A bicycle odometer (which measu7col;pso ei;is//2zcres distance traveled) is attached near the wheel hub and is designed for 27-ic;ols/o 2 /zispic;e 7nch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angularpvz+y.6 gxm*u /gfnm* velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the poin +*vmu *mp.ynf/ggxz6t have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be bq75vbxwj(ni-c4 v k5;vo f3wdescribed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a gi xe;5x ji(vnjiggpo,k 9/:+uwd x19y reater torque than a larger force? Explain.

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

Why is it more difficult to do a,dq e1r 7yc/zj sit-up with your hands behind your head than when your arms are stretcyz ,1/ e7dcjqrhed out in front of you? A diagram may help you to answer this.

A 21-speed bicycle hai7db r5 9wvq3ns seven sprockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large frontqwnrbvdi 5397 sprocket? Why?

Mammals that depend on being able cgzuc47k c: 2:bg7pr j6*jqlcto run fast have slender lower legs with flesh and muscle concentrated hiu7gjczqlc6b*c4pj ck:7 rg: 2gh, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 3 8-fblwtw0ti 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the net torque c50fah jfk;mg83slyocb*gxg)z) 6.p also zero? If the net torque on a system is z.3c gmyb*g5 g)kl ;xzfj 8ofaph s)c60ero, is the net force zero?

Two inclines have the same height but makgujkm y0,4)igy if+ eea d:m;m er;0b)e different angles with the horizontal. The same steel ball is rolled down each ie+ yf0 ieamgud)0 j:m)byi ,;k4gre;mncline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from re.u1bp*ojxyg 1 ri .6edst) 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 kinetic11.bg xj.d *o reiuy6p energy at the bottom?

A sphere and a cylinder have the sa1qfa1 9e 2zvklme radius and the same mass. They start from re 2laf 1kq9evz1st 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? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet ng:hb3c + k, bl lzesq2lgsq:p-/i6 e0most moving or rotating objects eventually slow down 3:,e/k-0:gcnple l hq+zsib l 2sqb 6gand stop. Explain.

If there were a great migration of people toward the Earth’s equator, how wou*ci fjn (t+ y qpsjt3*;m-v9ihld this affect -h;jmj 9ti (+tcpyvn3sq*f i*the length of the day?

Can the diver of Fig. 8–29 do a somersaui1di s 5mo tg5 6cpwf3bpqr.7/lt without having any initial rotation when she leave 5 6qbgst3fmpi1idpwo cr5/.7s the board?

The moment of inertia of a rotating solid diskr.2h)kq (/lf ab)0+plf; nst w( fp15jxhjkrx about an axis through its center of lfk rfa.xw1b ;( hhq(/ nx)5plsj+rf )jtpk02mass 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 mass (oa;;*fdw 81 c5iwswox lfmd(in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Expldx(8a 5fsoowf *wm ( l;wc;di1ain.

Two spheres look identical aw-;) x 9iod.s9(r mrj fko;cwrnd have the same mass. However, one is hollow and the other is solid. Describe an experiment tocmw j)d;.;-f s 9rwixoo 9r(kr determine which is which.

The angular velocity of a wheel rotating on u d3wk,fy9q0 ya horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the an, w09kuy3dq yfgular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turntable. hlvd g7jto,9eja;; z :What happens if you walk toward the cent9:ejgv dojzha,; 7l;ter?

A shortstop may leap inmjqnsx u -g49 :olzzjrtl7 e+3(1p m5-c4nryrto the air to catch a ball and throw it quickly. As he throws o5qm r9t +(m43njg ncr7j4 luzpsl ery:-zx1- the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the zyept5w5va:(p jv1i jkv00,b w4wy+ tlaw of conservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicoyp,w 4tatjwp500jvk 5 ivvwzy+1 :(be pter stable.

