DCP-04-071-50

> A frame is rotating in a circle with varying speed $v=(2t-4)~\rm{m/s}$ where $t$ is in second. An object is viewed from this frame. The pseudo force:

I'm very familiar with the application of pseudo forces like centrifugal and Coriolis force. But, I felt this question lacked some details and chose "Data is insufficient", but the answer says the pseudo force is "minimum at 2 s" i.e., option (b). Could you please provide any hints? I think without the data on object's position relative to the frame's axis of rotation, it's not possible to find the centrifugal force on the object.

In the rotating frame the object appears to be rotating at $v = -(2t - 4) m/s$

Ok sir.

So if the distance to the object is $r$ then there appears to be a centripetal force $F = v^2/r = (2t - 4)^2/r$

OK so far?

Ok sir.

So to find the minima/maxima we differentiate $dF/dt = 4(2t-4)$

Ok sir. So there's an extrema at $t=2$.

So we get a maximum/minimum at $t=2$.

Differentiate again to find out if it's a minimum of maximum $d^2F/dt^2 = +8$

So it's a minimum.

Ok sir. I understood this. But why isn't it "zero at 2 s"?

Hmm ...

It is zero at $t = 2$ ...

Yes sir.

Further, this question is from "only single correct answer" category and not "multiple correct answers" type, so we need to choose only one. I guess this question was mistyped in this section.

Or are we missing something else?

I think you're right. Both B and C are clearly correct.

Ok sir. Maybe the formula you provided for $F$ is only taking the centrifugal force into account. So if we want to justify (b), then we could say there might be some non-zero Coriolis force.

Does this reason look good?

But again another question, how do we know whether the Coriolis force is constant, or attaining maximum/minimum? Also the circular motion is also not uniform.

Oh, hang on.

There are two components to the (pseudo) acceleration, radial and tangential, and we have only calculated the radial component.

Ah. Fine. I hope it's constant however.

The tangential component is a constant $a = 2 m/s^2$.

Fine sir. So (b) is correct and the pseudo force is non-zero. Is this tangential component - the Coriolis force?

So we were correct that there is a minimum at $t = 2$, but the acceleration at that minimum is $a = \sqrt{a_r^2 + a_t^2} = \sqrt{0^2 + 2^2} = 2 m/s^2$.

So C is wrong.

Got it sir :-)

The Coriolis force is a different phenomenon.

Ok sir. So here, we're assuming the object under inspection is not moving in that rotating frame, right? There's no mention about this in the question however.

@GuruVishnu hi, sorry, I was on the phone.

The Coriolis force is seen when the radial distance is changing, and that isn't happenung here.

@JohnRennie No problem sir :-)

@JohnRennie Fine. Thank you for the clarification sir.

:-)