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The moment of inertia is a physical property of a rotating object that determines how much torque is needed to rotate the object about an axis. For a solid cylinder, like our flywheel, it can be thought of as the rotational equivalent of mass in linear motion. To calculate the moment of inertia for a solid cylinder, use the formula:\[ I = \frac{1}{2} mr^2 \]In this equation, \(m\) represents the mass of the cylinder and \(r\) is its radius. In our example, the mass is 500 kg, and the radius is 1 m. Plug these values into the formula, giving us:\[ I = \frac{1}{2} \times 500 \times (1.0)^2 = 250 \text{ kg m}^2 \]This tells us it takes 250 kg m² of rotational inertia to start or stop the flywheel’s spin. Understanding moment of inertia is crucial for designing systems that involve rotational motion.

Kinetic energy, in the context of a rotating object, measures the energy an object possesses due to its rotation. For rotational motion, the formula is given by:\[ KE = \frac{1}{2} I \omega^2 \]where \(I\) is the moment of inertia and \(\omega\) is the angular velocity. Our flywheel reaches an angular velocity of \(200 \pi \text{ rad/s}\). Using the moment of inertia calculated previously (250 kg m²), we find the kinetic energy:Calculate \((200 \pi)^2 = 40000 \pi^2\):\[ KE = 125 \times 40000 \pi^2 \]Substituting \(\pi \approx 3.1416\):\[ KE \approx 5000 \times (3.1416)^2 \approx 5000 \times 9.87 \approx 49350 \, \text{J} \]This calculated kinetic energy of 49350 Joules is the energy stored in the flywheel after it reaches its maximum speed. This energy will later be utilized by the truck for power consumption.

Power consumption refers to the rate at which energy is used up. In our scenario, the truck uses power at a rate of 8.0 kW, or 8000 W. To determine how long the truck can operate on the stored energy, divide the total kinetic energy by the power consumption:\[ t = \frac{KE}{P} = \frac{49350}{8000} \, \text{seconds} \]Perform the calculation:\[ t \approx 6.17 \, \text{seconds} \]Convert this operation time from seconds to minutes:\[ t \approx \frac{6.17}{60} \approx 0.1028 \, \text{minutes} \]This means the truck can run for just over 6 seconds on the stored energy. Understanding this helps in designing the system to ensure the energy demands match the application needs.

A solid cylinder is a three-dimensional object with two parallel circular bases connected by a curved surface. In physics, they often serve as models for rotating systems due to their uniform mass distribution. When considering rotational motion, the properties like mass, radius, and axis of rotation are crucial in determining a cylinder's moment of inertia.

• Mass: For the flywheel, this is given as 500 kg.
• Radius: Also provided as 1 m.
• Axis of Rotation: In this problem, the axis passes through the center of the base, making calculations simpler with the standard formulas.

Understanding solid cylinders from this perspective is vital not only for calculating properties like moment of inertia but also for optimizing energy storage and minimizing energy loss in rotating systems.