A bicycle odometer (which measures distvst/0ye x.;ctance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-in t/;c0xt ev.ysch wheels?

Suppose a disk rotates at constant 0 c5icp(bkpo d4rt/tkv,2od *angular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/otp*do kdrkc4t,v5i2( o /b0p cr tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be describegh e,u:eg 4 9zgt*5dtld by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque thsawj.b)* jbe;an 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 575*jvmn:gz8vfvaq p+r wh 7io 0bfy 2a sit-up with your hands behind your head than when your arms are str+ 8f*zvmyqiwf0arj7vp: 25o5nvghb7 etched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear whe z9cu5do-yn.ig1 w8udel 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 pedal, a small front sprocket or a large front sp8-.d5 zgc do u9y1niwurocket? Why?

Mammals that depend on being able to run fast have slender lower legs with e1uu7 oh(ze:m;cdd6t flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotationau m1dz:u6eh 7 odt(;cel dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long z( 8uhprje)i(, narrow beam?

If the net force on a system is zero, is the net tors:7pebm0fy 9 6ptm1i8z 8yej dque also zero? If the net torque on a syse1sy 9z:8pfmdet y 76im 0jb8ptem is zero, is the net force zero?

Two inclines have the same height but make different anglehai n) kya- bs9y.u) .7fuwcc:gxf5v2s with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explk afx2cy-ui h59f vs.w)yb7)n c:gau.ain.

Two solid spheres simultaneously start rr.dl (5bvs: f8qdzjhc (np06wolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Whidb:qlfv0zjs(dn5( c r 6hw8p.ch has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. They stvhl+ qnj m19bihyx:n3vm1 :9kq8n3 nwart from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic ene 1n:3 1nh8x+kqbm9m9l yniw n3vjqhv:rgy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most moving)hib y6a a1b/z or rotating objects eventually slow down and stop. Explainbaa16i zb/h )y.

If there were a great migration of people towar*vmdw.e6cm4(7h tufu 4lgim *d the Earth’s equator, how would this affect fvumc g*76e*(w4.ih lu mmd4tthe length of the day?

Can the diver of Fig. 8–29 do a somersault without having any initialsmqm5t )pfv 32 rotation when she leavf)t 2mm v35spqes the board?

The moment of inertia of a rotating solid 7sz/pnhl hz ;(disk about an axis through its center of mass is szlzh 7;(n/ hp$\frac{1}{2}WR^2$ (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 sittinj;we 1mkezofs 8ub w7 8)ug)1ns+zq,ag on a rotating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity ; e8 ks1s)+)8aunz1,eumqz j bwwgf7oincrease, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. zg+ m4c5nw2(oiv ibxspsui.f*9q( /caq5hq + However, one is hollow and th2a5iip(v9 fmqc.gzsb(wcn+/ o+x q ui h 5qs4*e other is solid. Describe an experiment to determine which is which.

The angular velocitydxd,x1nu1hw t(yv:o6) wsa1v of a wheel rotating on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describv 1ah,n61ox vxsu1 t (:y)dwdwe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a larf qm+8) t; xyp:fa(zvdge freely rotating turntable. What happt 8v(fxz; adfyq p+):mens if you walk toward the center?

A shortstop may leap into the air to catch a ball ah aom) lg+g-bdaqa(6 (nd throw it quickly. As he throws the ball, the upper part ob)omaalh6d +( -ggqa(f his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of con.5 f7*lyt/ bqwvmzw8 dq ,(0d:pnl xox)9tlrlservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second pz ld qt lpm(t,5or /vlwfnx q.* y9:)0lbdxw87ropeller can operate to keep the helicopter stable.

