23.4: Worked Examples
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Example 23.3: Rolling Without Slipping Oscillating Cylinder
Attach a solid cylinder of mass M and radius R to a horizontal massless spring with spring constant k so that it can roll without slipping along a horizontal surface. At time t , the center of mass of the cylinder is moving with speed Vcm and the spring is compressed vθ ,1 = ± 2gl(1− cosθ0 ) . a distance x from its equilibrium length. What is the period of simple harmonic motion for the center of mass of the cylinder?

Solution: At time t , the energy of the rolling cylinder and spring system is
E=12Mv2cm+12Icm(dθdt)2+12kx2
where x is the amount the spring has compressed, Icm=(1/2)MR2, and because it is rolling without slipping
dθdt=VcmR
Therefore the energy is
E=12MV2cm+14MR2(VcmR)2+12kx2=34MV2cm+12kx2
The energy is constant (no non-conservative force is doing work on the system) so
0=dEdt=342MVcmdVcmdt+12k2xdxdt=Vcm(32Md2xdt2+kx)
Because Vcm is non-zero most of the time, the displacement of the spring satisfies a simple harmonic oscillator equation
d2xdt2+2k3Mx=0
Hence the period is
T=2πω0=2π√3M2k
Example 23.4: U-Tube
A U-tube open at both ends is filled with an incompressible fluid of density ρ. The cross-sectional area A of the tube is uniform and the total length of the fluid in the tube is L . A piston is used to depress the height of the liquid column on one side by a distance x0 (raising the other side by the same distance) and then is quickly removed (Figure 23.10). What is the angular frequency of the ensuing simple harmonic motion? Neglect any resistive forces and at the walls of the U-tube.

Solution: We shall use conservation of energy. First choose as a zero for gravitational potential energy in the configuration where the water levels are equal on both sides of the tube. When the piston on one side depresses the fluid, it rises on the other. At a given instant in time when a portion of the fluid of mass Δm=ρAx is a height x above the equilibrium height (Figure 23.11), the potential energy of the fluid is given by
U=Δmgx=(ρAx)gx=ρAgx2
At that same instant the entire fluid of length L and mass m=ρAL is moving with speed v , so the kinetic energy is
K=12mv2=12ρALv2
Thus the total energy is
E=K+U=12ρALv2+ρAgx2
By neglecting resistive force, the mechanical energy of the fluid is constant. Therefore
0=dEdt=ρALvdvdt+2ρAgxdxdt
If we just consider the top of the fluid above the equilibrium position on the right arm in Figure 23.13, we rewrite Equation (23.4.10) as
0=dEdt=ρALvxdvxdt+2ρAgxdxdt
where vx=dx/dt. We now rewrite the energy condition using dvx/dt=d2x/dt2 as
0=vxρA(Ld2xdt2+2gx)
This condition is satisfied when vx=0 i.e. the equilibrium condition or when
0=Ld2xdt2+2gx
This last condition can be written as
d2xdt2=−2gLx
This last equation is the simple harmonic oscillator equation. Using the same mathematical techniques as we used for the spring-block system, the solution for the height of the fluid above the equilibrium position is given by
x(t)=Bcos(ω0t)+Csin(ω0t)
where
ω0=√2gL
is the angular frequency of oscillation. The x -component of the velocity of the fluid on the right-hand side of the U-tube is given by
vx(t)=dx(t)dt=−ω0Bsin(ω0t)+ω0Ccos(ω0t)
The coefficients B and C are determined by the initial conditions. At t=0 the height of the fluid is x(t=0)=B=x0. At t=0, the speed is zero so vx(t=0)=ω0C=0, hence C=0. The height of the fluid above the equilibrium position on the right hand-side of the U-tube as a function of time is thus
x(t)=x0cos(√2gLt)