Chapter 3: Linear Equations of Higher Order

Finally,

*«»- 5.2315).

The local maxima and minima of x(t) occur when

0 = x'(t)

' ' " '" - 5.2315) - :

(24)

- 5.2315)],

and thus when

tan(8? - 5.2315) = -0.75.

Because tan '(—0.75) ~ —0.6435, we want to find the first four positive values offsuch that 8? - 5.2315 is the sum of -0.6435 and an integral multiple of TT. These val-ues of r and the corresponding values of x computed with the aid of (24) are as fol-lows:

t (s)

x (m)

0.1808

-0.2725

0.5735

0.0258

0.9662

- 0.0024

1.3589

0.0002

We see that the oscillations are damped out very rapidly, with their amplitude decreas-ing by a factor of about 10 every half-cycle. See Problems 30 and 31 for a more gen-eral discussion of this phenomenon. •

3.4 Problems

1. Determine the period and frequency of the simple harmonicmotion of a 4-kg mass on the end of a spring with spring con-stant 16N/m.

2. Determine the period and frequency of the simple harmonicmotion of a body of mass 0.75 kg on the end of a spring withspring constant 48 N/m.

3. A mass of 3 kg is attached to the end of a spring that isstretched 20 cm by a force of 15 N. It is set in motion with initialposition *0 = 0 and initial velocity v0 = — 10 m/s. Find the am-plitude, period, and frequency of the resulting motion.

4. A body with mass 250 g is attached to the end of a springthat is stretched 25 cm by a force of 9 N. At time t = 0 the bodyis pulled 1 m to the right, stretching the spring, and set in motionwith an initial velocity of 5 m/s to the left, (a) Find x(t) in theform C cos(o)0? + a), (b) Find the amplitude and period of themotion of the body.

In Problems 5 through 8, assume that the differential equa-tion of a simple pendulum of length L is L8" + g6 — 0, whereg = GM/R2 is the gravitational acceleration at the location of

the pendulum (at distance R from the center of the earth; M de-notes the mass of the earth).

5. Two pendulums are of lengths Lt and L.,, and — when lo-cated at the respective distances R^ and R^ from the center of theearth — have periods p, andp-,. Show that

pt =

P2

6. A certain pendulum clock keeps perfect time in Paris, wherethe radius of the earth is R = 3956 (mi). But this clock loses 2min 40 s per day at a location on the equator. Use the result ofProblem 5 to find the amount of the equatorial bulge of theearth.

7. A pendulum of length 100.10 in., located at a point at sealevel where the radius of the earth is R = 3960 (mi), has thesame period as does a pendulum of length 100.00 in. atop anearby mountain. Use the result of Problem 5 to find the heightof the mountain.

8. Most grandfather clocks have pendulums with adjustable

Section 3.4: Mechanical Vibrations 171

lengths. One such clock loses 10 min per day when the length ofits pendulum is 30 in. With what length pendulum will this clockkeep perfect time?

9. Derive Eq. (5) describing the motion of a mass attached tothe bottom of a vertically suspended spring. (Suggestion: Firstdenote by x(t) the displacement of the mass below the un-stretched position of the spring; set up the differential equationfori. Then substitute v = x — XQ in this differential equation.)

10. Consider a floating cylindrical buoy with radius r, height h,and density p S 0.5 (recall that the density of water is 1 g/cm3).The buoy is initially suspended at rest with its bottom at the topsurface of the water and is released at time t = 0. Thereafter it isacted on by two forces: a downward gravitational force equal toits weight mg = pTrr2hg and an upward force of buoyancy equalto the weight Trr2xg of water displaced, where x = x(t) is thedepth of the bottom of the buoy beneath the surface at time t(Fig. 3.4.9). Conclude that the buoy undergoes simple harmonicmotion about the equilibrium position xe = ph with periodp = 2irVp/i/g. Compute p and the amplitude of the motion ifp = 0.5 g/cm3, h = 200 cm, and g = 980 cm/s2.