  Express the following w*m8p wg5/6(de7tnv4h onmh sangles in radians: (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 be4d g/dyx5g 0e lfo6vz.cause of an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters gdv54 o 60lgf/ .dxezy(in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,00u0llbgkxp2qpu f:4viwhqt .t93 l/;( 0 km from Earth. The beam diverges aqighp: v3llt/ xf42;l0u( kt.qbw p9ut an 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 ro3i0zajc ra6w0 by1+6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acc ic+o6 0b3jy0rwrza1 aeleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level fys/ 5(l0m qhhdd5bvr. ,nf er)loor 3.5 m to another child. If the ball makes 15.0 revoluq, 0b)d55dv rh/en(ry . fmshltions, what is its diameter?    m

  A bicycle with tires 68 cm in diame/ 0 ofs1*/fgjgtzizo4jw lp31ter travels 8.0 km. How many revolutions do th1 off4tg il zz1j/s*0/ gj3owpe wheels make?    $rev$

  (a) A grinding wheel 0.35 m in uoie4w,p,(l;x fbwi)diameter rotates at 2500 rpm. Calculate its angular vel),i f,ue (opi;4lwxb wocity 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 revolutioy:xuhg 51u4g zpsx.7v n in 4.0 s (Fig. 8–38). (a) p7yx.1gsu v5x4:gzh u What 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 tb kf41*-z(vd avdaend k7) rl3he Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a poinn6xsy2;c,gc* nzqr 0 ut
(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) must a centrifuge rotaq/i87zlu fmb3f g(u tr179s v8rb:)tmx 6ymd ute if a particle 7.0 cm from the axis of rotation is to experience an accel y13/bqfl:m(mzu8rx7if) m7 ugdt rstuv 986beration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center frornjljb z/ tva:2) dj(5m 130 rpm to 280 rpm in 4.0 l d2(jtj/avz) b:5rnj s. 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 radiushh-):ftc 2 qnxtc6w +s $R_1$ 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 spacecraft put themaock (wi0d c,1s,f2+mdp) homselves 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 b+dckdmp acwi2,0oh os,f1m() e thought of as a cylinder with 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 rest to 15,000 rpm in 220 s. Thr/ bfe /v(6ks1 bcxwi* fbrt-c,ough how mawbb,b/x ke-6rs*c1(c t/ff ivny revolutions did it turn in this time?    $rev$

  An automobile engine slows dow7*yij wq ml(+ey 4i;a*wdxy2w n from 4500 rpm to 1200 rpm in 2.5 s. Calculate
(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 stresses of flying highspeed jets in a whq- ahl47voy19intvp9 tg 2 k2firling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before fltgiy-29qv tao7h4k vpn 291reaching 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 rp2i 3kntdnpx/ *m to 360 rpm in 6.5 s. How far will a point on the edge of the wheel hik 3 tp/nn2*xdave traveled in this time?    m

  A cooling fan is turned off when it is ruip . ,:pxhht0wx8id0 wnning at 850rev/min It turns 1500 revolutions before it comip . 00hd:xixtwpwh,8 es to a 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 readej8k/5j-j9 u bpd2jvolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diamete p2jkjb8/ja5u de9- djr 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 myt8jqk 3 a6vl+ake 65 revolutions as the car reduces its speed uniformly from8+kya j6 lvq3t 95km/h to 45km/h The tires have a 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

  A 55-kg person riding a le1i. n,qo-lz7t )gabbike puts all her weight on each pedal when climbing a hill. The pedals rotate in a circle of rbo iq l)7ae-t.ng,1lzadius 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 cm wide. What is the)4qqiv:ova8s d 4p o2u magnitude of the torque if the force is exerteqvu )qv4 2dai:os8 p4od
(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 torque about the axle of t 7ggjdg4vls:i b1l o)9he wheel shown in Fig. 8–39. Assume thag s gi9b4l71jdov)gl:t 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 attached to the ends of a mass+b8 s.p79.2jvilyca 7ol rde- sddx,aless 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 sys,v27dro7l.8 d.dc b+lsya 9 p-eixsajtem.

