Give some examples of everyday vibrating objects. 1nnn.xue5 m8n*ic zf.Which exhibit SHM, at least appr8nzx5u in *fcnm.1ne. oximately?

Is the acceleration of a s3 vhc zy,(4amw8k b(umimple harmonic oscillator ever zero? If so, where?

Explain why the motion of a piston in as 8lh+wn1 ,bqpn automobile engine is approximately simple harmon l+pnb8,qwh 1sic.

Real springs have mass. z 2z:7 sdap-1lxlu 2ii//in*nl* cxiaWill the true period and frequency be larger or smaller than given by the equations for a mass oscillating on the end of an idpdnz :c*71iis xial 2laixl/ n-u2*z/ealized massless spring? Explain.

How could you double the maximum speed ob s- zk-.qxez1f a simple harmonic oscillator (SHO)?

A 5.0-kg trout is attached to the hook of a vertical spring scale, and thencilg(: wlna 22cny cx8h6-,bc is released. Describe the scale reading as a function of tima, nbc8xlwnyc2:cc(6l -h 2gie.

If a pendulum clock is accurate at sea level, will it gain or lose tixrkx.cnkp) ey4f(r0 zyzp7. 2+nm up:me when taken to high altitpfe.x p npnk+y zm r2z0u4.)rx7:cyk(ude? Why?

A tire swing hanging from a branch reaches naav3b;4cis4 w early to the ground (Fig. 11–48). How could you estimate the height of the branch using only a stopw;4b sviaaw 34catch?

Why can you make water slosh back and forth in a pan only if you shak3p1g mn3+ didme the pan at d 3dgpm+n31m ia certain frequency?

Give several everyday examples+tz tg/ zxqo5sh:o5,6cxweh/ of resonance.

Is a rattle in a car ever a resonance phenomenon? Explzn ghfq- i r h4:x.9urop,r2(iain.

Is the frequency of a simple periodic wave equal to the frequency of its sopces:q5;s9 n7ankm nsl7wg 1 ko0pt82 urce? Why or gneks7als pnp1k9 8tco:20s7q ; m5n wwhy not?

Explain the differenceu 4u3*zq4sx vu between the speed of a transverse wave traveling down a cord a4suv z4u *xuq3nd the speed of a tiny piece of the cord.

Why do the strings used for the lowest-frequenx 2ycz qsfaxpi pc/j1 7b202d ub30ci.cy notes on a piano normally have wire wrapped around thef3yz sid a 0i1cx 0c bcu27bp2/2jqp.xm?

What kind of waves do you think will travel down a horizontal metasik hrl6xs( (5c.i1g c; r2b9kwf 3exil rod if you strike its end (a) vertically from above and isr3ig;6ef (x 29wx5c1sbl rickk(h . (b) horizontally parallel to its length?

Since the density of air decreases with an increase inhbb +iy- 9f-ai,7 thak temperature, but the bulk modulus B is ne 7a-h kb,iiayt+fh-b9arly independent of temperature, how would you expect the speed of sound waves in air to vary with temperature?

Give two reasons why circular waterd)hx,)a isv,r waves decrease in amplitude as they travel away from the vsr, ))ah,dix source.

Two linear waves have the same amplilwn*(ns2(w id21pk5 p rjc0wetude and speed, and otherwise are identical, except one has hrl20pn*i ww 12en (wsc(jkp5dalf the wavelength of the other. Which transmits more energy? By what factor?

When a sinusoidal wave crosses the boundary between twoanuwtxkzv gvyqnhl:4b-* 6 1iug 3 0// sections of cord as in Fig. 11–33, the frequency does not change (although the wavellgn z6vn3gui y:1xh04vaqtu /b/*kw -ength and velocity do change). Explain why.

If a string is vibratin2wa9e aw+gt+f rl 2co2/fvbl 0dsjam( e+m24wg in three segments, are there any places you could touch it with a knife blade without distmfam cww2jgov2+r b 0+/(e2 fws24a+ae 9dtl lurbing the motion?

When a standing wave exists on v+ u j0xqk/omlncj81 8a string, the vibrations of incident and reflected waves cancel at the nodes. Does this mean that uco0n 8xljj1mv8q /+kenergy was destroyed? Explain.

If we knew that energy wa9t6 qgquos2 q-s being transmitted from one place to another, how might we determine whether the energy was being carried by particles (material bodies) o-69tqo gs2uqq r by waves?