FIGURE 3.4.9. The buoy of Problem 10

11. A cylindrical buoy weighing 100 Ib (thus of mass m =3.125 slugs in ft-lb-s (fps) units) floats in water with its axis ver-tical (as in Problem 10). When depressed slightly and released, itoscillates up and down four times every 10 s. Assume that fric-tion is negligible. Find the radius of the buoy.

12. Assume that the earth is a solid sphere of uniform density,with mass M and radius R = 3960 (mi). For a particle of mass mwithin the earth at distance r from the center of the earth, thegravitational force attracting m toward the center isF = - GMrm/r2, where Af. is the mass of the part of the earthwithin a sphere of radius r. (a) Show that Fr= — GMmr/R3.(b) Now suppose that a hole is drilled straight through the cen-ter of the earth, thus connecting two antipodal points on its sur-face. Let a particle of mass m be dropped at time t = 0 into thishole with initial speed zero, and let r(t) be its distance from thecenter of the earth at time t (Fig. 3.4.10). Conclude fromNewton's second law and part (a) that r"(t) = — k2r(t), where

FIGURE 3.4.10. A mass m falling down a hole throughthe center of the earth (Problem 12)

k2 = GMIR3 = gIR. (c) Take g = 32.2 ft/s2, and concludefrom part (b) that the particle undergoes simple harmonic mo-tion back and forth between the ends of the hole, with a periodof about 84 min. (d) Look up (or derive) the period of a satel-lite that just skims the surface of the earth; compare with the re-sults in part (c). How do you explain the coincidence? Or is it acoincidence? (e) With what speed (in miles per hour) does theparticle pass through the center of the earth? (f) Look up orderive the orbital velocity of a satellite that just skims the sur-face of the earth; compare the value with the result in part (e).How do you explain the coincidence? Or is it a coincidence?

The remaining problems in this section deal with freedamped motion. In Problems 13 through 19, a mass m isattached to both a spring (with given spring constant k) anda dashpot (with given damping constant c). The mass is set inmotion with initial position XQ and initial velocity z>0. Find theposition function x ( t ) and determine whether the motion is over-damped, critically, damped, or underdamped. If it is under-damped, write x(t) in the form Ce'p'cos(<u1f — a).

= 4;

: = 63;

k = 16;

k = 50;

k = 169;

k = 40;

fe= 125;

20. A 12-lb weight (mass m = 0.375 slugs in fps units) is at-tached both to a vertically suspended spring that it stretches6 in. and to a dashpot that provides 3 Ib of resistance for everyfoot per second of velocity, (a) If the weight is pulled down1 ft below its static equilibrium position and then released fromrest at time t = 0, find its position function x(t). (b) Find

13.

14.

15.

16.

17.

18.

19.

m

in

in

in

in

in

m

_ i2 '

= 3,

= 1,

= 2,

= 4

= 2,

= 1,

c

c

c

c

c

c

c

= 3,

= 30,

= 8,

= 12,

= 20,

= 16,

= 10,

*o =

*o

= 2, v

= 2, i

*0 = 5' V>

*o", x,

Xn0

; x,

= 0,

i = 4>

= 5, i

;, = 6,

o = 0

'J0 = 2

, = -10

yfl = -8

v0 = 16

v = 4ov = 50

172 Chapter 3: Linear Equations of Higher Order

the frequency, time-varying amplitude, and phase angle of themotion.

21. This problem deals with a highly simplified model of a carof weight 3200 Ib (mass m = 100 slugs in fps units). Assumethat the suspension system of the car acts like a single spring andits shock absorbers like a single dashpot, so that its vertical vi-brations satisfy Eq. (4) with appropriate values of the coeffi-cients. (a) Find the stiffness coefficient k of the spring if thecar undergoes free vibrations at 80 cycles per minute(cycles/min) when its shock absorbers are disconnected, (b)With the shock absorbers connected the car is set into vibrationby driving it over a bump, and the resulting damped vibrationshave a frequency of 78 cycles/min. After how long will thetime- varying amplitude be 1% of its initial value?

