A bicycle odometer (6h -weq jfdw +i/9nsd.edr/xu2*o u) pwhich measures distance traveled) is attached near the wheel hub and is designed for 27-inch whwr.e/q6uh+ p esdjx )9 o-/w*uidfnd2eels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on the rim ;oqj d,i jj6r8have radial and/or tangential acceleration? If the disk’s angular velocity incoq d,8i6j;j jrreases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be describeja.nr6b ; d 0/ zu0tefin79pf ;yo+b87yq lemsd by a single value of the angular velocity $\omega$ Explain.

Can a small force evero yn c9lw8th*go mh -bn:0-1hqpn0 m1i exert a greater torque than a larger force? Explain.

If a force $\vec{F}$ acts on an object such that its lever arm is zero, does it have any effect on the object’s motion? Explain.

Why is it more difficult to do a yv3 ib.-db./l;flw.nmc-tms h. ofo7sit-up with your hands behind your head than when your arms are oh w 7.yo;-3c-nt msv.f.b. ibmlfl d/stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprocketss1muevcd94 za /c h x9f.k47mt at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedak9m49mstu ax/cz 7v1d fh e4c.l, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lo98.7 lrqh jppyuel43dq:b tw9wer legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is aw4lrp 3q.9de8j:puhq7 l9t bydvantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow be5cdvoj-0p0 ((vk-uhqxm kvrj :39crcf6jpc 8am?

If the net force on a system (rqq6rcvwx.e60 y;h uis zero, is the net torque also zero? If the net torque xu qv(;yhrrwq.e066c on a system is zero, is the net force zero?

Two inclines have the sampl h.. 8xqvd0we height but make different angles with the horizontal. The same steel ball .ldxv0 h .qp8wis rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an :ygt85j1ywzu cx u-w1incline. One sphere has twice the radius and twice the uzyw8ug:1yt1 -wcj5 xmass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radid1mr ye+y0)n*n77 lipoo1 ea7sj n8wfus and the same mass. They start from rest at the top of an inclineoi7+ 771 e1n)dnneywasl80 pjro fym*. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and an:1v.hsx/ zszzy rh0 vvqi0) 6ngular momentum are conserved. Yet most moving or rotating objects eventually 6rn0 vzv i10x :)hzssyh/q. zvslow down and stop. Explain.

If there were a great mig9jztm 7i51l. ce)zxizration of people toward the Earth’s equator, how would this affect the le iti )z7z. jl1cmzxe59ngth of the day?

Can the diver of Fig. 8–m-z6igyd-a( .6 /in6uytamun 29 do a somersault without having any initial rotation when she leavuim-y.zamgi -un(ay6 6nd/6tes the board?

The moment of inertia of a rotating solid disk about an axis through its)zm.swwv) ,-zt1 fxybt,dkl3 center of mass v13,lfkt )-z.sxd,mtw)y bz wis $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating ( hjar mekvf/ z/66jw5stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or s5/f ve /mjj6wz6 rhka(tay the same? Explain.

Two spheres look identical and have the same mass. However, one is hol:ux;gw 0ic+4am si5j(a ty g-qgtq *w7juo:e8low and the other is solid. Describe an experi07sgagiotq5:(eqgx-a* cu;:t u+ iwy8jj wm4 ment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle4 u1t93c 1uw(9h;u7d hvyvmrizq4pzl points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top uhdu z4lti(qu7v1w yz;r9c9hm3p41v of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edger*fpg 4jk) o(g of a large freely rotating turntable. What happens if you (rog)jf *gk p4walk toward the center?