  If a particle undergoes SHM with amplitude 0.18 m, what is the total distance :z sb-twznt(: it( z wb-tznt::s travels in one period?    m

  An elastic cord is 65 cm long when a weight of 75 N hangs from it zawa2/p kp gwrna)910 983 pwo 5oslqzbut is 85 cm long when a weight of 180 N hangs from it. What is the “spring” constant k of ta k3 pw28p9aasr)l/op19w o nzwzg 05qhis elastic cord?    N/m

  The springs of a 1500-kg car compress 5.0 mm when its 68-kg driver gets into8feo+jbxth-kqp3p d6pcvz* e -l59a6 the driver’s seat. If the car goes over a bump, what will be tpp8cf-5 a3bo e+xl6*dt -6jze9vh kqphe frequency of vibrations?$\approx$    Hz

  A fisherman’s scale stretches 3.6 cm when a 2.7-kg fish hangs from it. (vvzev oohs/ vkn(,y. /l m3q/1a) What is the spri3ylk.v1v /o /zvs/(envq m h,ong stiffness constant $\approx$    N/m
(b) what will be the amplitude and frequency of vibration if the fish is pulled down 2.5 cm more and released so that it vibrates up and down? $\approx$    Hz (round to one decimal place)

  An elastic cord vibrates with a frequency of 3.0 Hz when a mass of 0qck ;t- kiu3+f.60 kg is hung from it. What is its frequency if only 0.38 kg hangs from c 3ftqiuk-+k;it?    Hz

Construct a Table indicating the position x of t)umpbirmoq9t u 6x/ 9+he mass in Fig. 11–2 at times t = bt9+)x9/ mq6orupim u 0,t = $\frac{1}{4}T$,$\frac{1}{2}T$,$\frac{3}{4}T$,T and $\frac{5}{4}T$,where T is the period of oscillation. On a graph of x vs. t, plot these six points. Now connect these points with a smooth curve. Based on these simple considerations, does your curve resemble that of a cosine or sine wave (Fig. 11–8a or 11–9)?

  A small fly of mass 0.25 g is l, b0ojtp7)ow5knf,3 +o qtqkcaught in a spider’s web. The web vibrates predominately with a frequency of ot, l73nj5qbt kfoqo)k,0w p+4.0 Hz.
(a) What is the value of the effective spring stiffness constant k for the web?$\approx$    N/m (round to two decimal place)
(b) At what frequency would you expect the web to vibrate if an insect of mass 0.50 g were trapped?    Hz

  A mass m at the end fghkfko 0 5d2 rgzo(h 4*,bj*zof a spring vibrates with a frequency of 0.88 Hz. When an additional 680z,ozkr(4h dfgg *f5*j0bh k2o-g mass is added to m, the frequency is 0.60 Hz. What is the value of m?    kg

  A 0.60-kg mass at the end of a spring vibrates 3.3l. ypk5cogi 2e2 j7oyo 0hag:0 times per second with an amplitude of 0.13 m. Dete: 7peogial o k2.yyjo gch0523rmine
(a) the velocity when it passes the equilibrium point,$\approx$    m/s(round to one decimal place)
(b) the velocity when it is 0.10 m from equilibrium, $\approx$ ±    m/s(round to one decimal place)
(c) the total energy of the system,$\approx$    J(round to one decimal place)
(d) the equation describing the motion of the mass, assuming that x was a maximum at x = (0.13π)cos(  πt)

  At what displacement fry lwij7.vq,-;sh x jyd0g0rl5om equilibrium is the speed of a SHO half the maximum valsyj7 .g,rxwd-q;l0i5l0vyh j ue?$\approx$    A

A mass attached to the end of a springx,pm, 5/lo fvekzb08m is stretched a distance $x_0$ from equilibrium and released. At what distance from equilibrium will it have acceleration equal to half its maximum acceleration?

  A mass of 2.62 kg stretches a vert1sc-ybi t/klc*h+ :nsical spring 0.315 m. If the spring is stretched an additional 0.130 m and released, how longcbs-y in/sl1*hc: +kt does it take to reach the (new) equilibrium position again?    s

  An object with mass 3.0 kg is attached to a spring with spring stiffness confh29u k:y0 jyystant k = 280 N/m and is executing simple harmonic motion. When the object is 0.020 m from its equ:j fyh9y0k2y uilibrium position, it is moving with a speed of 0.55 m/s
(a) Calculate the amplitude of the motion.    m
(b) Calculate the maximum velocity attained by the object.$\approx$    m/s (round to two decimal place)[Hint: Use conservation of energy.]

