A bicycle odometer (which measures distance traveled) is attachejf0(4l sghx8qo y p2w9d near the wheel hub and is designed for 27-inch wheels. What happens if you use it g jp0(lo29qwsxhfy48 on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does 99se sdomiuq(r3/masu 3 .sva6z3 u4 pa point on the rim have radial and/or tangential acceleration? If the disk’s m s96 zdusaeq(93sr3smiov au34 p.u/angular velocity increases 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 described by a single value of the angular velocity : ann tqzz(mc3/5cq)i $\omega$ Explain.

Can a small force ever exert a greater torque th/g64k xc+ dhce0 s+fgqan 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 sit-up with your hanppz ;lka 7je;/ds behind your head than when your arms are stretched out in front of you? A diagram m lj7pz; kepa/;ay help you to answer this.

A 21-speed bicycle has seven sprockets at the rear ku oy:m03i+vjfhb6cp: 3fa1y wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a lay+ub hkm3apj:036vfcy io 1 f:rge rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend ja/(uf p6bx+ .6legk gon being able to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotgje (gkpa+bfu x .l/66ational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow beam?9rejrx1lj+rc+izx 974 4 olrn

If the net force on a system is zero, is the net torque alsf7z avc *4lenm;6nag4 o zero? If the net torque on a system is zero, is the net evn n;6mzc* a aflg474force zero?

Two inclines have the same height but make different angles8 e49n 6u 1kanjqd te):4nkbbz with the horizontal. Thn6zank)jn b4qe8e 419du kbt:e same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultake71 7 hats9nzneously start rolling (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? Which has the greater speed there? Which has the greater total kinetic z 791an7ks ethenergy at the bottom?

A sphere and a cylinder have the same radius andy,eu cw,vl4y1,aajs u;f8k j 0 the same mass. They start 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 energy at the bottom? Which4 c s0eyw,yku,jj;f lauva,81 has the greater rotational KE?

We claim that momentum and angular momentummg :3q 4wa 2o;h*aklhj are conserved. Yet most moving or rotating objects eventually sl2haw:m lq;ah34g*o j kow down and stop. Explain.

If there were a great migration of people toward the Earth’s wb0y6(j ntk9qequator, how would this affect th 0nytjk b9w6q(e length of the day?

Can the diver of Fig. 8–29 do+f)k .2 :unbakxjbwodv yt0v,m2v/.a a somersault without having any initial rotation when sh,nfy advb mobx+j.2a wk:)t vv.u/k 02e leaves the board?

The moment of inertia of a rotating solid disk abouv dr(kvgo9 ,6xt an axis through its center of mgovd (rk ,v69xass is $\frac{1}{2}WR^2$ (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 x1d ckq:1asr-rotating stool holding a 2-kg mass in each outstretched hand. 1-kqr xcd:sa1If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, oux gel/*i5vuuty0 *vy6r cp:0qh7l h. ne is hollow and the other is solid. Descygvu*u0pvl*ir:h x0 lu.yt c 6e7/q5hribe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle pointf9i48+j /e m*az)khb-et blvqs west. In what direction ise k4/*zbafb ijl9 t mq8+)v-eh 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 of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a largez24 d 1i/ htbq.rloi0j freely rotating turntable. What happens if you walk toward40r tb qh1iio2j .lzd/ the center?

A shortstop may leap into the air xec5x.kf.z. 6utqb h6p)*gb nto catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotat.6ntk eh5.b xbpc .u qfz6g*x)es. 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 conservat )4rvtt/v5f ewla*j (dion of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second pt/d4( )5 fjtwveavl*rropeller can operate to keep the helicopter stable.

