Question

Can you explained all the steps and make it as much details as possible thank you:)

The disk is centered at the origin of the universe. The particle has a negligible mass and size and is not glued to the disk. While this problem is happening in a mathematical universe, in which laws of physics are not necessarily held, the particle can be imagined as a balloon that is tied to the disk by an imaginary string with a length of infinity, which is fully rigid with respect to rotation of the disk, however, it is fully flexible with respect to the wind and tornado. As a result, the velocity vector of the particle is constituted by the three components noted.

The notion of inequality in section d of the project considers the orientation of the rotation of the velocity vector that causes the curl. To clarify, the inequality in section d simply means: Under what condition the curl of the particle's velocity under a tornado, wind, and the rotating disk has the same rotation orientation as the curl of the particle's velocity under wind and the rotating disk with a necessarily higher magnitude?

Answer 1

The inquiry examines the rotational dynamics and the effect of external forces on a particle in a physics-based scenario, with a focus on velocity vector orientation, angular velocity, and resulting motion.

Step-by-step explanation:

The question deals with the rotational dynamics of a particle on a disk, influenced by external forces such as wind and a tornado. It utilizes concepts of angular velocity, tangential speed, centripetal acceleration, and angular momentum. To solve the inequality in section d, we need to consider the orientation of the rotation of the velocity vector caused by the curl. This is governed by the right-hand rule, which relates to angular velocity, direction of rotation, and the axis of rotation.

According to the information given, if a disk has a constant angular velocity ω, the tangential speed v of a particle at radius r from the disk's axis of rotation is given by v = rω. Hence, particles further from the axis move faster. When we add effects like wind or a tornado, the situation becomes more complex and a resultant velocity vector is formed, which is the vector sum of the individual velocities due to rotation, wind, and the tornado.

The particle's velocity vector under the complex motion can be analyzed by taking the curl of the velocity field. To meet the condition stated in the inequality, the additional effect of the tornado should cause the curl magnitude to increase while keeping the rotation orientation consistent with the pre-tornado state. This will depend on the relative strengths and directions of the wind, the tornado, and the rotation of the disk.