  It takes a force of 80o d-h7vsea+ g3l*)pdg.c jue3.0 N to compress the spring of a toy popgun 0.200 m to “load” a 0.180-kg ball. gsjlcd+7.d h-p a3e)v*go u3eWith what speed will the ball leave the gun?    m/s

  A mass sitting on a horizontal, frictionless surface is attac3 8u/paqx+ z q;ecxlbycp* 7a;hed to one end of a spring; the other end is fixed to a wall. 3.0 J of work is required to compress the spring by 0.12 m. If the mass is released from rest with the spring compressed, the mass experiences a maximum acqyb*8x;cq l ueaax7/3 ;pcz+pceleration of 15 $m/s^2$ Find the value of
(a) the spring stiffness constant    N/m
(b) the mass.$\approx$    kg

  A 0.60-kg mass vibrates according to the equation x = 0.45cos6.4t where x is iox z21q jo)nau)qf6+ cn meteuzfxacj n)q 1)o6o+q2rs and t is in seconds. Determine
(a) the amplitude,    m
(b) the frequency, $\approx$    J(round to two decimal place)
(c) the total energy, $\approx$    J(round to one decimal place)
(d) the kinetic energy and potential energies when $\approx$    J(round to one decimal place)

  At what displacement from equilibrium is the energy of a SHO half KE andy 3/y xyj*jzv9 half PE? ±y/j* xv jzyy39    A

  If one vibration has 7q1yaz xa d);2s.0 times the energy of a second, but their frequencies and masses are the same1)sx ;qazady 2, what is the ratio of their amplitudes?   

  A 2.00-kg pumpkin oscillates from a vertically hanging e(h my :q*sad9o dlz:4(exh2 jlight spring once every 0.65 s: h(9ledq( hze4oyjsmadx*2:.
(a) Write down the equation giving the pumpkin’s position y (+ upward) as a function of time t, assuming it started by being compressed 18 cm from the equilibrium position (where ), and released. y = (   m)cos($\frac{2\pi t}{0.65s}$)
(b) How long will it take to get to the equilibrium position for the first time?    s
(c) What will be the pumpkin’s maximum speed?    m/s
(d) What will be its maximum acceleration, and where will that first be attained?   $m/s^2 $

A block of mass m is supported bi 5d/uva. d gvu.fj8n3n*bp *sy two identical parallel vertical springs, each with spring stiffness constant k (Fig. 11–49). What will be the frequency of viba ujdsn*f.3 u5 8ip/dg.bvnv *ration?

  A 300-g mass vibrates according to the equation x l3m0oxkwq jtt945e 0 y:ysss2= 0.38sin6.5t where x is in meters and t is in seconds. Detey59:ys2t4mqkej wl0so03xt srmine
(a) the amplitude,    m
(b) the frequency,    Hz
(c) the period,    s
(d) the total energy, $\approx$    J
(e) the KE and PE when x is 9.0 cm.$E_{potential}$ =    J $E_{kinetic}$ =    J($E_{kinetic}$round to two decimal place)
(f) Draw a careful graph of x vs. t showing the correct amplitude and period.

  Figure 11–50 shows two examples of SHM, labeled A and B. Fo)/oqfnm2i y: jr each, what is
(a) the amplitude,$A_A$ =    m $A_B$ =    m
(b) the frequency, $f_A$ =    Hz $f_B$ =    Hz
(c) the period? $T_A$ =    s $T_B$ =    s
(d) Write the equations for both A and B in the form of a sine or cosine.$x_A$ = (    m) sin($\frac{\pi}{2}t$) $x_B$ = (    m)cos ($\pi$t)

  At a 755-g mass at rest on the end of a horizontal spring (k = 12o jb5byeaj*: *5:xgxd4 N/m) is struck by x55e *yjdbg :axo*b:ja hammer, which gives the mass an initial speed of 2.96m/s Determine
(a) the period and frequency of the motion,    s    Hz
(b) the amplitude,    m
(c) the maximum acceleration,    $m/s^2$
(d) the position as a function of time, x = (    m)sin(4.08$\pi$t)
(e) the total energy.    J

A vertical spring with spring stiffness constant 30t+y,3w1tm29 gp yix jj5 N/m vibrates with an amplitude of 28.0 cm when 0.260 kg hangs from it. The mass passes through the equilibrium point (x=0) with p1y ygwtx 9tj 23m+ipj,ositive velocity at
(a) What equation describes this motion as a function of time?
(b) At what times will the spring have its maximum and minimum extensions?

A mass m is connected to twou*lhlp.v*f6dgpv :. x springs, with spring stiffness constants $k_1$ and $k_2$ as shown in Fig. 11–51. Ignore friction. Show that the period is given by
T=2 $\pi \sqrt{\frac{m}{k_{1}+k_{2}}}$ .