A shortstop may leap into the air to catch a ballswp ata7 :esc-g915h/grh:xujt7j 0 p and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in w75txg h cpshu:rjgj/as19t7-0 ea :pthe opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discuss wf-ofamo-q5xehnh4h;a4 am -i dr/j,d-kn+ 9b hy a helicopter must have more than one rotor (or propeller). Discuss one or n n ia45;hj ,m ad-eq-o +fdh horm9kf/-a-4bxmore ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radi bqrb37 /nm /7) rq9tffl1nlhhans: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth becaudq3iti 6ta 5io-q9c3:fvxbx 8f oxn0,se of an amazing coincidence. Calculate, using thet-06ia x5:ivcxfq d ix39t,nf 8oq3 bo information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,002j)nazfyn z51 0 km from Earth. The beam diverges at a)1z2f5n y jzann angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 r trs;(du0f*t 6k/vhi)rco ax4 pm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleotr*i)tk /cvf; dau046xrh (sration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the ball mafcc/e,e* 8ria kes 15.0 rev/ ce*fa8 ec,riolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many zh- p,/p :hrjxrevolutions do t:hr, zjpxh /p-he wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates a u3t5sn5*fg3h rs0uvmt 2500 rpm. Calculate its angular velocitys0* 535vg rfnuts3umh in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.0 s (Figmek uzt00.odf.em,sm5el3fwv8l+4m . 8–38). (a) Whdmoev l em35+us 4me. tf0mzf,8wl0. kat is the linear speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbit around the Sunl vtnqhrcs)p91 ah2+-    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speeq3-pyw2) szj:8bis6y cfy d7pd of a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) mustsc0t;a,u v1xn sb0odxk;of18 a centrifuge rotate if a particle 7.0 cm from the axis of rotation is to experience an acceler s;ansdt fv8cb o0xox, k1;1u0ation of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about7obgc 6*f.cs j 9)mco exm(3zr its center from 130 rpm to 280 rpm in 4.0 sjczx6csrg(mme* ).o93 fo7bc. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radius etimek-ky*or* mgr)o,3 8 m/v $R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard the Apollo spacecraftod ez m4*b76cg put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder wzmg bo *7dc4e6ith a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from res 6 t:hkwea8uv,t to 15,000 rpm in 220 s. Through how many wev:8uk,h6t arevolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calcufj2cc48j d 3h8a5 askpz6aq*y late
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stressesyhw1 jyn64 8l2vpih(teo7 i4q of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reachingepj46qt 4yi1yohl(i v n 78w2h its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm 0.g+0rrqlx p8jvknaz 3 *wzg 9to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in this time8jpg qz n90krx .+vr3a0g*zlw?    m

  A cooling fan is turned off when it is running at 850r+w tj*e m-/ z*w3xrxagev/min It turns 1500 revolutions before it comes to a 3tmwra zg/*ej- w xx*+stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 roi:opnp h(b) 5eo; 0ibevolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have h;bo oe5p(b i:pi)o0na diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions a3 eotuo,(z x8vs the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diame8 3 vouz(x,oteter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each pedal when climb( wf93o6 : zwkbsq*cm q0q:tyving a hill. The pedals rotate i(voc3s9 6f :tqyw0z*mq :qbwkn a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 d o)o 8mo.8y4wa(g dhyomkym*)9 .6q ts-zbbocm wide. What is the magnitude of the torque if the fory to)o.)wmbody(8 byo*zm8oqm4s a d6 hkg9.- ce is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torqu ,mgm h y/xw0xy.0+p (7o4phtqinooo2e about the axle of the wheel shown in Fig. 8–39. Assume to0 pnmgph0tyi qox h ox ymo,.4(7/+w2hat a friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are atf,4ig1jn2 l/v7o a5ji;o) knk*zrkltn -i+fktached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this systemvk1ik t2 rlf+lzn*j/)k7 fi-k o4a g n,in;5jo.

