Bead sliding on a rotating rod

[shankarrg]

Bead is at rest on a thin rod pivoted at one end. Bead is about a cm from the pivoted end of the rod. Rod now starts rotating with an uniform angular velocity w rad/sec.

1.What curve does the bead trace from the point of view of an inertial observer?

Here what i think... solution of the differential equation m*d^2(r)/dt^2= mrw^2

In the radial direction

r=Ae^(-wt)+Be^(-wt) , where A and B can be found out by the initial conditions. A=a/2, B=a/2

and in angular direction, theta =w*t

I think the curve would be cycloid. How to infer cycloid from the above equation

Regards
shankar

 

[Chestermiller]

Is this a homework problem?

Chet

 

[shankarrg]

No i was reading non inertial frames chapter in introduction to mechanics david kleppner there is an worked out example about finding the above equation i understood that but i started wondering about the curve it makes

 

[HomogenousCow]

http://en.wikipedia.org/wiki/Logarithmic_spiral

The implicit curve r(theta) is a logarithmic spiral.

 

[Chestermiller]

Please show us how you arrived at your differential equation, and how you solved it. The solution as you've written it doesn't look correct (with the same exponent in both terms).

Chet

 

[HomogenousCow]

Please show us how you arrived at your differential equation, and how you solved it. The solution as you've written it doesn't look correct (with the same exponent in both terms).

Chet

One of the exponents is supposed to be positive.

 

[Chestermiller]

One of the exponents is supposed to be positive.

Yes. Otherwise, the solution looks OK (assuming that the radius has been scaled to the initial radius). We are talking about hyperbolic cosine here.

chet

 

[shankarrg]

Oh sorry was unaware that i made a typo . Yes i meant one exponent is positive and radius is scaled

Thanks for your reply chet