  A 25.0-g bullet strikes a 0.600-kg block attached to 3bo*xbesz4u7l v3 *bq a fixed horizontal spring whose spring stif73zbs3oqe *b lb*vx4u fness constant is $ 7.7\times10^{3 }$ N/m The block is set into vibration with an amplitude of 21.5 cm. What was the speed of the bullet before impact if the bullet and block move together after impact?    m/s

  A bungee jumper with mass 65.0 kbmzv k dt30s( i-/ (x+.vxq1xojcrw.zg2doq5 g jumps from a high bridge. After reaching his lowest point, he oscillates up (z3 kmxv /d20qr-d. 1 xwsb (+.5zxqgvji ocotand down, hitting a low point eight more times in 38.0 s. He finally comes to rest 25.0 m below the level of the bridge. Calculate the spring stiffness constant and the unstretched length of the bungee cord.
spring stiffness constant$\approx$    N/m(Round to the nearest integer)
unstretched length of the bungee cord    m

  A pendulum makes 36 vibratio g)dc 5m5b8tvmgq12ykns in exactly 60 s. What is its
(a) period,    s/cycle
(b) frequency?   Hz

  How long must a simple pendulum be if it is to make exactly one swing per second5wsp3ta-s7 bw0tth x: xic8t .? (That is, one complete vibration taket0:w xi5sp hwb st7txtc.a3 8-s exactly 2.0 s.)    m

  A pendulum has a period of 0.80 s on Earth. What is its period on Mars, whe,:nn e uwjs8*,b:kldpi l *n/1)rbu rpre the acceleration of gravity is about 0.37 that on p*/jlln*b k,,brn:1:w ui8 rp duens)Earth?    s

  What is the period of a sii4rno60 66tz eqw+wdpmple pendulum 80 cm long
(a) on the Earth,    s
(b) when it is in a freely falling elevator?

  The length of a simple pendulum is 0.760 m, the pendulo1e s3he k0w- ey+ nvcit5n0ygs-0-wvum bob has a mass of 365 grams, and it is rs10ykv +w0-eshn 0ti-3ge o wcn-y5veeleased at an angle of 12.0 $^{\circ} $ to the vertical.
(a) With what frequency does it vibrate? Assume SHM. $\approx$    Hz(round to three decimal place)
(b) What is the pendulum bob’s speed when it passes through the lowest point of the swing?    m/s
(c) What is the total energy stored in this oscillation, assuming no losses?    J

  Your grandfather clock’s pendulum has a length of 0.99308u** cbr4z5 fyj*fhr v m. If the clock loses half a minute per day, 5bvu8jy*f*r c*h4zrf how should you adjust the length of the pendulum? shortened by    mm.

Derive a formula for the maximumbexl +jzxz0,j 5mc,4wlp .w9 c speed $v_{max}$ of a simple pendulum bob in terms of g, the length L, and the angle of swing $\theta_{0}$

  A clock pendulum oscillates at a frequency of 2.5 Hz. At it is wyd5m3xy- pcg.r9 sh3x0m8 nwreleased from rest starting at an angle 3 hwpcxnrs89 05wxy3mmdy.-g of 15 $^{\circ} $ to the vertical. Ignoring friction, what will be the position (angle) of the pendulum at
(a) t = 0.25 s ,    $^{\circ} $
(b) t = 1.60 s ,    $^{\circ} $
(c) t = 500 s ?    $^{\circ} $
[Hint: Do not confuse the angle of swing $\theta$ of the pendulum with the angle that appears as the argument of the cosine.]

  A fisherman notices thaowlr oo 21 dsyc1n7yhi(qq7:3t wave crests pass the bow of his anchored boat every 3.0 s. He measures the distance between two crests to be 6.5 m. How fast ar yio qd3yhw1(oo 7cnl :s217qre the waves traveling?    m/s

  A sound wave in air 1hx-p(w1q )nprn6gi zevk+6ehas a frequency of 262 Hz and travels with a speed of 343 m/s How far apart (6p hxrw -nzqve+pg ik6)n11e are the wave crests (compressions)?    m

  (a) AM radio signals have frequencies between 551c4t8yycski ,0 kHz and 1600 kHz (kilohertz) and travel wkys i ,c4tc1y8ith a speed of $ 3\times10^{8 }$ m/s What are the wavelengths of these signals?    m to    m(b) On FM, the frequencies range from 88.0 MHz to 108 MHz (megahertz) and travel at the same speed; what are their wavelengths?   m to    m

  Calculate the speed of pvst *581l fl ao2.u,t7yiyzylongitudinal waves in
(a) water,    m/s
(b) granite,    m/s
(c) steel.    m/s

  Two solid rods have the same elastic modulus, but t98bm /t ac6tj zr5 3;-yz(issobhsw ,one is twice as dense as the other. In which rod will the speed of longitudinal waves be greh9m rbzwz -;t o(,s8y5/s63 icbjtstaater, and by what factor?   

