A bicycle odometer (which measures e*4*ns h od+4grnjat1distance traveled) is attached near the wheel hub and is designed for 27-inch wheels.4*ngd+rt4 1h nsa *jeo What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does gli, minf26.zf1j4d; nep 1 ava point on the rim have radial and/or tangential acceleration? If the disk’s62nm1i,l g4dz v.np;f ej1ifa 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 describef :cg0myf vae *:x(h 2w9stcn4d by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Explain.)gic4evpw ea z57z /q9

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 tpr54jdf8al x)nh h+0rw y) aw.o do a sit-up with your hands behind your head than when your arms are0 p n8yl.w afah)w)rd4+ hjr5x stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets ofe *fi. 7f*jvr2zwkv1s/ c.uat 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 ped* 7cswe1 ff.vv/*jfi u k2.rzoal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on be;gvm -hv(g,za ing able to run fast have slender lower legs with flesh and muscle concentrated high, close to the bodzhg-, ;vva(m gy (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35xo 5ie) r(d-wq) carry a long, narrow beam?

If the net force on a system is zero, is the net torque also zerh;4hw r*u ae ve -(zb6(d2cw:loahp9zo? If the net torque on a system is zero, is the zah2h6(ehwla9rz:4cpv w-( ou* ;edb net force zero?

Two inclines have the same heightegbk xzd(,(iif -:k +a but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball z+-ibkeig( ,dkx :fa( at the bottom be greater? Explain.

Two solid spheres simultaneously start lzps +(-t- m4vul7wrurolling (from rest) down an incline. One sphere has twice the radius and w+sz- tvlm7- pl(u4 urtwice 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 energy at the bottom?

A sphere and a cylindejcg wk6q zipv+k+;8/pr have the same radius and the same mass. They start from rest at the top of an incligp+wv;k6/ k zj+ cqip8ne. 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 : mv )rkjg6r2yangular momentum are conserved. Yet most moving or rotating objejv)6y rr mkg2:cts eventually slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equator, how would t:eu+ sq -0kzp9lod, cthis affect the pz,su-le0ot 9d +kc:q length of the day?

Can the diver of Fig. 8–29 do a somersault without havl ix* s-w-vi.m y-c*heing any initial rotation whhxm-.sl**iw e- ci-yven she leaves the board?

The moment of inertia of a rotating solid disk about an axis through its cenj6 lzj8- mnpdx). p0iuter 0jdlij. z6p-)8pmuxnof mass 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 rotating stool holding a 2-kg masxm 7,j umq4;poo,,m0czbkp9 fs in each outstretched hand. If you suddenly drop,fp,kq;9 z m7uo,mpcmjo0bx4 the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However,k xm(d.t(ixdj; reg)4xv2 hg; one is hollow and the(x2gj4r( m;)t.evx ; kixhg dd other is solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle p b5wpkj vf/mrak500m4 oints 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 of the wheel. Is the angular speed increasing orvbja0k5r5fpmm /0k w4 decreasing?

Suppose you are standing on the edge of a lar4ryo j(i e3 x7l)wxjg3y, kys0ge freely rotating turntable. What happens if you wal ys3)y(kgw0,lji7oyr ex 43xj k toward the center?

A shortstop may leap into the air to catch a ball and throw it quickly. A1q2 eg mdkpe9q:8fc5ts he throws the ball, the upper part of his body rotates. If you look quickly you will notict1 8m qgkp2e:5dfecq9e that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discuss why 7jmx w9vs,rdiji4k ;0a helicopter must have more than one rotor (or propeller). Discuss one or more ways,0i4d7sx rvmjkwi j;9 the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: by*sh 5+ uy3 qi bl,hlk8m.,qz(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 because of an amazing coincidence. Ca.5fkq/di i iz8lculate, using the information inside the Fronti zi8kqif5 /.d 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, 380,000 km f rt lyolta9f;7 6-fgb0rom Earth. The beam diverges at anttfl go0f;796l r-ayb 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 o7 xb+ io:ul*hbd+um7qf 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angul 7b+bi*mo +:ld7u xquhar acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3 x5frt0h;*sy-t8jg zc.5 m to another child. If the ball makes 15.0 revolutions, whatgr j8fy-hc*z;t5sxt 0 is its diameter?    m

