Question

In experiments at the Princeton Plasma Physics Laboratory, a plasma of hydrogen atoms is heated to over 500 million degrees Celsius (about 25 times hotter than the center of the Sun) and confined for tens of milliseconds by powerful magnetic fields ( 100,000 times greater than the Earth's magnetic field). For each experimental run, a huge amount of energy is required over a fraction of a second, which translates into a power requirement that would cause a blackout if electricity from the normal grid were to be used to power the experiment. Instead, kinetic energy is stored in a colossal flywheel, which is a spinning solid cylinder with a radius of \(3.00 \mathrm{~m}\) and mass of \(1.18 \cdot 10^{6} \mathrm{~kg}\). Electrical energy from the power grid starts the flywheel spinning, and it takes 10.0 min to reach an angular speed of \(1.95 \mathrm{rad} / \mathrm{s}\). Once the flywheel reaches this angular speed, all of its energy can be drawn off very quickly to power an experimental run. What is the mechanical energy stored in the flywheel when it spins at \(1.95 \mathrm{rad} / \mathrm{s}\) ? What is the average torque required to accelerate the flywheel from rest to \(1.95 \mathrm{rad} / \mathrm{s}\) in \(10.0 \mathrm{~min} ?\)

Short answer

Answer: The mechanical energy stored in the flywheel is \(1.02 \cdot 10^{10} \mathrm{J}\), and the average torque required to accelerate it is \(1.725 \cdot 10^4 \mathrm{N \cdot m}\).

Step-by-step solution

Find the Moment of Inertia for Flywheel

We know the flywheel is a spinning solid cylinder with a mass of \(1.18 \cdot 10^{6} \mathrm{~kg}\) and a radius of \(3.00 \mathrm{~m}\). The formula for the moment of inertia (I) for a solid cylinder is: \(I=\frac{1}{2} m r^2\). So, substituting values we get: \(I = \frac{1}{2} (1.18 \cdot 10^{6})(3)^2 = 5.31 \cdot 10^6 \mathrm{kg \cdot m^2}\).

Compute the Mechanical Energy Stored in the Flywheel

To find the mechanical energy stored in the flywheel, we can use the formula for kinetic energy, which is: \(E_{k} = \frac{1}{2} I \omega^2\). Substituting the values, we get: \(E_k = \frac{1}{2}(5.31 \cdot 10^6)(1.95)^2 = 1.02 \cdot 10^{10} \mathrm{J}\).

Compute the Angular Acceleration

Next, let's find the angular acceleration (\(\alpha\)) of the flywheel. The formula for angular acceleration is: \(\alpha = \frac{\Delta \omega}{\Delta t}\). We know the initial angular velocity (\(\omega_{0}\)) is 0, and the final angular velocity (\(\omega\)) is 1.95 rad/s. The time during which the change in angular velocity occurs is 10.0 minutes, which is (10 * 60 sec) = 600 seconds. So the angular acceleration is: \(\alpha = \frac{1.95}{600} = 3.25 \cdot 10^{-3} \mathrm{rad/s^2}\).

Compute the Average Torque

Now, we can find the average torque required to accelerate the flywheel from rest to 1.95 rad/s in 10.0 minutes. The formula for torque is: \(\tau = I\alpha\). Substituting the values, we get: \(\tau = (5.31 \cdot 10^6)(3.25 \cdot 10^{-3}) = 1.725 \cdot 10^4 \mathrm{N \cdot m}\). To summarize, the mechanical energy stored in the flywheel when it spins at 1.95 rad/s is \(1.02 \cdot 10^{10} \mathrm{J}\), and the average torque required to accelerate the flywheel from rest to 1.95 rad/s in 10.0 minutes is \(1.725 \cdot 10^4 \mathrm{N \cdot m}\).

Key concepts

Moment of Inertia

Picture the flywheel as an enormous, spinning disc—it's not just mass that dictates its resistance to rotational changes, but also how that mass is distributed. This is where the concept of moment of inertia comes into play. It's a measure of an object's resistance to changes in its rotation, depending on the mass and how far it's spread from the axis of rotation.

For solid cylinders like our flywheel, the moment of inertia (\(I\)) can be mathematically expressed as \(I=\frac{1}{2} m r^2\), where \(m\) is the mass and \(r\) the radius. What's surprising here is the role of the radius—it's squared. This means that even a slight increase in \(r\) significantly increases the \(I\), hence requiring more force to change its rotational speed.

Mechanical Energy

The stalwart flywheel doesn't just sit there—it holds onto energy! Mechanical energy, in this context, is essentially kinetic energy due to the flywheel's rotation. You can visualize it as the energy that would be needed to stop the wheel or to get it spinning up from a standstill. Calculating this energy involves the moment of inertia and the angular speed (\(\omega\)) with the formula \(E_{k} = \frac{1}{2} I \omega^2\).

This mechanical energy is the flywheel's currency, storing and then 'spending' it to power equipment quickly, a trait particularly useful when avoiding blackouts caused by sudden, high power demand—like our plasma physics experiment. The faster the flywheel spins, or the greater its mass and size, the more energy it can store.

Angular Acceleration

What about the change of the flywheel's rotation over time, or how quickly it spins up? Here we meet angular acceleration (\(\alpha\)), a measure of the rate of change in angular velocity. This acceleration isn't constant when a real-world flywheel powers up, but for our scenario, we assume it to be steady for simplicity.

The formula \(\alpha = \frac{\Delta \omega}{\Delta t}\) helps us find the angular acceleration by dividing the change in angular velocity (\(\Delta \omega\)) by the time (\(\Delta t\)) taken for that change. Significant angular acceleration means the flywheel can reach its required speed swiftly, critical for rapid energy deployment.

Torque

Now, to get our flywheel up to speed—literally—we'll need to apply a force. This force working at a distance from the rotation axis is called torque (\(\tau\)). It's the rotational equivalent of a linear force. Think of it as the twist you apply when opening a jar; you're exerting torque.

The connection between torque, moment of inertia, and angular acceleration is pivotal (pun intended). The torque formula \(\tau = I\alpha\) binds them together, demonstrating that the greater the moment of inertia or the faster the angular acceleration required, the larger the torque needed. In mechanical systems like the flywheel, torque is what gets things moving and is vital for controlling the speed of rotation.