Vibrations and Waves MP205, Assignment 4

Please hand in your solutions before 5pm Wednesday 11th March, starred questions willbe done out in tutorial and do not need to be handed up.If you have any questions about this assignment, please ask your tutor,Una Nı Ruairc, (uniruairc@thphys.nuim.ie), Office 1.6, Dept. Mathematical Physics.

1. (a) Show that the frequency of vibration under adiabatic conditions of a columnof gas confined to a cylindrical tube, closed at one end, with a well-fitting butfreely moving piston of mass m is given by:

ω =

√Aγp

lm

.

(b) A steel ball of diameter 2cm oscillates vertically in a 12-liter glass tube con-taining air at atmospheric pressure (as pictured below). Verify that the periodof oscillation should be about 1 sec. (Assume adiabatic pressure change withγ = 1.4. Density of steel = 7600kgm−3.)

* 2. The motion of a linear oscillator may be represented by means of a graph in whichx is shown on the x-axis and dx/dt on the y-axis. (ie a point on the graph will beof the form (x, dx/dt)) The history of the oscillator is then a curve.

(a) Show that for an undamped oscillator this curve is an ellipse.

(b) Show (at least qualitatively) that if a damping term is introduced one gets acurve spiraling into the origin.

3. Verify that x = Ae−αt cosωt is a possible solution of the equation

d2x

dt2+ γ

dx

dt+ ω2

0x = 0,

and find α and ω in terms of γ and ω0.

* 4. An object of mass 0.2 kg is hung from a spring whose spring constant is 80 N/m.The object is subject to a resistive force given by −bv, where v is its velocity inmeters per second.

(a) Set up the differential equation of motion for free oscillations of the system.

(b) If the damped frequency is√

3/2 of the undamped frequency, what is the valueof the constant b?

(c) What is the Q of the system?

5. Many oscillatory systems, although the loss or dissipation mechanism is not anal-ogous to viscous damping, show an exponential decrease in their stored averageenergy with time E = E0e

−γt. A Q for such oscillators may be defined using thedefinition Q = ω0

γ, where ω0 is the natural angular frequency.

(a) When the note ”middle C” on the piano is struck, its energy of oscillationdecreases to one half its initial value in about 1 sec. The frequency of middleC is 256 Hz. What is the Q of the system?

(b) If the note an octave higher (512 Hz) takes about the same time for its energyto decay, what is its Q?

(c) A free, damped harmonic oscillator, consisting of a mass m = 0.1kg movingin a viscous liquid of damped coefficient b (Fviscous = −bv), and attached to aspring of spring constant k = 0.9Nm−1, is observed as it performs oscillatorymotion. Its average energy decays to 1

eof its initial value in 4 sec. What is the

Q of the oscillator? What is the value of b?

6. A U-tube has vertical arms of radii r and 2r, connected by a horizontal tube oflength ` whose radius increases linearly from r to 2r. The U-tube contains liquidup to a height h in each arm. The liquid is set oscillating, and at a given instantthe liquid in the narrower arm is a distance y above the equilibrium level.

* (a) Show that the potential energy of the liquid is given by U = 58gρπr2y2.

* (b) Show that the kinetic energy of a small slice of liquid in the horizontal arm(see the diagram) is given by

dK =1

2ρ

πr2dx

(1 + x/`)2

(dy

dt

)2

.

(Note that, if liquid is not to pile up anywhere, the product velocity × crosssection must have the same value everywhere along the tube.)

(c) Using the result of part (b), show that the total kinetic energy of all the movingliquid is given by

K =1

4ρπr2

(`+

5

2h

)(dy

dt

)2

.

(Ignore any nastiness at the corners.)Hint: Work out the kinetic energy for the 3 sections separately.

(d) From (a) and (c) calculate the period of oscillations of ` = 5h/2.Hint: Look at the standard form for the total Energy, can we manipulate ourexpression into this form?