A bicycle odometer (which measures distance traveled) is attached t7fq mb1q har z,7+ltfk.o-o 4near the wheel hub and is designed for 27-inch wqq4,thfma7r 1k7f-tlo o.+ zbheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant cwbgz ( 83k8jycdj58kangular 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/or tangential accelergj 5cjcwykk(8 88 3dbzation? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a y)cbw4qio41tl ). 9su n gsfy7single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torqvdxsc(,: ln;tkee 21 gue than a larger force? Explain.

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

Why is it more difficult to do a sit-up withllj;d.pt 2: 7f jsll 1ffc.,tj your hands behind your head than when : jlpd jjct;.1l,l .f2t7sf flyour arms are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the pedal9 7y3kyi edhyf, /quxs7b2h;boh s-h: 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 lar s7fsb:d-7h hyuyx/3b ,oq ;9eh2khy ige front sprocket? Why?

Mammals that depend on being able1o+ag 9xv -pxb9j*jta to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why jg*x1o9v - jxt+ 9bapathis distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a l nmv.0uwk oa.1c)b.ib*9 t rruong, narrow beam?

If the net force on a system is zero, is thev 5uryul;b)mkxg 8t9. net torque also zero? If the net torque on a syukxul9;5 .v)br ygt8m stem is zero, is the net force zero?

Two inclines have the same height but make different angles with the horizontal. mr1y5lvt7t8w The same steel ball is rolled down each incline. On which incline will tt lvwr m15t87yhe speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling w+n1zol8yx6e u tk (4j(from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reachzolu t6 1+ek(4yxn8j wes 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 cylinderdq+2) y q:kcrl have the same radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first?c:+)2r kdqylq 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 angular momentum are coqsrimg.eu3tgj 1,14f6 0edt ynserved. Yet most moving or rotemi.r6 t1tdggq 1y,ufs 430jeating objects eventually slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equator, how wx r98a f)azb;dould this affect the length of the d)r9;f ba8xzd aay?

Can the diver of Fig. 8–29 do a somersault wc3uz hzqa )xry )o1p rt:-:d-i 8t(t u/i:xlnaithout having any initial rotation when she leaves r-)i8c 1o l pr/zz ha3t )::(ntiux qxu-:daytthe board?

The moment of inertia of a rotating solid disk about akzbooi228 f5 rz +w9ian axis through its center of mas 9kzo8orbfwi25+a2 iz s 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 rotatixq+/. sid i5qvl j8u5png stool holding a 2-kg mass in each outstretched hand. If you suddenly drovq8pli+5i/d u xjs .5qp the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hoilrpsz5xqp 6 9)em8,q llow and the other is solid. Describe an experiment to determine m)lpe6,q i9 sq5px 8rzwhich is which.

The angular velocity of a wheeqjsl90 x4 7wkm7*)oimbzq. n pl 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 po *zq)ib. wmk njp79sl740x qomints 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 large fre*0h-c nufyik 1ely rotating turntable. What hapc* 0 -khin1ufypens if you walk toward the center?

A shortstop may leap into the aidz s r4q0kv ftj*sohgyu(n91mp;+ ,6 mr to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips anq4ro;mkuh , fmsyd* p t169jsz+nvg(0d legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentu70rrdrw,iv- qm, discuss why a helicoi7q-r,0v rdrw pter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (a) 3vl 5w8krcuxx7)krxn+2 hs+t (0 $^{\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 coi cbbm q5p ay.ifi:78g)ncidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of tyfb :ia8.qb)75mi cp ghe Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000zy:gw- sv.j:voo+- ny km from Earth. The beam diverges at an ang-+ -ojon vyv.:sw:ygz le $\theta$ (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 lp; s z-a;lz4y3oosfri2(/ mlof 6500 rpm. When the motor is turned off during opersloz iy-l r;zsf 4a(/m2;po3lation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m 6hrj u9o;nql7to another child. If the ball makes 15.0 revolutions, what is its di9r;hu q7 nj6olameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutioyn)qqo/3 5ic ins do the wheels make? y5cni3qi)/oq    $rev$

  (a) A grinding wheel 0.35 m in diameter ro*.+uml;pja f un9u*s(flky rox amu666b 5b(ntates at 2500 rpm. Calculate its angular velocik9fusanfy;6. l( bx u6(5upnrmm* a6bl ou+*jty 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 o zz,2xakt f7fm7l/ v7xne complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child sealxavxm ,72 f z7k/t7zfted 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 angula(dn,josyrw : if-. xg*r velocity 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 *qrkr5l6)hhqvqr. /+pgd3w ea point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a partkb79n,aua5r; atfx6n3,s l nuicle 7.0 cm from the axis of rotation is to experienr6a7 xnb;5f9n3ka,tun s ual,ce an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel acceler e7ld5kig 57nqates uniformly about its center from 130 rpm to 280 rpm5l77ke 5gid nq in 4.0 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 radius 0dv*-rsuhh dx-ym2 k/$R_1$ 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 spacg;c)lexs2v 9nx01ju u ecraft 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 with a diametenxsu9lu;x0 gv)1 2 cejr 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 tmpz 2u* 2v t60a-qc4xsyznty)lf:2 prpm in 220 s. Through how-s6zm4)x220 utcva :qnp2p*yl fyt z t many revolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5wwh,9q 08fenv 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 whirling “hussws 4q u n*18reci (6ltdk0*aman centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before rea6s1uaq8 cd40t*n(*kwir l essching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 360 rpm i- :t8odsuz xd+n 6.5 s. How far will a point on the edge of the wheel h+: uszo-d xd8tave traveled in this time?    m