  A bicycle with tires 68 cm in dia4v; jy0 ww;eoameter travels 8.0 km. How many revolutions do the wheels ov4j ; wa;0weymake?    $rev$

  (a) A grinding wheel 0.35 m in d h.t*+/zl:t. ka ucitmp5n gs/iameter rotates at 2500 rpm. Calculate its angular velocity iuapi +m sg*t.hl :5/k.c t/zntn $rad/s$ $\omega$ =    $rad/sec$
(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 ink8n2 fxsqr59o 4.0 s (Fig. 8–38). (a) What is the linear speed ofx52 98knqso rf 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 ib3icle (6nzb:4l.fma ts orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a wfzzw00 a qbmvb6 6(+npoint
(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 centrxirg20f( bt2tifuge rotate if a particle 7.0 cm from the axis of rotation is to expri2xgb (t02tf erience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpm to 28ql1d8 zx(sd4/ i;nzdd *,qnq:l o; ggr0 rpm in 4.0 sn/(rll z:dqs; ; d qd1qn doz8,x4gi*g. 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 radiumzd4p:;sv* ok u)h qc6s $R_1$ 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, ymq ;32bi0 j(ak;gfa0psncd0cgd9, t astronauts aboard 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 rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder tmks0f; db(cijp2g yq 0d9,;g3n c0a awith 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 15,2uszl3 xib.:s000 rpm in 220 s. Through how many revolutions did it turn in this time2i sx3. szl:ub?    $rev$

  An automobile engine slows down from 4s;0d j+ (zfncc500 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 high4) wwrnx/5ambkd 23 uvspeed jets in a whirling “human centrifb) u/5n3v42xmkd arw wuge,” 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 diameter accelerat;v,rqo0 : ozcies uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a povz0i oco ;r:q,int on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turnu ocg1cw te +0ft-n)8ps 1500 revolutions before it comes to at1pcn0+ tgwc) euf- 8o 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 sp9m+ vtl 7l x-c6am fk:sxx;0tneed uniformly from 95km/h to 45km/h The tires have a diameter of 0.869ma cl0t+x l-7fstxk:n; vx m0 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 reduces its speed unifopj(/ lsj2lj4ch0xh ihl+t:a) cj( e) zrmly from 95km/h to 45+h (4c:hh/ jjzj)its(p)0 jlel2x calkm/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 4mf(vk0es 74crh0b ria bike puts all her weight on each pedal when climbing a hill. The pedals rotate i (vb40c rism74erk h0fn 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 cm wide. What is the m l9pg6;bn)c hsagnitude of the torque if th b lh;ps96ngc)e force 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 torque about the axle of the wheel shown in Fig.gy04xa z, ;hhe 8–39. Assume that a friction torque of 0x , 0yzhgae;h4.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attache .aq6etm- fjz*d 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 toje.tzf a- 6q*mrque on this system.