  Express the following angles in radians: (x7lov8 (z0x vna) 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 because of an amazing m 4pest.w*k6k0 q q9zw4ba 1.gvu l*kzcoincidence. Calculate, using the information inside the Front Cover, the angular diameters uk4t0 46b as zg.we9wq *vk pqkl1*.mz(in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed a6 tz)1 mq+ry jm.3ismyt the Moon, 380,000 km from Earth. The beam diverges at an an3 6yym.zsq ir )tm1mj+gle $\theta$ (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 rpm. When the motobg9(*k u pg 1lqu7 wrt;ae6h ..ls4ejrr is turned off during operation, the blades slow to rest in 3.0 guhpb e4 *klrtl6. s( g9aq.;7uwj 1res. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a levwhs*dhqv;mv7( jw y ;e18pl+fel floor 3.5 m to another child. If the ball makes 15.0 revolutions, what is its diamet;f7 pl1h8ywm*w;+j hv (veqd ser?    m

  A bicycle with tires 6 :k l2mh-aex7t9n(jd n8 cm in diameter travels 8.0 km. How many revolutions do the wheels ma2d7ha-mtx 9jn:ekn ( lke?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate i,1i vj+qa6s sx *dtw54d*zzfj ts anguljts z zxqj*fv51*+ws4d6, i daar velocity 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 completeh;zm8u jir(3kqgab :1m3 ( foq revolution in 4.0 s (Fig. 8–38). (a) What is the linear sp o k ;r8m :((jmhb33gazi1fquqeed 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 velod(*qiw7c6:o s4) zgtg - r(q kkmyfm;mcity of the 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 pointd8ltu4 n-j:x+ 2zg0nxde mbk+
(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 rotate if a particle 7.0 cm from .hdv fv7d8v x3the axis of rotation is to experience an a3f.vv dhd7 8xvcceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about iwd-k .jr41d mc4e856jmy9zhw +egpv g ts center from 130 rpm to 280 rpm in 4.0 s. 1yg -d9dgjph 4rkz8emj vm5w6.c+w4e 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 radiusntgpak* 5s,1)*sn juh $R_1$ 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 xmj+ im3n v4ayupfu-ys o1r8sz6f6 v.jn() 1z the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotatiua. )mv(zf 3j1uzx+pymo4 fs1v nj6isry8-n6on to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be 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 fromz (/t6k.t:ku1;crb cs gff+z1jgd; jq rest to 15,000 rpm in 220 s. Through how many revolub:6 r cfd/ tj (qs.k;jz11g+kugcf;zttions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. *k(.y hdl 6gz/vblh9k 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 hgd,ra9v ,i 2din a whirling “humanv a,d,r2d h9ig centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching 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 diametep,)o,earj +gp ew/)szr accelerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge o)a +g/ ,ewrp)seozp,jf the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 1500 r- se; 1v2u2zlq5 pl ll;ikna*)zax.j tevolution*ke. z pzlvlts -2) aaqu5n2ij;ll1 ;xs before it comes 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 revolutions as the car reduces its speed uniformro;)5 s h .dsnu5ecg1mly from 95km/h to 45km/h The tires hav. d51;esmun)hog scr5e 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

  The tires of a car make 65 revolutions as the car c5aba( 11bclm12/d+flbdb bc*nn 9ir reduces its speed uniformly fromicd c2 nla*d 1bfm11 b5b(cr9/+ablnb 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 ,p / amsp;* ktsi2cff ,*i2fyq1nuic 9bike puts all her weight on each pedal when climbing a hill. The pedals rotate in a circle omsnf/iip 1, c;t2q29kysafc*pf ,u i* f 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 the9wkct)n9 e(ne* z1cl(g ssxc+ end of a door 74 cm wide. What is the magnitude of the torque if the force is exerts1cx( neklt)c*g99nec(+ wszed
(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 torzx6,sg gl jn-0* lqv )ptk7qz6que about the axle of the wheel shown in Fig. 8–39. Assume thatv 0x )tlggk pnzsz-q6,l6 7*qj 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 massless rod which pivq//fzzh ;./stkpt3yx vyh)p)oka i 14ots as sho ayq ;4tzzkk/h/3/ f.tpvhs) x1iy)opwn 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 system.

