Question

98. A bobbin rolls without slipping over a horizontal surface so that the velocity $v$ of the end of the thread (point $A$) is directed along the horizontal. A board hinged at point $B$ leans against the bobbin as shown. The inner and outer radii of the bobbin are $r$ and $R$ respectively. Determine the angular velocity $\omega$ of the board as a function of angle $\theta$.

Answer 1

\omega = (v(1+cos θ))/r.

Answer 2

We use the geometric constraint that the distance from the bobbin’s center O = (X,R) to the board is R. Writing this as   –X sin θ + R cos θ = R,  we solve for X and differentiate with respect to time. Equating dX/dt (from the constraint) with the horizontal speed of the bobbin, V = (vR)/r (obtained from the no–slip condition together with v = ω_b r) yields, after simplification and using a half–angle identity, the result   ω = (v(1+cos θ))/r.