time} = \frac{\text{Energy}}{\text{Power}} \]

Given that the flywheel holds 1.80 x 10^7 J of energy, and the bus requires an average power of 1.86 x 10^4 W, the bus can operate for approximately 968 seconds between stops.
These calculations demonstrate how effective flywheel energy storage can be in managing the energy needs of real-world applications, such as public transportation.