  The bolts on the cylinder headil,0 kwadartd3e.b+ueqw ;8. of an engine require tightening to a torque of 38 8wdi 3a.be ,etld ;.kq awu+0r$m \cdot N$ 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 0.6.p gp+,1478q j7yihru+8 welx lmqyba48 m when the axis of roti xh7a 8l+.gb+pyq4r yqpj8u 1l m,7weation is through its center.    $kg \cdot m^2$

  Calculate the moment of igi )vgibar c7urg6/i:i; s8c82wq cx1nertia of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combine8as867gcrrc :wii 1;gg)i c/bx qvui2d mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod ispl:s,+-tjpk 1d dbmr 3fs y3d; rotated in a horizontal circle of radius 1.2 m. Calc 3dm-s btf+,plsd3j;rk1dp:yulate
(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 w+(glkr-zzyf2e/i5z at constant angular speed (Fig. 8–42). The friction force between her hands zewz/(y f r5-lz2ig+k 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 objects shonan-+ x.g:st)uf1b bwwn in Fig. 8–43 aboux +1 sgfwunab-.n ):tbt
(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 7uixpf2 q 5:8 kv 3c8rp:tsicwwhose total 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 1o( jwzfpt0er 6oszt4 **sd- xrate, engineers fire four tangential rockets as shown in Fig. 8–ss0tjpr6(zdo-4*teox w 1fz*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 with a radius of 8.50 cm and a masf5cvo0*x9 (wbq 3kj pv(y2 dqos of 0.580 kg. Ca 392jk0pq*( x5q cvoovyb(w fdlculate
(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 mitd(o- b( /,da y*yktfrom 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 hand-driven merry-go-round -z6v8(i ph(uzw7 b 6ysmfx;yl 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 (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque requilx6;zh w8b(-f7zmy uis (v 6ypred 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 rpbn cnx:xt.bz 4 8ji4(pm is shut off and is eventually brought 48 .iznctxj nb4:p(b xuniformly 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. 8–45l;ut: x/ e x8cm7zew5d 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 acuv.y p2px d+.ftion of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly+d .uvyfp.2px 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 thin rod, as shown in Fig.05 yxx n(dz*mo 8–40dz*xym (5xo n6.
(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 of tw 5jqcdk vjc(s63t )sz9o 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 fr8+ykm9 0n bvvf(.hmx k8 k3cxyhy 6sd(om rest within four full turns (revolutions) and releases it at a spee3k8skv 8mn. dchh6yyfxxb0 yk(9(+ mvd 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 4b f-sa-dtx-jmc3 ,nm moment of inertia 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 torque of -z4m6 tl*r clj280 $m \cdot N$ 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 wiygw 3kw ; e6bad,c2a6prg*ur- di ;larrw);: ethout slipping down a lanep da;k6-wr *2uggb ;,;ir a6l 3d)rc eewwrya: at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth wi3g vw(- elp)ug5; 10*4jw u sgjbnsmy0x-y zpzth respect to the Sun as the sum of two terms,zwx )-gj *n03pbye- l0szy v;1m(w supgugj54
(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 an3k/fbasv:cdgi 6pi7 + d a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rateib 73pd a6fvick:/gs+ of 1.00 revolution 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 rtlip yq9lp82bv x*-i *est and rolls without slippitpl9ly28*qi vi*b p-xng 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 babxmk/qd4 kgv 0 q.6h*alanced vertically on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservatio/4q.x kd6 k*v hgmq0abn of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg balh+yjy )w*k/o0hu9 lz87vhssqb xw 8q/l rotating on the end of a thin string in a circle of radius 1.10 m at an anguxo87q9lhk)bjs+8 *shvq y/ uw/ 0hwyzlar speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular mom )-b pp2k 0x1ibxpjtk5q(sfhu :xh,v 3entum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm wi)pp- q2ku1jkbx3t vf 5xsp0:h( x,bhhen 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 is rotating agqvq a/(flv)d oy7u8 8t a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases to o(lvqdu)7gyaq/ 8 vf80.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce her moment of 4g9ka/)py mfi inertia by akay4/g9)pim f factor of about 3.5 when changing from the 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 rotation rate from an initial rate z v r1jl5dc70 6hqqan*of 1.0 rev every 2.0 s to a final jqr6 qhc1a*lnzd7v50 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 throey6ckjzu0 qa( 6cfu6. ugh 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 throk(ca6ufeuqjz06. c6y ws 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 ake633x lxh-:b1we jjy 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 th 8u* ao84l0wxavv5e mbe 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 ofe1 8rtsnv3j;y 5r0f ad inertia I is dropped onto an identical disk rotating at angular spee50;a8tjsv 3rr1 fdn eyd $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at ; 5bvjgva)+:yh964mnb ccmlv 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 rotating merryp4 c4ekw )fs+(u6l )yffgaxe :-go-round platform of radius 3.0 m and moment : e)4fu+ xg )akcs6(le pwffy4of 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-go-roundupjcq(cn36 +s6 mmz7r is rotating freely with an angular velocity of 0mm6sc (u3jzq 7+p6cnr .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 c3oi(0(z4o fnr8p vll about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be 3(l ozifv0c(84r olpn?(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),muil7 oa nk7 in 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 qakdfjw6 d,)e 6qk gid8 3ej1.q0sd2 b$ 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 platforsmqx3+w vklmw *p :-i2t-g8lr m, initially at rest, that can rotate freely without friction. The moment of inertia2 swwk 8 m-qglxml-vi:*rpt 3+ 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 diameter merrwg3f :px(lmi4y-go-round turntable that :l g4p3fxwi(mis mounted on frictionless bearings 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 enh5d9u ar ndfv +q58rj(d of 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, 5ujq(9+nr8 vhf5a rddFig. 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 mass move?