  A cord of mass 0.65 kg i*2ib1 k; idisw/ h3lphs stretched between two supports 28 m apart. If the tension in t3sw/dk 2i ;lihh1pib *he cord is 150 N, how long will it take a pulse to travel from one support to the other?    s

  A ski gondola is connected to the top of a hill by a steel cable oky /yd-+s,mz wf length 620 m and diameter 1.5 cm. As the gondola comes to the end of its run, it bumps into the terminal ayd k/ z-,+swmynd sends a wave pulse along the cable. It is observed that it took 16 s for the pulse to return.
(a) What is the speed of the pulse? (Round to the nearest integer)$\approx$    m/s
(b) What is the tension in the cable?    N

  A sailor strikes the side of his ship juz,t c:mmw,u:azj9e f2kzq /epl/ w(r+st below the surface of the sea. He hears the echo of the wave reflectez u9/c + /aw,,:e(ptqlzm:ezm2 fkj wrd from the ocean floor directly below 3.0 s later. How deep is the ocean at this point?    m

  P and S waves from an earthquake travel at differen ob l.*p5b,:zgpw , msvrj.)pdt speeds, and this difference helps in locating the earthquake5gos)pl : pr,wbpdm,b.zjv.* “epicenter” (where the disturbance took place).
(a) Assuming typical speeds of 8.5km/s and 5.5km/s for P and S waves, respectively, how far away did the earthquake occur if a particular seismic station detects the arrival of these two types of waves 2.0 min apart?    km
(b) Is one seismic station sufficient to determine the position of the epicenter? Explain.

  An earthquake-produced surface wave can be approximated by 4+ )atcj+nxj)ikc daszd/ +t(a sinusoidal transverse wave. Assuming a frequency of 0.50 Hz (typical of earthquakes, which actually include a mixture of frequencies), what amplitude is needed so that objects begin to leave contact t+sdi4 cja)(nd z+c j/txk )a+with the ground? [Hint: Set the acceleration a>g]    m

  What is the ratio of (a) the intensities, and (b) the amu*zng.zl-o5du( pz /f-b (qj pplitudes, of an earthquake P wave passing through the Earth and detected at two points 10 km and 20 km from jl(fzonu(qp.pz/-5 ug- z db*the source?(a)    (b)   

  The intensity of an earthquaxt /tcw*eut:v7iw4 cz ,0w1kfke wave passing through the Earth is measured to be uxtc ew0/w 7 wz:4ivft1c *,tk$ 2\times10^{6 }J/m^2$ at a distance of 48 km from the source.
(a) What was its intensity when it passed a point only 1.0 km from the source?(round to one decimal place)$\approx$    $ \times10^{9 }W/m^2$
(b) At what rate did energy pass through an area of $5m^2$ at 1.0 km?    $ \times10^{10 }W$

  Two earthquake waves oc +qjs4arz*e0 giy ig;l1cf 75+ zzlg-f the same frequency travel through the same portion of the Earth, but one is carrying twice the energy. What is the ratio of the amplitudes of the twgz1*-0;ei rg7z5g s flczj yl 4ac+qi+o waves?   

  Two waves traveling along a stretched string have the wmb uhb:z(q-3x;3v h9wo ol5tsame frequency, but one transports three times the power of the other. What is the ratio35u wh:-vo hblw 93oz(btxmq ; of the amplitudes of the two waves?   

  A bug on the surface of a pond is observed to move up and dow,*hb,z )vdoye m))o qan a total vertical distance of 6.0 cm, from the lowest to the highest point, as a wave passes. If the ripples decrease to 4.5 cm, by what facoz) o),d,hm by)v*eqator does the bug’s maximum KE change?   

The two pulses shown in Fig. 11–52 are moving toward each oth3kawh hn2,np29o u4b ser. (a) Sketch the shape of the string at the moment they directly overlap. (b) Sketch the shape of the string a 2kawnn349 osb,u2hhp few moments later. (c) In Fig. 11–36a, at the moment the pulses pass each other, the string is straight. What has happened to the energy at this moment?

  If a violin string vibrates at 440 Hz as its fundamental frequency, what are kx;bosc w;too d ;p6f7k2eq4 :the frequo; ;6wtd:skox ;e 7bf 4ocpq2kencies of the first four harmonics?$f_1$ =    Hz $f_2$=    Hz $f_3 $ =    Hz $f_4 $=    Hz

  A violin string vibrates at 294 Hz wog5kf14*bd)-y.q y1g qbeurawf;k j*hen unfingered. At what frequency will it vibrate if it is fingered one-third of the way down from the end? (That is, only two-thirds of the string v*w4.dbaqyb k ;o5-*k y fjre1f ggq)1uibrates as a standing wave.)    Hz