  The bolts on the cylinder head of an engine reqa8/es6.jvgt wp son-fqb )45luire tightening to a torque of 38ap jf os85nv6.ql- wbes/tg4) $m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg spher5+ :cd; fupncgfrl.i* e of radius 0.648 m when the axis of rotatcgi.lupfc d;rn+f * :5ion is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. Toyatd1 200/7 iuwa qrehe rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ige ayduqi2a0w07to/r1 nored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotaoxyj+ )+4fcca. +oua tted in a horizontal circle of radiusca)o xcatufj o.y++4+ 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rota6ofdpi*6i3x+ h j jmm-ting at constant angular speed hd3m6 o- f+ji*mx ip6j(Fig. 8–42). The friction force between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inert jo-f;/dh1v1n wh *tuzia of the array of point objects shown in Fig. 8–43 about*h - d1jnv;wf/t1h uzo
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen atoms whose total x9etyr )ah:n9 mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the correct rate, enpjk8eut jm2wx 7(rkfg 823) algineers fire four tangentxe2kwark mj8upfg j tl(2738 )ial rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder witv cau*6e z5w6o8iq- mqh a radius of 8.50 cm and a mass of 0.580 kg. Calcv w-c6aomqe5i*u q 86zulate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerating it fro:oj8ue2ic46 ku 9m gjum rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small handxsy3cal* kt6k 0f5 52kbr id9+eb+x rt-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round 5x+5l d bkbyc3ts9rax*r0i+ktk62ef is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is eve y7w:xcrks*sw45* rgljse -c/ntually brought * x/ck*r s4 swly7w:r-j5 sgceuniformly to rest by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–4ybw)02vn x8 bjxvpma h1x5 8c05 accelerates a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the actbir g 2*jl5yuao; z(gnci8s+uk 8e;u-ion of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is acceleralu ubks*cu8y;oe 5 iin a;2gr+jz8g (-ted uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long t82r qtqmy,f*7s0 e c6rn4 jknthin rod, as shown in Fig. 8–46cqf 8,24t *y7rs6tn0mqekjn r.
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consists51nh 3g wm8pwqob i6ryqr( yd3:cc(c) of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest within faxa4-vh4 *m cw6oyr,vour full turns (revolutions) and releases it at a speed of 28r,m h6 ov-vwxyaa 4c4* $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment of inehd1 r-0p +.rekk svypm7)*sf nrtia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torlgh5./ d 3egxique of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls without slipping down a lkgyss(607d lbw :hk( bdx p59sane at 3.9dx 70s6 yh:dbkw5slgs(b p(k3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with lzzqm/0 x+pi3(-auwr respect to the Sun as the sum of two x zi 3z(+w/p0-mlrauq terms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mask*i4/nus+ tl 4;niytg s of 1640 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assumls4n* yngi+k t;/4ti ue it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest andrb-+/ecr(i c h rolls without slipping down ihcb r +c/-r(ea 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

  Two masses, $m_1$ = 18 kg and $m_2$ = 26.5 kg are connected by a rope that hangs over a pulley (as in Fig. 8–47). The pulley is a uniform cylinder of radius 0.260 m and mass 7.50 kg. Initially, is on the ground and $m_2$ rests 3.00 m above the ground. If the system is now released, use conservation of energy to determine the speed of $m_2$ just before it strikes the ground. Assume the pulley is frictionless.    $m/s$

