Find the equations of motion for a particle in a frame rotating with variable angular velocity \(\boldsymbol{\omega}\), and show that there is another apparent force of the form \(-m \dot{\boldsymbol{\omega}} \wedge \boldsymbol{r}\). Discuss the physical origin of this force.

If you've ever been on a merry-go-round, you might have felt like you were being pushed outward as it spun. That sensation is due to what we refer to as the centrifugal force. It's important to note that centrifugal force is not a 'real' force in physics, but rather an apparent force that is felt by an object in a rotating frame of reference. It arises because of inertia, the property that makes objects resist changes in their state of motion. In a rotating reference frame, this force appears as a reaction to the centripetal force that is necessary to keep an object moving in a circle. The formula for this force is \( -m\boldsymbol{\omega}\times(\boldsymbol{\omega}\times\boldsymbol{r'}) \) where \( \boldsymbol{r'} \) is the position in the rotating frame and \( \boldsymbol{\omega} \) is the frame's angular velocity.

In our exercise, the centrifugal force is one of the components that need to be considered to understand the equations of motion for a particle in a rotating frame.

Now let's move on to another fascinating apparent force: the Coriolis force. This effect is observed in rotating systems and can be quite counterintuitive. It is responsible for deflections in trajectories of objects in motion within a rotating frame. For instance, Earth's rotation causes wind patterns to curve, which is a significant factor in meteorology. The Coriolis force is given by \( -2m(\boldsymbol{\omega}\times\boldsymbol{v'}) \) where \( \boldsymbol{v'} \) is the velocity relative to the rotating frame.

In our problem, the Coriolis force contributes to the total apparent force on a particle in a rotating frame when considering its motion equations. The particle's path appears curved as this force acts perpendicular to its velocity.

The Euler force, often less discussed than the Coriolis or centrifugal forces, is nonetheless an essential part of the dynamics in non-uniformly rotating systems. It appears in response to changes in the rotation rate of the frame, specifically when the angular velocity of the frame is not constant. In the context of our exercise, we find the additional force term \( -m \dot{\boldsymbol{\omega}} \wedge \boldsymbol{r'} \) that stems from a varying angular velocity, \( \dot{\boldsymbol{\omega}} \). This force, akin to the Euler force in classical mechanics, manifests during time-dependent rotational movements, leading to effects like precession or nutation in spinning objects.

Angular velocity, denoted by \( \boldsymbol{\omega} \), is a vector quantity that represents how fast an object rotates or revolves relative to another point, usually the center of a circle or another axis. In a sense, it's the rotational equivalent of linear velocity. In our exercise, we deal with a frame that is rotating with a variable angular velocity, meaning \( \boldsymbol{\omega} \) changes with time. This change contributes to the Euler force and must be included in the equations of motion for the particle. Understanding angular velocity is crucial because it directly influences the magnitude and direction of the centrifugal, Coriolis, and Euler forces in a rotating system.