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The moment of inertia, often symbolized as \( I \), denotes an object's resistance to rotational motion. For a solid cylinder rotating about its central axis, the moment of inertia is calculated using the formula: \( I = \frac{1}{2} m r^2 \). This formula considers the cylinder’s mass \( m \) and radius \( r \). For our satellite example, given that the mass is 1210 kg and the diameter is 1.21 meters, the radius is therefore half of the diameter, which is 0.605 meters. Plugging these values into the formula, we obtain: \( I = \frac{1}{2} \times 1210 \text{kg} \times (0.605 \text{m})^2 \) Calculating inside the parentheses first: \( (0.605 \text{m})^2 = 0.366025 \text{m}^2 \) Now multiply by the other values: \( I = \frac{1}{2} \times 1210 \times 0.366025 = 221.45 \text{kg} \cdot \text{m}^2 \) This tells us the amount of resistance the satellite has to changes in its rotational motion.

Rotational kinetic energy is the energy an object possesses due to its rotation. It’s given by the formula: \[ K = \frac{1}{2} I \omega^2 \] where \( I \) is the moment of inertia and \( \omega \) is the angular velocity. To solve for the rotational kinetic energy of our communication satellite, we first need to convert its rotational speed from revolutions per second to radians per second. Given the rotation rate of 1.52 revolutions per second: \(1\text{ rev} = 2 \pi \text{ radians} \), thus \( \omega = 1.52 \times 2 \pi = 1.52 \times 6.2832 = 9.5483 \text{ rad/s} \) Now, plug \( I \) and \( \omega \) into the energy formula: \( K = \frac{1}{2} \times 221.45 \times (9.5483)^2 \)First, calculate the square of \( \omega \): \( (9.5483)^2 = 91.15 \text{rad}^2/\text{s}^2 \) Then multiply through: \( K = \frac{1}{2} \times 221.45 \times 91.15 = 10096.6 \text{ J} \) Thus, the satellite's rotational kinetic energy is approximately 10096.6 Joules.

Understanding the physics behind a solid cylinder is crucial when dealing with various physical scenarios, such as our satellite. Here’s a brief breakdown: **Properties** - A solid cylinder has uniform mass distribution throughout its volume. - Its rotational inertia is a measure of how the mass is distributed in relation to its axis of rotation. **Key Equations** - Moment of inertia: \( I = \frac{1}{2} m r^2 \) - Rotational kinetic energy: \( K = \frac{1}{2} I \omega^2 \)For our scenario, given the mass, radius, and rotational speed of the cylinder, we used these key concepts to determine both the moment of inertia and the rotational kinetic energy. This exemplifies how fundamental principles in physics can directly apply to real-world objects like communication satellites. In nutshell, understanding these concepts not only helps in calculating values effectively but also in comprehending the underlying principles and their broader applications in physics.