Centripetal accelerations with two parallel axes - the Scrambler

[fezzik]

Centripetal accelerations with two parallel axes -- the "Scrambler"

Homework Statement

This should be a pretty quick and straightforward question to answer. The problem regards centripetal forces on an object rotating simultaneously around two parallel axes, like in the "Scrambler" amusement park ride. Basically, one object has uniform circular motion around a minor axis, while this minor axis simultaneously rotates around a major axis (both axes are in the same plane). (Check out the animation at http://www.mrwaynesclass.com/teacher/circular/cd/scrambler/home.html and the problems at http://vip.vast.org/Circ_Lab/big_act/index.htm ).

Homework Equations

The Attempt at a Solution

It seems to me you can find the net centripetal acceleration on the rider at point C (where the rider gets closest to the major axis) by finding (1) the centripetal acceleration relative to the major axis and (2) the centripetal acceleration relative to the minor axis, and then (3) subtracting one from the other (since, at C, the two accelerations are in opposite directions). It also seems to make sense that, when (1) calculating the centripetal acceleration relative to the major axis, the radius is given by the distance between C and the major axis, and the velocity is given by the tangential velocity of the major arm (and not by the absolute tangential velocity of the rider) at C -- is that correct?

Thanks!

 

[fezzik]

Maybe someone can recommend an online reference that explains this kind of situation? (I.e., how to calculate the total centripetal acceleration of an object rotating around a centre of rotation, while that centre of rotation itself rotates around another centre of rotation -- both centres of rotation in the same plane.)

 

[fezzik]

Is there another category in the forum where I should be posting this question instead?

 

[ehild]

What is the question? To find the centripetal acceleration in the inertial frame of reference in general, or at point C? Your result looks correct, that the the centripetal acceleration is the difference between that with respect to the major axis and the one with respect to the minor axis.

I would solve the general problem by writing out the position vector of the rider in the inertial frame of reference, and would take the second derivative with respect to time to get the acceleration. Than I would determine the radial component with respect to the main axis.

ehild

 

[fezzik]

Thanks! Well, it's a bit over my head to try to figure out what function describes the position of the rider, so it sounds like I can just take the approach I described (basically, (1) ignoring the rotation around the minor axis in order to find the rider's acceleration around the major axis alone; then (2) ignoring the rotation around the major axis, that is, assuming it is fixed, in order to find the rider's acceleration around the minor axis alone; then, (3) finding the difference between the two accelerations). For clarification, I am trying to find the total centripetal acceleration on the rider when he/she's at point C, when viewed in an inertial frame of reference.

Also for clarification, the main thing I had wanted to confirm was that when (1) calculating the centripetal acceleration relative to the major axis alone, the velocity is given by the tangential velocity of the major arm (and not by the absolute tangential velocity of the rider) at C) -- and it sounds like you have confirmed that that approach is correct.

 

[ehild]

For me it is easier to think in Descartes coordinates:
You have two systems, A with origin O and B with origin O'. The position vector of O' is R(t) in system A. You have an object, and its position is r'(t) in system B. In system A, the position vector is

r(t) = R(t) + r'(t)

O' moves along a circle of radius R around O with angular velocity Ω, so its coordinates are:

X=Rcos(Ωt), Y=Rsin(Ωt).

In system B, the rider moves along a circle with radius r around O' with angular velocity ω', its coordinates are

x=rcos(ω't), y=rsin(ω't).

The coordinates will be

x=Rcos(Ωt)+rcos(ω't), y=Rsin(Ωt)+rsin(ω't)

in system A.

The components of acceleration:

ax=-[RcosΩ²(Ωt)+rω'²cos(ω't)],

ay=-[RsinΩ²(Ωt)+rω'²sin(ω't)].

The centripetal acceleration in A is the radial component of the vector a, the dot product of a with the radial unit vector, r/r.

ehild

 

[fezzik]

Thanks for the detailed description, that's great -- I'll take some time to digest it.