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Moment of inertia is a fundamental concept in rotational motion. It is essentially the rotational equivalent of mass in linear motion. It describes how difficult it is to change an object's rotational speed. The larger the moment of inertia, the harder it is to rotate the object.

For different shapes, the formula for moment of inertia, denoted as \( I \), varies. For a solid sphere like the asteroid in the problem, rotating about an axis through its center, the formula is:

• \( I = \frac{2}{5} M r^2 \)

where:

• \( M \) is the mass of the object in kilograms
• \( r \) is the radius in meters

This formula is derived from integrating all the infinitesimal mass elements over the volume of the sphere, considering how far each element is from the axis of rotation.Knowing the moment of inertia is crucial for understanding how much torque is needed to achieve a desired angular acceleration, setting the stage for calculating the angular velocity, and ultimately solving the given problem.

Angular velocity is the rate at which an object rotates or revolves. It tells us how fast the rotational motion is happening. The concept is akin to linear velocity in translational motion but in a rotational sense.

Measured in radians per second (rad/s), angular velocity is typically denoted by \( \omega \). In the problem, the asteroid rotates four times a day, so we first need to convert this to a standard unit of radians per second:

• One complete revolution equals \( 2\pi \) radians.
• With four revolutions per day, the angular velocity \( \omega_i \) is calculated as:\[ \omega_i = \frac{4 \times 2\pi}{24 \times 3600} \] rad/s.

This conversion is vital because rotational kinematic equations and other calculations typically require angular velocity in rad/s.Understanding angular velocity is key to solving problems involving rotational motion, helping determine how long and how much force is needed to cause a change in the rotational state.

Torque is the rotational equivalent of force. It measures how much a force acting on an object causes it to rotate. When a force is applied tangentially, like in the case of the astronaut's "tug" on the asteroid, it creates torque.

The formula for torque, symbolized by \( \tau \), is given by:

• \( \tau = F \cdot r \)

where:

• \( F \) is the force applied in Newtons
• \( r \) is the distance from the axis of rotation, or the radius

In this example, a force of 285 N applied at a radius of 123 m results in:

• \( \tau = 285 \times 123 \)

This calculated torque is then used to find the angular acceleration using the formula \( \tau = I \cdot \alpha \).By understanding torque, we can better comprehend how rotational motion is controlled and altered, making it possible to predict the rotation of objects under various forces.