  A 2.30-m-long pole is balanced vertically on its tip. It st*x*qa49 o7)yn r qaxcgarts to fall and its lower end does not slip. What will be the speed of the upper end of the p*g qq)*ocrxay4x9a 7nole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum s 0:qlj- 3wdg33r g)e g9kqvxp/xa4c rof a 0.210-kg ball rotating on the end of a thin strx0v-: 3dra3rqw3p4jq/ l9cseg gg )xking in a circle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2 z/ hwm5dk1;qc.8-kg uniform cylindrical grinding wheel of radius 51hc; wkzqm d/18 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform 9 6z caf yjypr,f8tv2 p4.ad:zthat is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decrea2yfpza9j6r,.v:z c ftp4y8 da ses to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29nlh 98cs )kkif)xoc99) can reduce her moment of inertia by a factor of about 3.5 when changing from the nfcoh xkk8 9)i99cs)l straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotatiow++s( x3 l0 iykz;/guqg- -qsstn5bwx n rate from an initial rate of 1.0 rev every 2.0 s to a final ratxu-b5-0kgs; qwq +w i/zsts lx(n g3y+e of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis throwa6-vn+:jv mkbd ou )ih 1os83b99trsugh its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wu m1i -w 3 9avjst8bk)or9:+vsobdnh6heel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinning at 3.5vr 7ahkqu9.of lw) ;y4uk 9u:f $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of thoqh (+50c+;matox xzfe Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindriw3vo(5gmq ur3 cal disk of moment of inertia I is dropped onto an identical disk rotating at angular sp(r53oq 3umw vgeed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns9r v;/p ;qukx) q;scv wv/uas3 at 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the cente1g6ryygrod.c4 0.s mbr of a rotating merry-go-round platform of radius 3.0 m and moment oys.6 b 4dy go0r1.rmcgf inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-gokl2zux;- 5yoe b8lrt k;u+td;-round is rotating freely with an angular velocity of 0.ebtol x k;ztdu+u;ly 5;rk28-8 $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, losing about half w0jsjb5nd,4 r its mass in the process, and winding up with a radius 1.0% of its existing radius. Asr bwjd5s4j ,n0suming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds p2ny/2p)wr2:a lqgy gin excess of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass nn:qz68j;q9 :)srk;h sjx0 t,8 h gtulq 9kmnw$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, that can rotate freelyn dmsc9)m u m8 w+dn.l.ggxqb3kih3;8 without fribx3 hkguqn)9milm.gn 3.; 8+cs mwdd8ction. The moment of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diametthz5.ix )e.hb; khiijb lh8x6jg90p; er merry-go-round turntable that is mounted on frictionless bearinhxjg9j. .hl;x0hk iti8)e 65bz hip; bgs and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of ms./ xs7 fulx(the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of m(m 7ul/sxxf .sass move?

  The Moon orbits the Earth such that the same side always faces the Earth. D8ycqc604u jq netermine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Mcq j864n 0uycqoon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates 9yxu3r v zl(r*y 3.xl8g m5kgrfrom rest at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of60 cs kikv6:qf radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular:k0 isk6f qv6c acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two kj 7 ettn9eae-npb265solid cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is re pnkt79a jte2ene- b56leased from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of bgo;w) )rbnr-u /wpw 9the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the s2jj6 qayc-jjhz6rl 79ize of our Sun, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assu6clrzy79- 26qjhj ajjme that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollu r7y)wogdx /3ob3mtf0x whu1nts5d0* tion automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass ot1dof0)uhw3dmwgs3* yx / b nr0xot75f 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates aniu+dg2q i1s735qe6dm q skbw4 $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) ip4/htqc -6za ys rolling on a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and 19m sths8ck h4gb+na9length L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tibn 49gshts9a8k1 h+cmp of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has raw rptp;t4* k0cdius R. It is standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step w tt4;p0kprc *against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling wittx*7w-j:o+o zt cwu1 lh speed v=4.2m/s on a flat road is making a turn with a radiuso-:t1lo*wu+j7w xztc The forces acting on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling(kxiv n,p5f9 zy.a+/ w88f y2(ocz xm 8rcrwhv of length 1.5 m and begins whirling the rock in a nearly horizontal circle ab+ (ixhfzovvyp/,an r5f8y.2kxw(r c cz8m 9w8ove his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms a j 2uy)i(lj,pms thin rods, making reasonable estimates for the dimensions. ,p 2i(j yjlmu)Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of twoeo8a28hwhy4 4ql/s xb cylindrical plates, of mass4el4w y oa28h8b x/qsh $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the l, p(w lcr2t;npooped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume ; ,tr nlcw(2pp$r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assume 8bb i c)4ns9kt:6 fv 6j.sdfbk$r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car rede0 a3uj54kd6c. ef.q etfdf0l uces its speed uniformly from 90km/h to 60km/h The tires have a d4dda f 5c 3ke 6tlfujq00ee.f.iameter of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s