  A particular string resonates in four loops at a frequency of 280 Hz. N,mx.y7kr qoe fxgf0 /d 46er6vame at least three other frequencies at whic.qm rfx dvx6 g6 r7e/ke4o,0fyh it will resonate. $f_1$ =    Hz $f_2$ =    Hz $f_3 $ =    Hz $f_5 $ =    Hz

  The velocity of waves on a string is 92m/s If the frequency of standinga0d(po ;8ankdgr3 pt5 h4kp 0g waves is 475 Hz, how far apart are two adjacent nodes;5aor k 4n3d0hg (pgap08dkpt ?   m

  If two successive overtones of a ve3g plh*6:rm-fm w2-ccnf/) yt at-egibrating string are 280 Hz and 350 Hz, what is the frequency of the fu emh:ygw recn-a l3- -*g2m)6f pttc/fndamental?    Hz

  A guitar string is 90 cm 5xi gm(n3lqkz20ae:ys5mf(z,- xwf mlong and has a mass of 3.6 g. The distance from the bridge to the support post is L = 62cm and the string is under a tension of 520 N. What are the frequencies of the fundamental and first twomal,q:zxgz35 (m5mif 20xf( ekwsyn- overtones?    Hz

  A particular guitar strwnkm -f3hds s5(zz a/7ing is supposed to vibrate at 200 Hz, but it is measured to vibrate at 205 Hz. By what percent shoulzfhn(7/ -w5mz dkass3 d the tension in the string be changed to correct the frequency?So the three frequencies are,    Hz    Hz    Hz to 2 significant figures.

  A particular guitar string is supposed tq )rcm0sg8l)ml()aeucn e + -so vibrate at 200 Hz, but it is measured to vibrate at 205 Hz. By what percent should the tension in the string be changed to correct the frequency? decreased byu0s(m+)qn relce-) csag )l8m    %

  One end of a horizontal string is attached g-09gi4h4 s5hoa,mdrpe o iq 9 gf:)rkto a small-amplitude mechanical 60-Hz vibrator. The string’s mass per umsdr -pro ohe94ki9)q:gg,5g0f 4 hainit length is $3.9 \times10^{ -4}$ kg/m The string passes over a pulley, a distance L = 1.50 m away, and weights are hung from this end, Fig. 11–53. What mass m must be hung from this end of the string to produce
(a) one loop,(round to one decimal place) $\approx$    kg
(b) two loops,    kg
(c) five loops of a standing wave? Assume the string at the vibrator is a node, which is nearly true.   kg

  In Problem 59, the length of the string may be adjusted9bd g u(ehht.3bi31lq by moving the pulley. If the hanging mass m is fixed at 0.080 kg, how many different standing wave patterns may be achieved by varying L betweenehu t3.glqh 139d i(bb 10 cm and 1.5 m?   

  When you slosh the water back and forth in a tub at ju1f29gc m6u)uh/gdj l ust the right frequency, the water alternately rises and falls at each end, remaining relatively calm at the center. Suppose the frequency to produce such a standing wave in a 65-cm-wide tub is 0.85 /u6g1lju9d) 2fhg ucmHz. What is the speed of the water wave?    m/s

  An earthquake P wave traveling9pou y.f p.isnw 9j+m0 at k.0 km/s strikes a boundary within the Earth betweepo.y9p0+u. fns mjiw9n two kinds of material. If it approaches the boundary at an incident angle of 47 $^{\circ} $ and the angle of refraction is 35 $^{\circ} $, what is the speed in the second medium?    km/s

  Water waves approach an underwa tw;jakqj9p ; 5zm3rdjqc0 -8hter “shelf” where the velocity changes from 2.8 m/s to 2.1 m/s If the incident wapcmj9 jj08 ; qa-krzthw;dq53ve crests make a 34$^{\circ} $ angle with the shelf, what will be the angle of refraction?    $^{\circ} $

  A sound wave is traveling in warqor/8 qem io.o.lmh,(m air when it hits a layer of cold, dense air. If the sound wave hits the cold air interface at o 8qmem,(l.oio.r /hq an angle of 25$^{\circ} $, what is the angle of refraction? Assume that the cold air temperature is $ - 10 ^{\circ} C$. and the warm air temperature is $ + 10 ^{\circ} C$. The speed of sound as a function of temperature can be approximated by v=(331+0.60T) m/s where T is in $ -10 ^{\circ} C$.    $^{\circ} $

  A longitudinal earthquake wave strikes a bo 8u:qwx0e.e2ny e *sasundary between two types of rock at a0: 2*e qywn8ssaxu e.e 38 $^{\circ} $ angle. As the wave crosses the boundary, the specific gravity of the rock changes from 3.6 to 2.8. Assuming that the elastic modulus is the same for both types of rock, determine the angle of refraction.    $^{\circ} $