  The bolts on the cylinder head of an engine require tightening to ; (hv1c sx+wpva torque of 38 vxh; +p( vscw1$m \cdot N$ 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 sphere of radius 0.648 m when theh)xya+a bv1vsy l*7t/ axis of rotation is 1vsb*ay+l/ vxt7 hy)athrough its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm ink qo99k *w14xem9zlwmau- yd* diameter. The rim and tire have a combined masd 9kq y94woewaz*mum*x1l9- ks of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram bal4jhw6s-c;l6et 3 c.b jgcz,fm l on the end of a thin, light rod is rotated in a horizontal circle of radius 1.2-.hcg, f;36 ljccwb etmzs46j 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 rotating at constant angula9w lxnb od3 2po:;qpe-r speeo:3qx -onpd p;9l2bwed (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 inertia of the array of point 7hweq8,+edc wtqbe:) objects shown in Fig. 8–43 abe+eq8qwc w:t7 ,deh)bout
(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 mass is xn(s*8ff nbw4 s;m3ff$5.3 \times10^{ -26}$ 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 satellia9rt7ij4ez de g34 nn1te spinning at the correct rate, engineers fire four tangential rockets as n3n4dez19g ie4 at7rj 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 r k56-0lp 5zb9ipro lej(ololnj0t ,0a uniform cylinder with a radius of 8.50 cm and a mass of 0.580 kg. Calc05 ,lo0(e lli6- bpon0 rjjtpolzrk59ulate
(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 from rest to , pc;v9ccr(zh 07n xam3 $rev/s$ 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 tangentt( hdcmtnf* vc(s*p )+ially on a small hand-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 is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children ( )cttn fh+(*(dspv*c meach 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 og hr9glg7ur-j/t9o, zakv s 34ff and is eventually brought sr kh9,49jv3o/rgtu-l gz 7aguniformly 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. 8–45 accelerates a 3.6-kg ba*k6s .ypoq 2lyq;oj8nll 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 action of the foz/jiib(4ryw (rearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The 4br ji/w ((ziyball is accelerated 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 consider3 wxs 4l9j,wnred a long thin rod, as shown in Fig. 8–4n 9ws3w,xjlr46.
(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 consists f n4/3 z eza)3b(goayvof 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 t 2jh:h/rx. smfhe hammer from rest within four full turns (revolutions) and releash.m xf/2h j:sres it at a speed of 28 $m/s$ 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 inertia oq )8m6skwo-f pf $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develda qbkv:0tq ei/4y+c3yshr );ops a torque 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 dow4g9f- xp zm:/6ray(og 6yew evn a lan6f yw mp:gvey6zax4 /r-o e(9ge at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of thef . 6zh;+bhyip Earth with respect to the Sun as the sum of two tp.i6h+ hf ;ybzerms,
(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 mass of 1640 kg f.;wusn/5d ;skwopki )tdf9- 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 rpikw-t;;fkw9f5 ou d d/)n. ssevolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and roll2k jv1sd;xf s;s without slipping;vskdj21 fs ;x down a 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 ssot zont t.anxwoaew/h/ 2 f-+a5)x6 +tarts to fall and its lower end doesa+not/w5 o t2xoh a+z.)-exfw/n ats6 not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a8xdk8vzcvj :9ch+ :xt 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an angular speed of 1c vkt:xz9+v8cj:xdh 80.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindr2e jrf (l *rtvj um0qq(t7w7;cical grinding wheel of rrtu(qwevcrtlj*2 0 j7q;f7m( adius 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 that i0-zfdzk a02j,z w0 f07cyjijps rotating at a rate of 1.3rev/s If he raises his arms to a horizontal positi kf,d72pjj 0ajiy0zw-zcf z 00on, Fig. 8–48, the speed of rotation decreases 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.hpdm dm 2;s1 66:cmffr 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight posi; 1c 6pmmh2s rf6dfd:mtion 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 increaseesnl49,*raqi)+wq sw di j2k7 her spin rotation rate from an initial rate of 1.0 rev every 2.0 s to a finalsi7wj2kde + wq)*4na ls9ri,q rate 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 through its center at abn6l bh d yoh17ey+(,:o) xvwp frequency of 1.5rev/s The wheel can be conside,7 np(:ohehbx )vboy 6 y1wdl+red 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 wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of c1htu h3:p i9m;el :lffs 5xx+a figure skater spinning at 3.5 $rev/s$ 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 tq;1r0m* gx xafhe 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 cylindrical disk of moment of inertia I/p-n* r5fznhsso b*h 4 is dropped onto an identical disk rotating at ang h pfnz-r*ns/45hb*o sular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns w8i0u6qlkg +axnx:m-14 lqo pat 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 center of a rotatin5mfs*1 7f-j(cersz3p pbnid8g merry-go-round platform of radius 3.0 m and moment of e f3d j*7r(zn81mf ip5cbps-sinertia 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-go-round is rotating freely with an angular ve6mux43 3men5 m1gn xw(wko8rylocity of 0.8 4 mxou3n31k(rwy5xg m8 mnw 6e$rad/s$ 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 collapsp* 3zkh kru-9 d6jdp,hes into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% ofkk*hj 3pz9pu,dd- r 6h its existing radius. Assuming 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 in excess ofllc h:8 olbxib-h)-s : 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 flwai. d1 c3r*j+j296l:q bzq)dqo tv $ 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 zn-ya6l yd7fhs mlh;;;c 0hlfe s 15y.rest, that can rotate freely without friction. The moment of inertia of the person;nfzhal0;yyy 7f. 5 dlsle6c;-1mh h s 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 -d rs s3.is3oa6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has3 i.r-dss3 oas 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 o(,e5cxo o170mz yq klzf 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 slippingme y(z,7q05o cxzk1lo. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always p-tz/m):n curnjb 4q2 faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latt4 :n2b )tuqzrm-cj pn/er case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerate* )v oupk43vits from 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 of radius 0.20 m lb 4ct.i 7t1xpllq:g: iv 5z-yacquires a rotational ra7g t v lptl-5i:4.cyl:zix1bqte of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050fae3 cyzhz 3pavpl;z7:x /(h9 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-;fvxy(pazh3a7 hpl ez/:z 3c9long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of the rear wheel 6 n2 ukg6en (hvrq1n+egcx )-w($\omega_R$) 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 size of a +dl,mh/1p;(.-afu wfyw tgfour 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? Assume that the star is a uniform sphere at all times, and thaywlf(,fwdh.g mt/+1ap f u- a;t the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a x; c6kjww-hz7fv+ut e id7*gh l9 xy79low-pollution automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 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 needingtw9g-e lv7dhy + ij*xw; fh 7czuxk769 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 illustraqzc*6sz ol7/l tes an $H_2O$ 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 (hoa3m8 eqb 4 ;bpoke+;g: jf+uqon9bn q+(ow)jxop) is 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 5ez6s.ut*wmbxayj6 *9gf0 jtr9 s4doand length 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 tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Figyb.a*j6d jx9 gwmt* 5 f 0ztse6s4uro9. 8–21g.]

A wheel of mass M has radius R. It in q) delyu;qqh,5;v f;s standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has l qe;,u ;qq5;)dfnyh vheight h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a t2oni * dt-p4.lw9 ldj:br5yb)vqaz d 8urn with a radius The forcey b4qantr8ldvpb5wd 9jo. iz 2l:*d)-s 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 of length 1.5 m and begq;w;u:m39mp z en/xgcy g1 kezwy;a,)ins whirling the rock in a nearly horizontal circle zx m)cpuznk yeqw9 ;g1g / e3ym:wa,;;above 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 6 d xg,n*pepn9cylinder and her arms as thin rods, making reasonable expn* ne9,p6dgstimates for the dimensions. 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 yb60jqvv v6.ffuef:f(ij6(l two cylindrical plates, of ma (f6j 6 ibjf0(y.fevquv6:lf vss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 rad- .xdp g0th qkdfm8*v*ius r rolls along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height g.t 80*kp*m-f dhvxqdh that the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, buel -cy+.na4 i*agq/oo t do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the carvyah (c*8/rzb reduces its speed uniformly from 90km/h azhv/ *(8bycr to 60km/h The tires have a diameter 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