A circular object with moment of inertiaI around its axis of symmetry, mass M an

Question

A circular object with moment of inertiaI around its axis of symmetry, mass M and radius R rollswithout slipping along a horizontal surface. It is connectedby a spring with constant k to a wall. The object is observedto have maximum velocity vmax.

a) Find , the angular frequency withwhich the object oscillates.

b) Find the maximum displacementxmax of theobject from its equilibrium position.

c) We have been coy in specifying the exactshape of the object. What is the angular frequency if theobject is: i) a solid cylinder; ii) a cylindrical shell; iii) asphere; and iv) something with I -> 0, appropriate if allthe mass were concentrated on the axis.

Explanation / Answer

(a) Let s be the rotational speed of the object. So, 0.5*k*A*A - 0.5*k*x*x = 0.5*m*v*v + 0.5*I*s*s s = v/R for no slipping. So, v*v = [kA2 - kx2]/(m +I/R2) and v = [(kA2 - kx2)/(m +I/R2)] = *[A*A - x*x] So, angular frequency of oscillation = =[k/(m + I/R2)] (b) For v = vmax, x = 0 So, vmax = [kA2/(m +I/R2)] So, At maximum displacement, v = 0. So, A = vmax*[(m+I/R2)/k] (c) For the various cases,    I/R2 values are respectively m/2, m, 2m/5,0.    So, values are : C*(k/m)where C = (2/3), (1/2), (5/7), 1.

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