  A satellite dish is about 0.5 m in diameter. According to th bi) eun,5gbk3e user’s manual, the dish has to be pointed in the direction of the satellite, but an error of aboib)nbk u3 5g,eut 2 $^{\circ} $ is allowed without loss of reception. Estimate the wavelength of the electromagnetic waves received by the dish.    m

  A tsunami of wavelength 250 km and velocity 750 km/h travels across the Pacific by;2eoa- f) 5b dh/llsOcean. As it approaches Hawaii, people observe an unusual decrease of sea level in the harbors. Approximately how much time do they have to run to safety? (Ibo5;y-fl)es2l/b ad hn the absence of knowledge and warning, people have died during tsunamis, some of them attracted to the shore to see stranded fishes and boats.)    min

  An energy-absorbing cax68ru,adv lk6 r bumper has a spring stiffness constant of 550 kN/m Find the maximum compression of the bumper if the car, with mass 1500 kg, collides with a wall at a speed of 2.2 m/s (approximatexk8rvl6d6 a, uly 5 mi/h). [Hint: Use conservation of energy.]    m

  A 65-kg person jumps from a window to a fire net 18 m below, which stretches thge5cxbke0g:y3 b+6 bot nwdxxht--g9 h+ +u)ye net 1.1 m. Assume t9g+-bgyd: x5nx+6kg+ e h 3 u)we x0ot-hcbbtyhat the net behaves like a simple spring, and
(a) calculate how much it would stretch if the same person were lying in it.    m
(b) How much would it stretch if the person jumped from 35 m?    m

  A mass m is gently placed on the end of a freely hanging sprinzzmur zkn 616q,7 x0udg. The mass then falls 33 cm before it stops and begins to rise. What is the freqn06rqm6,ux kud z1z7z uency of the oscillation?    Hz

  A 950-kg car strikes a huge spring at a spe-ge5sg q )g.v9o/ h vyb,8pc4kssmw1hed of 22 m/s (Fig. 11–54), compressing the spring 5.0 m. (a) What is the springs-s g 4 mh,p/15)hgvqgcwyo.e9 kvs8b stiffness constant of the spring?    N/m (b) How long is the car in contact with the spring before it bounces off in the opposite direction?    s

  When you walk with a cup of coffee (diameter 8 cm) at just thq1f6 cat/,z2 hm/a xfz*v ,njje right pace of about 1 hz*,tzfaafcj,xv mq/ n1/j 62 step per second, the coffee sloshes more and more until eventually it starts to spill over the top (Fig. 11–55). Estimate the speed of waves in the coffee.    cm/s

  The ripples in a certain groove 10.8 cm from the center 6dq ;e noo,kxm+ 6v)mqof a 33-rpm phonograph record have a wavelength of 1.70 mm. Whm6 nv6x) q,k oqo;edm+at will be the frequency of the sound emitted?    Hz

  A 2.00-kg mass vibrates according to th4x jg,pos b6n9e equation x=0.650cos7.40t where x is in meters and t in seconds. Do6 xg4 npb,s9jetermine
(a) the amplitude,    m
(b) the frequency,(round to two decimal place)$\approx$    Hz
(c) the total energy,(round to one decimal place)$\approx$    J
(d) the kinetic energy and potential energy when x=0.260 m.(round to one decimal place) PE$\approx$    J KE =    J

  A simple pendulum oscillates with frequency f. What is its frequew 7r8rg -0uf- w:cjrn ps)j1nzsk59nnncy if it accelerates at 0.50g (a) upnnf s )jcww-zg- 9:jun8s50pr7n k1rrward,    f
(b) downward?    f

  A 220-kg wooden raft floats on a lake. When a 75-kg man3t3pvu3 vqhja9qr:ji *5* p qj stands on the raft, it sinks 4.0 cm deeper into the water. When he steps off, th qj3i5:p3 * qvth3*jj9uqvap re raft vibrates for a while.
(a) What is the frequency of vibration?(round to one decimal place) $\approx$    Hz
(b) What is the total energy of vibration (ignoring damping)?(Round to the nearest intege) $\approx$    J

  Two strings on a musicawzs a5jlpxr06 pd 3:u bw17rabo.7hr:l instrument are tuned to play at 392 Hz (G) and 440 Hz (Aljwu1raoda6 37 bp0z5 :phr 7xw:rs.b).
(a) What are the frequencies of the first two overtones for each string? G : $f_2$=    Hz $f_3$=    Hz A : $f_2$=    Hz $f_3$=    Hz
(b) If the two strings have the same length and are under the same tension, what is the ratio of their masses ($m_{\mathrm{G}}/m_{\mathrm{A}}$)   
(c) If the strings instead have the same mass per unit length and are under the same tension, what is the ratio of their lengths($L_{\mathrm{G}}/L_{\mathrm{A}}$)   
(d) If their masses and lengths are the same, what must be the ratio of the tensions in the two strings?   

