Question

In a mathematical universe and in the time t=t 0​ , a disk starts rotating with a constant angular velocity of w disk =a 1^ +b 1 ^ +c 1k^ , over which there is a randomly-located particle P with the position vector of rP =x i^ +y ^ +z k^ . We know that under such a motion, the velocity vector of the particle P can be found as V P∣disk = w disk ​ × rP . This sole rotating condition continues until the time t=t 1​ , when a harmonic wind field with the velocity vector of V wind =a 2​ cosz ^ + b 2​ cosx ^ +c 2​ cosy k^ starts to blow over the rotating disk.

Answer 1

This physics problem involves a particle on a rotating disk with a constant angular velocity and later influenced by a wind field, combining rotational motion with harmonic motion.

Step-by-step explanation:

The question involves analyzing the motion of a particle on a rotating disk initially experiencing a constant angular velocity, and then being influenced by an additional wind field. The particle's velocity in relation to the disk (VP|disk) is determined by the cross product of the disk's angular velocity vector (wdisk) and the particle's position vector (rP). Since the disk rotates with a constant angular velocity, each point on the disk, and therefore the particle, will have a tangential speed that increases with its distance from the axis of rotation. Once the wind starts blowing over the disk at t=t1, the particle's velocity is influenced by both the rotating disk and the wind field. A significant concept to understand here is the simple harmonic motion observed when projecting the circular motion of the particle onto an axis, relating to the x-component of the velocity following a cosine function in time.