  The bolts on the cylinder head of an enginet yy n5ri/b3(;,tikiq require tightening to a torque of 38 yi3t5b yq(/rk tn ;ii,$m \cdot N$ 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 the0jqn8 n5oiu6* lp0ib 5yt xvx0 axis of rotation is through itbju xotyvnl685nqx i5ip * 000s center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle35f q ;x.zqf(pwc /ab :igf8vq wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. Thefb : qc 3. wfqp5g/z;i8aqf(xv mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, l*jy- ;9o brxdqh(/jp1xv i x5dight rod is rotated in a horizontal circle of radius 1.2 m. Calculaj1; q(/ dbx-r y5*d9pvjhi xxote
(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 rotati)s.dqs 0u)kp uls*6wb+gx -z7qnaqxp2c l,.nng at constant angular speed (Fig. 8–42). The friction force between heux7g-p. sk l6s )w ql2nz.ac+,) qbndqs* 0xpur 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 objects shown in F0stvaqc): ,uu ;uj s.fvhl-b9 ig. 8–43 abou-f) ,bvtv:cq lhuus.;0 sj9uat
(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 w quw 3ub8.t7dx gn-wm*hose 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, uniforfvz/b3i q)e,ljughvf +:9 b:ym cylindrical satellite spinning at the correct rate, engineers fire four tangential 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 :i)/j:vy9bfqz v uhl,3bgfe+ satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a unr; ud6ukbomm .048hij iform cylinder with a radius of 8.50 cm and a mass of 0.580 kgo0j.ikm 6u 48hm b;rdu. Calculate
(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 frmc2:zyl*4iwn om 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 han;oa53;a;ek ybkjp su (d-driven merry-go-round and is able to accelerate it from rest to a frequ(;;ab yeu jkpk;aos5 3ency 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 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 mp:c.v 5+a3lijgd:p f off and is eventually brought uniform m.gf v 3il+p:apd:c5jly 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 ball at 7 0qd:tjjqj0heqkv y8 f,u *5d+$m/s^2$ 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 forearm, whic,yinqmq6c9* ku-yk +fh rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 1k 6u,* fykiqc+ qmyn9-0 $m/s$ 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 lontu r f7j 3nru4dos+4xw9 n/o-vg thin rod, as shown in Fig. 8–46 / onowx3nr44vd +j-7u9ftusr.
(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 conyd 77mnknh3 w2 u/60kfszenh e 97i0xgsists 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 hao* q eb0;9chgdmmer from rest within four full turns (revolu0dc ;*gbh9o eqtions) and releases 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 of ahxn ,v g.h2:(exl;hu $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine deh0 fp0y5ax i5evelops 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 x0yyk c dd k90u*oev:hs:6wb/kg and radius 9.0 cm rolls without slipping down a lane :c ykbhydux6:vde9s/wo0 k*0 at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as thegr8ncorwm v rk8stskd: g9p87 :49;r t sum of two te9gc : kd8p7v9rsors:8nm t4;8gkw rr trms,
(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 and a radius of 7.50 lhr9-++6gs gn4qmk go m. How much net work is r6s q+-g +4grlkn9 gmohequired to accelerate it from rest to a rotation rate 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 fr9owymg l*lm6huo6o n v0(q 7p)om rest and rolls without somg 0nv(wmlp6 *hy 7 lq)oo9u6lipping 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 veraoj*s 08z k gi5akn;)ltically on its tip. It starts to fall and its lower;z)k j a g5*ail80nosk end does 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 a 0.210-kg ball rotating on the end of a thin sybrx ,1,sxj2 1h i4ucntring in a circle of radius 1.1, rxy14hn, ci2sbju1x0 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular -c fl -id)h;v;vo( +gpb/of0dn2ptmrmomentum of a 2.8-kg uniform cylindrical grinding wheel of radiusp /n-v )oidc2mfh(-vog b fpdlt;+r; 0 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 platf 5m;e :gn/dtraorm that is rotating at a rate of 1.3rev/s If he raises his arms to a horiz5:gm /dte;r anontal position, 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 int6g gq)lfsmc/ t3f18 j(6 uehk Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the strmfeg cu3ls6 1/jh6t)gt f8k(q aight 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 .f 8u 3xubs57vg5)rgm9xdesiinitial rate of 1.0 rev every 2.0 s to e bvifr.s3 g78u 59mxu5 xsg)da final 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 arte*a :g-fvzz8 nrh7v(ound a vertical axis through 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-kgaznv -zh*v ert8:(7fg 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 a figure skater spinnidsm9gm, 78ur mng 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 momentumhqhz u c1 fuxs ajwrc(387qu(:vg0j06 of the 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 is dropped onto an identic-z+u6bk ou jtqh3*x v(j,4ks val disk rotating at angular spq6b-vjk*3s+,jxku tov( zh4ueed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns a63 ivhw/ (lm gxouka1mgy,)p9t 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 ut0,mbtfb*q -of a rotating merry-go-round platforu t*f -qtb,0bmm of radius 3.0 m and moment of 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-round is rotating freely with aq.hrd / t);)vaujl:wq n angular velocity of 0..h/a)vlw: rjq;tqu)d 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,upsv f)nin)n4;j)s kow n 8y+fzl-i5 white dwarf, losing about half its mass in the process, and winding up with a radi 5z)4v ion)w8+l;nfy, jnis fknsu- p)us 1.0% of 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 inr,kctla,- +sov1*qhpl* rc( w 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 hj7- gq6cldmn 4 i)mm0zdif28$ 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 xhozj n03 f9z/rotate freely without friction. The moment of inertia of txho9 zj 3/nz0fhe 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 7(t 0)mc;ik *zkmnj-gw;vfkx(0x le wa 6.5-m diameter merry-go-round turntable that is gte((if;)* k0x7cklnmxk-w 0; mvw jzmounted 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+45k3h*pu bcs uik1y i qpqpl(12az, b rolls on the ground with the end 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, 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 mass m1uys ah2zp4bqlbiip,q(3c+1k 5 *ukpove?

