5. ROTATIONAL OSCILLATIONS IN SEVERAL CONFIGURATIONS This problem deals with thr

Question

5. ROTATIONAL OSCILLATIONS IN SEVERAL CONFIGURATIONS This problem deals with three oscillatory scenarios with the goal of seeing similarities in oscil- latory motion across different situations. (a) Consider a cylindrical disc of mass M and radius R that sits on a surface and has a spring (with spring constant, k) connected to its center. The other end of the spring is connected to a vertical wall, such that the spring is horizontal. This is shown in Fig. 2. Show that for small os- cillations, in which the wheel rolls without slipping, the period of oscillation for this configuration 3M (3) 2k Assume the disc is set into motion by an initial displacement of the spring end by a small amount M, R FIGURE 2. The oscillating wheel for problem 5 part a) rm. (b) Consider a bicycle wheel (mass M, and radius R) that is fixed in place, but can rotate about an axle through its center. One end of a spring is attached to the wheel at a point a distance r directly above the center. The other end of the spring is attached to a wall such that the spring is horizontal. Find the period of oscillation for the configuration (assume small oscillations). The set-up is shown in Fig. 3. (c) Consider the same disc from part a) hung from a thin, massless cable such that the cable is

Explanation / Answer

Part a.

P.E of spring is=kx2/2

K.E of translation of clylindrical disc.=Mv2/2

K.E of Rotation of clylindrical disc=Mv2/4

Total energy of system is,E=kx2/2+Mv2/2+Mv2/4

Applying enegy conservation.

dE/dt=0

kx.dx/dt+3Mv.dv/2dt=0

d2x/dt2= -2kx/3M

compare above equation with simple harmonic motion's equation,

d2x/dt2= -omega2x

omega2=2k/3M

As T=2pi/omega

Hence

T=2pi(3M/2k)1/2

Part b.

Time period is same in this case as part a.

Part c.

In dynamic equlibrium

Id2theta/dt2= k.theta

where I is moment of inertia of disc.

and theta is displacement angle.

and k is torsion constant.

d2theta/dt2= k.theta/I

compare this equation with S.H.M equation

T=2pi(I/k)1/2

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