Modern power grids are incorporating massive mechanical flywheels to store excess energy. When the grid produces more power than is consumed (e.g., from solar panels at noon), the excess electricity drives a motor to spin up a heavy carbon-fiber rotor housed in a vacuum chamber. The electrical energy is converted into rotational kinetic energy ($K_R = \frac{1}{2}I\omega^2$). When the grid needs a sudden burst of power, the spinning flywheel acts as a generator, converting its massive stored kinetic energy back into electricity. The design challenge is maximizing the Moment of Inertia ($I$) and angular velocity ($\omega$) without the rotor breaking apart from centrifugal stresses.

When the main engine of a helicopter spins the large overhead rotor blades clockwise, it applies a massive torque to the blades. According to Newton's Third Law (and conservation of angular momentum), the blades exert an equal and opposite torque on the helicopter's body, causing the fuselage to spin counter-clockwise. To prevent the helicopter from spinning out of control, a smaller tail rotor is installed. This tail rotor creates a horizontal thrust force at a distance from the center of mass, generating a counter-torque that exactly cancels the torque from the main rotor, keeping the helicopter stable.

Two small masses, $m_1 = 3 \text{ kg}$ and $m_2 = 5 \text{ kg}$, are attached to a massless rod. They are located at $x = -2 \text{ m}$ and $x = 4 \text{ m}$ respectively on the x-axis. Calculate the moment of inertia of this system about the y-axis.

A uniform solid sphere has a mass of $M = 10 \text{ kg}$ and a radius of $R = 0.5 \text{ m}$. Its moment of inertia through its center of mass is $I_{cm} = \frac{2}{5}MR^2$. Find its moment of inertia if it is spun around an axis tangent to its surface.

Derive the moment of inertia of a uniform thin rod of mass $M$ and length $L$ rotating about an axis perpendicular to the rod passing through one of its ends (let the end be at $x=0$).

A wrench $0.3 \text{ m}$ long is used to loosen a bolt. A force of $50 \text{ N}$ is applied at the end of the wrench at an angle of $60^\circ$ relative to the wrench handle. What is the magnitude of the torque applied?

A solid cylindrical pulley has a mass of $4 \text{ kg}$ and a radius of $0.2 \text{ m}$ ($I = \frac{1}{2}MR^2$). A rope is wrapped around it, and a constant pull of $10 \text{ N}$ is applied to the rope. Assuming the pulley rotates freely, what is its angular acceleration?

A block of mass $m = 2 \text{ kg}$ hangs from a string wrapped around a solid cylindrical pulley of mass $M = 6 \text{ kg}$ and radius $R = 0.3 \text{ m}$. The block is released from rest. Find the downward acceleration of the block. ($g = 9.8 \text{ m/s}^2$)

A figure skater is spinning at $3.0 \text{ rad/s}$ with her arms extended. Her moment of inertia in this pose is $2.4 \text{ kg}\cdot\text{m}^2$. She pulls her arms in, reducing her moment of inertia to $0.8 \text{ kg}\cdot\text{m}^2$. What is her new angular velocity? (Ignore friction).

A stationary merry-go-round has a radius of $2 \text{ m}$ and a moment of inertia of $500 \text{ kg}\cdot\text{m}^2$. A $40 \text{ kg}$ child runs tangentially at $5 \text{ m/s}$ and jumps onto the very edge of the merry-go-round. What is the angular velocity of the system immediately after?

Disk 1 ($I_1 = 0.04 \text{ kg}\cdot\text{m}^2$) is spinning clockwise at $40 \text{ rad/s}$. It is dropped onto Disk 2 ($I_2 = 0.06 \text{ kg}\cdot\text{m}^2$) which is initially at rest. Friction between them causes them to eventually spin together at the same rate. Find their final angular velocity and the percentage of rotational kinetic energy lost.