  The Moon orbits the Earth such ty wo h8nhc/8-nhat the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Mo/hyhw 8 n8oc-non as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates fn-2uli:dv64by rk vc 9rom 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 uniformbg:sw.g,hgm; cylinder of radius 0.20 m acquires a rotational rate of from rest over .m;g,sb g gw:ha 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, uu ,3jyvee (fxpc6,2e 2:ua qneach of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.005 uqy(nf e6: ev3auu2c 2x,p,ej0 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 released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the a7b:5 ic cnblz2 : r*l* 6y/ytsae*cbnzngular 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 Sun, but with mass 8.0 tiwz: faz:a4hj .1qh bpuf;h c,4mes 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 that the l;aw. ca:h u4fphzf1bz4hq: ,j ost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to uxowbo/p,t 8ro -l2 agi .yk0v9se 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 tra/bv82kxpt-lr . i y90 ,oowogavel 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 anavjscrdyb*e8 5ris m :17+7fb $H_2O$ 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 horizontal surfahgd*eo.gpj209ex 7 f/e p/ rxvn u23vnce 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 lengthr27poljtpu )pnzl4; , L can pivot freely (i.e., we ignore friction) ab2lr;lt7j)pz 4ppnuo,out 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 Fig. 8–21g.]

A wheel of mass M has radius R. It is standing verd- hp fvylrgvr, 92(v.tically on the floor, and we want to exert a horizontal force F at its axle so that it will cli 2(hr-. ,9dv rpvgylvfmb 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 9q p* p zlstoip)8qm q9+2mnf;a flat road is making a turn with a +pt9z)8nmqm 29i oqqpsl;*pf radius 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 of length 1.5 m and begins whifn9;* pceqf8irling the rock in a nearly horizontal circle above his head, acc f8q iepc*f9n;elerating 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 b d t 6j114p;ckjc(f(wthw6y d7grs xc;ody as a solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms7c ;sg6h tdkw xwtjdc 61c(1pf4jy ;r(, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of twplq2 ki) 9i bj018xexeo cylindrical plates, of mj)k ib9xe8 p0e l2ixq1ass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 lg fa0voy+l 0 0tgmfy0*ooped 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 higl+m0* y0vtf g00ygfaohest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, butp 98is 9obyv8o do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uniforml,o0tl; lqkbo f6cfw90; lf/hw y from 90km/h to 60km/h The tlfo;;c0w9hfq,6 o l/t 0kb lfwires 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