Problems 22 through 32 deal with a mass-spring-dashpotsystem having position function x(t) satisfying Eq. (4). We writex0 = x(0) and l'0 = x'(0) and recall that/; = cl(2m), co* = k/m,and io\ WQ — p1. The system is critically damped, over-damped, or underdamped, as specified in each problem.

22. (Critically damped) Show in this case that

x(t) = (x0px0t)

23. (Critically damped) Deduce from Problem 22 that the masspasses through x = 0 at some instant t > 0 if and only if x0 andv0 + px0 have opposite signs.

24. (Critically damped) Deduce from Problem 22 that x(t) hasa local maximum or minimum at some instant t > 0 if and onlyif VQ and v0 + px0 have the same sign.

25. (Overdamped) Show in this case that

x(t) = — [(V0 - r2x0) er>< - (V0 - r,x0) e'*'],

where r, , r, = — p ± Vp^ — WQ and y = (r, — r2)/2 > 0.

26. (Overdamped) If jc0 = 0, deduce from Problem 25 that

x(t) = — e-

27. (Overdamped) Prove that in this case the mass can passthrough its equilibrium position x = 0 at most once.

28. (Underdamped) Show that in this case

x(t) =

29. (Underdamped) If the damping constant c is small in com-parison with VSmA:, apply the binomial series to show that

8m*

30. (Underdamped) Show that the local maxima and minima ofx(t) = Ce~pt cos(o)lt — a) occur where

tan (to,/ — a) = .to,

Conclude that ?2 — /, = 27r/w, if two consecutive maxima occurat times t, and ?2.

31. (Underdamped) Let x{ and x2 be two consecutive local max-imum values of x(t). Deduce from the result of Problem 30 that

..-=,- ,_ *i _ 2"P

The constant A = 2-!rp/(o] is called the logarithmic decrementof the oscillation. Note also that c = mw\\lir becausep = c/(2m).

Note: The result of Problem 31 provides an accuratemethod for measuring the viscosity of a fluid, which is an impor-tant parameter in fluid dynamics, but which is not easy to mea-sure directly. According to Stokes' drag law, a spherical body ofradius a moving at a (relatively slow) speed v through a fluid ofviscosity m experiences a resistive force FR = 6Tr/jj2V. Thus if aspherical mass on a spring is immersed in the fluid and set inmotion, this drag resistance damps its oscillations with dampingconstant c = GTTO/J.. The frequency to! and logarithmic decre-ment A of the oscillations can be measured by direct observa-tion. The final formula in Problem 31 then gives c and hence theviscosity of the fluid.

32. (Underdamped) A body weighing 100 Ib (mass m = 3.125slugs in fps units) is oscillating attached to a spring and a dash-pot. Its first two maximum displacements of 6.73 in. and 1.46 in.are observed to occur at times 0.34 s and 1.17 s, respectively.Compute the damping constant (in pound-seconds per foot) andspring constant (in pounds per foot).

Differential Equations and DeterminismGiven a mass m, a dashpot constant c, and a spring constant

k, Theorem 2 of Section 3.1 implies that the equation

mx" + ex' + kx = 0(25)

has a unique solution for t = 0 satisfying given initial conditionsx(0) = xa, jc'(0) = v0. Thus the future motion of an ideal mass-spring-dashpot system is completely determined by the differen-tial equation and the initial conditions. Of course in a real physi-cal system it is impossible to measure the parameters m, c, andk precisely. Problems 33 through 36 explore the resulting uncer-tainty in predicting the future behavior of a physical system.

33. Suppose that m = 1, c = 2, and k = 1 in Eq. (25). Show-that the solution with jc(0) = 0 and jt'(O) = 1 is

x,(t) = te~'.