Centrifugal and circular motion

[clavin]

ell let's say there's a cabin and its rotating about its axisand we have a block of mass m
placed (not at the the centre of the cabin)
and there's some friction between the the block and the surface
now let's say the speed of the cabin is increased a lil high so that friction can no more balance it
to make the block rotate it in a circular motion
so now if i see from the cabin frame then there's the centrifugal force which pushes the block to the wall
so now my qs is
firstly have i stated the trajectory correct that in the case friction cannot balance then the block will move outward in a straight line to towards the wall of the cabin

2nd qs. in the cabin frame i used the centrifugal force to see the motion
but how do i know the blocks motion in ground frame?
in ground frame there is no centrifugal force so why now is the block moving outward?


*edit

well i forgot that in the above qs there will be coroilis force acting
so let me change the situation
lets say i have a circular disk rotating about it axis and a grove(channel) from the centre to the outer end of the disk
and i have placed a particle in the grove.

now again the same qs how to find the force that makes the particle move outward from the ground frame

 

[Doc Al]

so now if i see from the cabin frame then there's the centrifugal force which pushes the block to the wall
so now my qs is
firstly have i stated the trajectory correct that in the case friction cannot balance then the block will move outward in a straight line to towards the wall of the cabin

Once it starts moving there will also be a coriolis force to contend with, thus the trajectory won't be a straight radial line.

2nd qs. in the cabin frame i used the centrifugal force to see the motion
but how do i know the blocks motion in ground frame?
in ground frame there is no centrifugal force so why now is the block moving outward?

Because it has a tangential velocity and insufficient force to keep it moving in a circle.

 

[clavin]

ok tnx for that
i forgot about the coriolis force
anyways that's not in my sylabus
so i am going to edit the qs a bit
please refer to the first post

 

[RoyalCat]

One way to resolve the 'bead-in-a-groove' problem is to brute-force it with vector algebra. This approach is, however, less than satisfactory.

Rewriting the vector identity for acceleration in terms of radial and polar coordinates:

$$\vec a_{radial}=[\frac{d^2 r}{dt^2}-r(\frac{d\theta}{dt})^2]\hat r$$

$$\vec a_{tangential}=\frac{1}{r}[\frac{d}{dt}(r^2\frac{d\theta}{dt})]\hat \theta$$

These identities are far from trivial, and personally, I had to look them up, but they are valid in any frame of reference.

Let's assume for the sake of simplicity that there's no friction, and that the normal force can only act perpendicular to the grooves. That means that the bead will only experience a normal force acting in the direction of the tangential velocity in the laboratory frame, while the groove rotates with constant angular velocity $$\omega$$

Using Newton's second law:
$$a_{radial}=0$$ (No forces in the radial direction!)
$$a_{tangential}=0$$ (The angular velocity is constant!)

Plugging all we know into the two identities for the accelerations:
$$\vec\frac{d^2 r}{dt^2} = \omega ^2 \vec r$$
$$\vec N=2m\omega\frac{dr}{dt}\hat \theta$$

These sorts of problems break down what we usually perceive as radial and tangential accelerations. For instance, we were just shown that there is 0 "radial" acceleration, but there is an acceleration in the radial direction!
Things get wonky here.