Fictitious Forces: Science Fiction & Fantasy

[greendog77]

[N/A]

 

[hilbert2]

Yes, in the rotating frame the "centrifugal force" has magnitude $m\omega^{2} x$, and the equation of motion of the ball is

$\frac{d^{2}x}{dt^{2}}=-(2k-m\omega^{2})x$

Therefore, the frequency of oscillation in radians/second is $\sqrt{\frac{2k-m\omega^{2}}{m}}$ and in periods/second it is $\frac{1}{2\pi}\sqrt{\frac{2k-m\omega^{2}}{m}}$. The period of oscillation is the reciprocal of the latter expression: $T=2\pi\sqrt{\frac{m}{2k-m\omega^{2}}}$. You only made a small error in how frequency, angular frequency and period are related to each other. You can avoid this kind of mistakes by checking that your answer has correct dimensions.

 

[greendog77]

I was going more for the concept of fictitious forces. What I'm confused about is that if you leave a small mass in a larger sphere, and rotate the sphere with angular velocity w, the small mass would travel in a straight line to the edge of the sphere in the inertial frame of the sphere. However, this isn't really true, as the mass simply travels in a circle in the inertial frame of the sphere. What's wrong here?

 

[hilbert2]

It seems you're confusing the terms "inertial" and "non-inertial" here. The rotating frame, where centrifugal effects arise, is non-inertial, not inertial.

If you have a mass at rest inside a rotating sphere, the sphere exerts no force on the mass. When you transform to the rotating rest frame of the sphere, there is an apparent inward centripetal force on the mass and it travels on a circular trajectory. In the "ball-inside-tube" problem, the ball was constrained to have same angular velocity as the tube (the walls of the tube exert a force on the ball) and the situation was opposite, leading to a centrifugal force.

 

[greendog77]

Hmm so what is the fictitious force for the rotating sphere scenario?

 

[hilbert2]

If you take the equation of motion of the mass in rotating sphere, and transform to a rotating coordinate system, a term appears that describes an inward centripetal force. That is the fictitious force in this situation.

 

[greendog77]

I thought the four base fictitious forces were centrifugal, linear, coriolis, and azimuthal. So does that mean fictitious forces don't necessarily have to fall within those categories? Under what circumstances do they fall under those categories? You mentioned the ball is constrained to have the same angular velocity as the tube. Why does that cause it to have a centrifugal force instead of a centripetal fictitious force?

 

[hilbert2]

In the rotating sphere scenario, the fictitious force falls in the centrifugal category, it just has a negative value in that case.

You probably haven't studied coordinate transformations yet. The confusion disappears when you learn how to actually transform the eqs of motion and see how they look like in different non-inertial frames.

 

[A.T.]

Hmm so what is the fictitious force for the rotating sphere scenario?

The inertially moving mass moves straight in the non-rotating frame. In the roating frame is moves on a cruved path. This is attributed to fictitious forces (centrifugal & coriolis).

https://www.youtube.com/watch?v=49JwbrXcPjc

 

[dauto]

In this example the fictitious force is a combination of centrifuge (away from the center) and Coriolis (towards the center. The Coriolis is larger than the centrifuge giving a net fictitious force towards the center which turns out to exactly what is needed to provide the centripetal force (towards the center) required to keep the object in circular motion.

 

[greendog77]

Thank you all! This makes complete sense to me now.