  The Moon orbits the Earth s5qm-9k bfu, ch8ua8zg ;zjd7w uch that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angulzj ; zh5 ,8cufu8wq -bgadkm97ar momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest j0z rvo5r f.sgwhof.j529g. v 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 f*s qtodfge*. k( cp23of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculatsc*o*fpetf23 d g(kq .e the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of twoi 5supo1ul-e6 solid 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 uoe i-lu165spdiameter 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 released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is ru 8g/ -jt*,k2 vp ny*cm2h,iu7bfkplthe angular speed of the 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 size of our S/uzx fez x4x3c3a2 72bq. vakxun, 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, losxb2 au37/43 ac evxzz.xkxfq2ing 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 that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use energa1zd. 8csb+ psy stored in a heavy rotating flywheel. Suppose such a car has a8dbs+. spa1z c 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 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 illustratm*crf8k:e9 m/jtg s)mes 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 (hoop) is rolling on a horiz50 ms7pllkmy8 ontal 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 length L can pivot freely (i.e., we igno5b8wah gcscu3bh75j4 re 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 57h j4s aub5c83ghb wcof gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing verticaj1,fohxte* i1-z hvz6lly on the floor, and we want to exert a horizontal force F at its axle so that it wo- zt* vez1hj1,xfi 6hill climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a turn with/t(px.g w3 ixe a radius The forces acting o./xgpwite x( 3n 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 begins whirliikt, , r0 0pdwqu4ik*xng the rock in a nearly horizontalkxu ,qkw,4*ii d0p0tr circle 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 cylinder and her armrnjqa6-t;/ h*hvo d5ts as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speedsh*or dh;/q5 -ttn6vja for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of(3v+-bh uu:bply1 kimm)h)qw two cylindrical plates, of mass mh+l)31qh (m) -u kpuiyb:vb w$M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 looped ro/0nf9zemj-*2 n 2ghfcgbt.rwugh 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 th wj.n/rt9g0m h bf*f2-ncz2gee highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assumel ddv gl99ww+4 $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions asbj k:,i 1w :i2lv3 ulgq,bw0jk the car reduces its speed uniformly from 90km/h l:vlq3jiwk b2 : ,gbwuj1,k0ito 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