  A cooling fan is turned off when it is running r*.9 zcyi1r imp5u ,x2 2mvgxaat 850rev/min It turns 1500 revolutions before it comes to a stypmirx2za r c,vi.2u*15gmx 9 op.
(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 uniforzxmet/ wldgue/ m:kk g , )p3(.wgg9*zmly from 95km/h to 45km/h The tirekzm*g g)glk z9 u//xpegm3:(tdew ,w.s 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

  The tires of a car madmwdxbm4cd*n (1ut je2i4(l, ke 65 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires htx 42l(d dbju(ie*,m mcdn w41ave 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 ridinv-wkv3 i g a8a4fmlm:5g a bike puts all her weight on each pedal when climbing a hill. The p4a8mfmw -gvk 5a3l vi:edals rotate in 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 4+i, sr 9y:erkjmtx+ wof 55 N on the end of a door 74 cm wide. What is the magnitude of the torque if the foeyj4+9x :rt +i,w rkmsrce 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 a(tig h 1zvkmlp y/2;jn f52o3fg,hu-of the wheel shown in Fig. 8–39. Assume that a g- 5ivz 2h1p l ;/utgnf,ahkfj23 (yomfriction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass muqmt5j,/kr x mu7e*hnc ue8,3, are attached to the ends of a massless rod which pivots m7,u3/,meqjxe u8rt 5*nu c hkas 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 system.

  The bolts on the cylinder head of an engine re1 6k7jbmf8xwaquire tightening to a torque of 387bxfaj 18w6km $m \cdot N$ 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 sbrpxcojsbr. hg02nq/a.j d1 nf f;wjmp.-.-0phere of radius 0.648 m when the axis of rxb a.ho bm.r cjp.-q2d.f; 0g0n- 1/sjjrw pfnotation is through its center.    $kg \cdot m^2$