Consider a sine wave traveling down the stretched two-part c5q gzvh9:qy)gc( e 0djord of Fig. 11–33. Determine a formula (a) for the ratio of the speeds of the wave in the heavy section versus that in the lighter section, and (b) for the ratio of the wavelengths in the q(y5 dgevq:zg )9h0cjtwo sections. (The frequency is the same in both sections. Why?) (c) Is the wavelength greater in the heavier section of cord or the lighter?

  A tuning fork vibrates at a frequency of 264 Hz, and ofrz-pnek24e jq 8-xm2z 3v8kgq(o,p the tip of each prong moves 1.8 mm to e(q4xz8eje82pg,z q-k-mkvp 3o n f2orither side of center. Calculate
(a) the maximum speed    m/s
(b) the maximum acceleration of the tip of a prong.    $m/s^2$

A diving board oscillates with sej(*,)dpomp knf )v mxo; g2p0imple harmonic motion of frequency 1.5 cycles per second. What is the maximum amplitude with which the end of the board can vibre;(vnm)dppm0k2 fjxo, g *)opate in order that a pebble placed there (Fig. 11–56) will not lose contact with the board during the oscillation?

A string can have a “free” end if that en6i;uq fry8n7mfn dx a5 7lv+cg69 x:ypd is attached to a ring that can slide without friction on a vertical pole (Fig. 11–57). Determine the wavelengths r:q89p uifn67 l5a fcy+7mngvxxy6d; of the resonant vibrations of such a string with one end fixed and the other free.

  A “seconds” pendulum has a period of exactlcd6he;zn08(fj/x s: 7q gkfaiy 2.000 s — each one-way swing takes 1.000 s7q6:hjd/kn ;cfgafz8i( ex 0s.
(a) What is the length of a seconds pendulum in Austin, Texas, where $g=9.793\mathrm{m/s}^{2}$    m
(b) If the pendulum is moved to Paris, where $g=9.809\mathrm{m/s}^{2}$ by how many millimeters must we lengthen the pendulum?    mm
(c) What would be the length of a seconds pendulum on the Moon, where $g=1.62\mathrm{m/s}^{2}$    m

A mass hanging from a spring can oscillate in the vertical direction or can sb8ouqhrn5, *u9ml+ + :jj xpsewing as a pendulum of small amplitude, but not both at the same time. Which one is longer, the period of the vertical oscillations or the period of the horizontal swings, and by what amounthb9n: 5 lo xr8m++jp* uuqesj,? [Hint: Let $l_0$ be the length of the unstretched spring, and L be its length with the mass attached at rest.]

  A block with mass $M=5.0\mathrm{kg}$ rests on a frictionless table and is attached by a horizontal spring $\begin{pmatrix}k=130\mathrm{N/m}\end{pmatrix}$ to a wall. A second block, of mass $m=1.25\mathrm{kg}$ rests on top of M. The coefficient of static friction between the two blocks is 0.30. What is the maximum possible amplitude of oscillation such that m will not slip off M?    m

  A 10.0-m-long wire of mass 123 g is stretched under a tension of (btt oehh v8x2(v.b5gbx1u )n 255 N. A pulse is generated at one end, and 20.0 ms later a second pulse is generated at the opposite env (25v bb)th(n .u1exoxhtg8bd. Where will the two pulses first meet?    m

A block of mass M is suspended from a ceiling by a spring with spring stiffnelllv4: bv f,6wss constant k. A penny of mass m is placed on top of the block. What is the maximum amplitude of os, v6:llv4blw fcillations that will allow the penny to just stay on top of the block? (Assume $m \ll M$)

  A crane has hoisted a 1200-kg car at the junkyard. The steel crane cable +alz v-66ii2gofh( xv is 22 m long and has a diameter of 6.4 mm. Axlv f i6v2zoh-g6(i+a breeze starts the car bouncing at the end of the cable. What is the period of the bouncing? [Hint: Refer to Table 9–1.]    s

  A block of jello rests on a plate as shown in Fig. 11–58 (which also giv lsx*nb+xlpw; f b7o*.5ess r5es the dimensions of the block). You push it sideways as shown, and then you let go. The jello springs back and begins to vibrate. In analogy to a mass vibrating on a springs75p sln fbxs+o*ler*. x; b5w, estimate the frequency of this vibration, given that the shear modulus (Section 9–5) of jello is 520 $N/m^2$ and its density is 1300 $kg/m^3$    Hz