Problem: a unifiorm hoop of mass m and radius r rolls without slipping on a fixed...

Problem: a unifiorm hoop of mass m and radius r rolls without slipping on a fixed cylinder of radius R. if the hoop is stats rolling from rest on top of the bigger cylinder, use the method of Lagrange multipliers to find the point at which the hoop fall off the cylinder.

Question: I know how to derive Lagrange equeation. but to use the method of Lagrange multipliers, i have to finde constrain.

solution says that f1, f2 are constrain. i don't know why constrain should be f2.

f1=R+r-L=0: L is the distance from center of cylinder to center of hoop

f2= L(d_Thet/d_t) +r(d_Phi/d_t)

But i think, to not slipp...Hoop rotation distance is same as hoop travel distance on the cylinder surface.

So, i thought f2 as like below;

f2=r ( d_Thet/d_t)- R(d_Phi/d_t)=0

( d_Thet) is rotation angle of hoop, (d_Phi) is sweep anglular by hoop on surface of cylinder.

Am i wrong?

Solutions

Expert Solution

genius_generous answered 1 year ago

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