A bicycle wheel (with very light spokes) of mass M and radius R is mounted on a

Question

A bicycle wheel (with very light spokes) of mass M and radius R is mounted on a light axel through its center. The axle is attached to a horizontal spring of spring constant k, and the wheel is constrained to roll back and forth, without slipping, as shown. Your job is to figure out if this device is a simple harmonic oscillator, and to derive a formula for the period of the motion.

<a href="http://s893.photobucket.com/user/yonnyg19/media/physsmall.jpg.html" target="_blank"><img src="http://i893.photobucket.com/albums/ac132/yonnyg19/physsmall.jpg" border="0" alt=" photo physsmall.jpg"/></a>

(a) Write a statement of conservation of energy for this system.

(b) Differentiate your expression for energy with respect to time to get an equation of
motion, that is, a relation between position and acceleration.

(c) Compare your equation of motion to the equation of motion for a simple
harmonic oscillator. Is this device a simple harmonic oscillator? Justify your
answer briefly.

d) Derive an expression for the period of the motion of this oscillator.

Explanation / Answer

a)conserving energy,

0.5kx^2+0.5mv^2+0.6Iw^2=constant

or 0.5kx^2+0.5mv^2+0.5*0.5mv^2=constant

or 0.5kx^2+0.5*1.5*mv^2

b)differentiating,

0.5*2*kxv +0.5*2*1.5*va*m=0

or kx+1.5ma=0

but a=-w^2x

so,

kx=1.5*Mw^2x

or w=(k/(1.5m))^1/2

d)period=2pi/w

=2pi*(1.5m/k)^1/2

c)yes it is a simple harmonic oscillator since the acceleration is proportional to the negative of the displacement.

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