Please help In the figure above, a solid cylinder of mass M attached to a horizo

Question

Please help

In the figure above, a solid cylinder of mass M attached to a horizontal spring (stiffness k, negligible mass) rolls without slipping along a horizontal surface. If the system is released from rest when the spring is stretched by an amount A, find the maximum potential energy and the maximum kinetic energy, which is the sum of the translational and the rotational kinetic energies of the cylinder as it passes through the equilibrium position. Find the period of the simple harmonic motion that the cylinder's center of mass executes in terms of the given k and M. To do this, write an expression for the total mechanical energy of the system when the center of mass is at some arbitrary point xcm, then take the time derivative of this expression to find a relation between acceleration d2xcm/dt2 and displacement xcm (a coordinate system with the origin an equilibrium yields this). This second-order ODE has the pattern that we expect for all simple harmonic oscillators (see 15-7 and 15-8 in the book).

Explanation / Answer

a)maximum potential energy=0.5kA^2

b)conserving energy,

max PE= max KE

so max KE=0.5kA^2

c)applyinh law of conservation of energy for small oscillartion,

0.5kA^2=0.5kx^2+0.5mv^2+0.5*0.4mv^2

differentiating this with respect to t,

0=kxv+mva+0.5mva

or kx=-1.4 ma

so,

-kx=-1.4mw^2x

or w=(k/(1.4m))^1/2

so T=2pi/w

=2pi*(1.4m/w)^1/2

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