  Calculate the moment,k*vqnemdnam)cq w1,c:(5k b b;j d q5 of inertia of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignow*qb c1mdk d,en5)n5 qb:; a,jvk( qmcred (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thiu6h ktp q h yc0tql.;h.34se)gn, light rod is rotated in a horizontal cis ut.3 ch)ly qhte;g6qhk. p04rcle of radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotatinoci:jp1h9qo ;g at constant angular speed (Fig. 8–42). T: ho;op1 jiq9che 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 u9dpz,2gbbz2f j/td; of inertia of the array of point objects shown in Fig. 8–43 abof, jtg9;du2zz d2p/b but
(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 cofi jl n+en197nnsists of two oxygen atoms whose 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 correckgh*2 9 t/o6 -rlbnvdot rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and2/6 gk dloo9*hvnrbt- 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 mass of q,xg3k*i:n5gke+ u vn0.580 kg. vngik:5ge*+,q3unx kCalculate
(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 rapmzb - 65tn2zest 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 hv ke 1-*uvhj,8f no5*t cy.+za2xqxrband-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 (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglectinnxuraf+2t1-k* zhv8 y*,vx qc5 boe.j g frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut i+r-;n qzike/ off and is eventually brought uniformly to rest by a frici-ierz qk+;/ntional 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 accetm kr 9:o.ftfh+5xh b,lerates 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 szg tee17b9l-;c j cugj ri+8y2olely by the action of the forearm, which rotates about the elbow joint under the action of the tric leuec9z1 bi crj+7-g2;jygt8 eps muscle, Fig. 8–45. The ball 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 cons8aeb/ nj9f jo.idered a long thin rod, as shown in Fig. 8–46 a.jjf8noeb/ 9.
(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 two jmc3px* o6wd61lj/ed masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest within four l ms3le; bb7 )6w5xpuefull turns (revolutions) and releases it at a speed of 2m eu53bl6l)px7ws e b;8 $m/s$ 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 osxaq, r3/8fmef 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 280 -fym qav /wa2qt0.z5s $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg a :46q*mp)ayw;0pk,u jy tcqn fnd radius 9.0 cm rolls without slipping down a lafu4y;,y:cm)wpj pn06*kaqtq ne at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth wit x -;2mztw6uj:s v ua0ob:hd8hh respect to the Sun as the sum of twh2 a;tv ud:uhox6 8bzm-j:sw 0o terms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius of 7.50 mcho.(2)crcltc nin)1v( z ,mw9+kv bs. How much net work is required to accelerate it from rest to a ronvv 9)c(ht(kwniocs1cmc, z.)2l+ r btation 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 k.fh,,ecq ) fax4(fdsh*f nx8dg starts from rest and rolls without slippxxa,*fe)(d4fc sf , qdfh 8.nhing 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 s/ :c2 i.piqt/qr3ycrx tarts to fall and its lower end does not slip. What will be the speed of the uppe/crqqt x/p.2y:3icri r 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 rotatin6:ix j96 +hugru4l;izngy da d/4)jwa l88 gtzg on the end of a thin string in a circle of radius 1.10 m at an angular speed o x+d:gz)t6jgd6a 4y/l;zri8 4lha9 j gi n8uuwf 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angulardt8nwtz6 wt):q.n+lb momentum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rota+zl t n)8tdw6nq.wt: bting 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 rotatbl u)dz(+ +xispr2dp 1ing at a rate of 1.3rev/s If he raises his idspbp21+d+l )zr (uxarms to a horizontal 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 in Fpm) c/dcb,)d . hgtbi1ig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 roih,c1 mpdb)d)bc g./ ttations 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 .u/ydpr q 2/.lplbc,irotation rate from an initial rate of 1.0 rev every 2.0 s to a final rate of 3 y,ld.plr pbui/. c2q/$rev/s$ 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 aroundyd46f8q1/zitglp- ,x 2cd d cn 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. Thel/c xdinqpf42zyd8c,- g6td1 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 a figure skater sp6rcq +s(*ty ig j(tjl2bx(g7ginning 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 momf qsbgqi, *)5uentum 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 wyvqv)6 c7gbgb 6 .5zyinertia I is dropped onto an identical disk rotating at angular spe6vy5yg.gvq)7 cbb z6wed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns )t,hf 7 vi dvn1*+airlat 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 merry-go-ror g:js/arc12fund platform of radius 3.0 m and r/cjar 1sf 2g: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 an angular velocityb cae89z m-u6v of 0.8 z98emu vab 6c-$rad/s$ 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, ./.x7 lbj q(gbp7vabylosing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the labp g (.lxvy77qj /bb.ost 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 of 1 / /vy.nljzr5 ,p:kmub20 $km/h$ 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 k5ug;s i2 tt9xg4*g8pj qtv9y$ 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 f q,rla;;h)lf9 9sw()w bixoipg9 .jl, initially at rest, that can rotate freely without friction. The moment of inertia),al;9 jx9ql(orpwb ffl s;i w9i g)h. 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 edt 9du ng/ .vrovw6k1x 2gff78sge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionlessk 98 nvt1 rsvg7f6d.u xowgf/2 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 grouhjbk0 /tbxel+vk 5c8b0 a(/dend with the end of the rope lying on the top edge of the spool. A per 0cjbl 8a /b 50(hebv+kkt/edxson 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 move?

  The Moon orbits the Earth suchta3metum91 +bb 8 ;vcm 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 angmvmc1 mb;btua3et +98ular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from res shj7c 2s64 )*dul+ moyh 6naiwit/;tvt 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 sog4ne)ec (j9z h:p f6shape of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0ns) 4zc(h6 g:opje9ef -s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made m:bl w7o 6wx efik6mb/5wjb//of two solid cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of m6fw/jl / bwm:ibmb5 x6w7/oekass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is theo.4w*bzg3h o n 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 sicl aq9tf6s 6x(i u i61w*zv*enze of our Sun, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-*n v *cu6e6f6i1w xqts9i za(lquarters 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 e5i qud.r t5p,8:kxl)to;nk eautomobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has ekqp;d:)8r i otelu,t nk55.xa 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 illustrates aoiv 1-,+ngl8x 0xomt dn $H_2O$ 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 (hz j 1w1sf:sqnec;ii6u9h7)yj oop) 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 and length L can pfkof5s9i ct7 /ivot 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 gr9o7 t 5kisf/fcavity 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 standi51d20jqhu j .jiq3dha ng 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 height h, whe03dj2a 1uqdq j h5.jihre h

  A bicyclist traveling with speed v=4.2m/s on a flat road is mak w2e;ga,gv :qging a turn with a radius The forces acting on the ag2: geq,vgw; 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 inb (gweid 4c;bqw9 bf+d )7a6tmto a sling of length 1.5 m and begins whirling the rock in a nearly hb t aw7) f; dd+b(wmbgeiq46c9orizontal 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 aqgnt*+y/p pr2u9gv 0ow/s:s 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 arms, and wit/gr+vqg wu/y* t0a:29 snpsoph arms held tightly against her body.   

  You are designing a clutch assembly which consists of -gg4/e2onxnw..oh c htwo cylindrical plates, of mash-/g on e2x .h4wonc.gs $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 ( bl3zkr;di36lwak, ulooped rough track of Fig. 8–58. What is the minimum value of the vertical heulk(,a3; z3wdi lbrk 6ight h 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, but do not assur bw 9e6n5av12 l :;tud5cjtbiasg6s9me $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reducby s ;u3zp;7qxes its speed uniformly from 90km/h to 60km/h The tires have a diameter of up ;z;bq7 s3xy0.90 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