  Express the following anglesy bwvkrit:pgb 0j a *d.ko 25+6r6mlp* in radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth becauswy ( k*d.z;:hf2j vf)g/zzrdme of an amazing coincidence. Calculate, using the inf;fkhj)d/ r.z2w gfd*v m :(zzyormation inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed nmf0s :r z3 1jhnede58a9e ,vb5f,lnp at the Moon, 380,000 km from Earth. The beam diverges at an m5a: nhldn,efb, e n9epf3v5sr1j 08zangle $\theta$ (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 m*0wweum/z(amotor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the b/wz0mm a*u(e wlades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another chi dy 818 86clbkmx0rmdrld. If the ball makes 15.0 revolutrly cm88d1 0dbrxk m86ions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter tra5p* i19e)1ngpad h qegvels 8.0 km. How many revolutions do the wheels make? peq 11)p hdgaein 9*5g   $rev$

  (a) A grinding wheel 0.35 m in diametx3ndgz: /a5e2p0xqt j bljz1,kr ,x 2yer rotates at 2500 rpm. Calculate its angular velocity inz2px:gq2a5xe,tn1kx lz/r , j jd 3b0y $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.0 s (Fig. 8–382nsr0h9: :( a ro (g5zrkk3icpjwyg b.). (a) Wj9p3(w.i arg 0zk( r 5yo:skhg2:rcnbhat is the linear speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) y laxc 9 7dx;r.;graq,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 poile/q-zvy q:8o nt
(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 particlef,4l,y z4qqpb gc 53pk 7.0 cm from the axis of rotation is to experience an acceleration of 1005gqz 4 q3l,ypp f,k4bc,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpm t6+r(8r1r lwed5a gxo no 280 rpm in 4.0 s. Determine15 al exd8 rrng6o(rw+
(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 cwy6;boseh 1k(al)+p $R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard the Apollo spacecraft put themselv cq*ze)u.x(okr0 1abd8wu d6oes into a slow rotation t)* .db 1x06e oukq(dcozrwu8 ao 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 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 a;9msbq 1 v1tg, k7ibyuniformly from rest to 15,000 rpm in 220 s. Through how many revolutions diq7g1;vi a,ms9t1bk y bd it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in ak*qnt)3/c5 0jk vjbrwl +mc* 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 fora 7;a ),rh aqyzoto07 kl7f.del4 h6yb the stresses of flying highspeed jets in a whirling “human centrifuged74lf06 q ,ha.ytyza7 ; oobk )7rlhae,” 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 accelerates unifordx:hipaq 6 zqv: j2 (8jjfrh)+mly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled qpxj6vq8 h fa+hi2::)j j( zrdin this time?    m

  A cooling fan is turned off when it is r) 0ezdahz amof:3a+af0z j1q+unning at 850rev/min It turns 1500 revolutions before it comes to a stop.zf em+hdfaj1)a0a+qo03zza :
(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 makjeg 09uvo 9wv0 ebo,5ge 65 revolutions as the car reduces its speed uniformly from 95km/h to 45jge9ou5 wo9ge0 v v,0bkm/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

  The tires of a car make 655 u.sob6f6q,5ps yiwp revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter6 wpsibfq5 po,.y6u5 s of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each pedal when +ho tf.1o2i xx0d+l*/xouo mr;(lb fqclimbing a hill. The pedals rotate in a circxh /.1d2qo0ftxoimu ( bfr *l +ox;o+lle 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 o fxm;1psjmu).:v(uzb 9prq1 vp1nfqp+men/ (f 55 N on the end of a door 74 cm wide. What is the magnitude of the torque if the fvjup1v u1ne91 rmz:m)(qpp ( nbpf+ qf.; smx/orce 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 Fighyz r 8pu7kttbo pdi :73 7nan2 5(+/vcc;hfxb. 8–39. Assume that a friction torque of 0a fc t7nr8py 35;p/h+xk n7u : 27cotdz(bihbv.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 uq7*k; v-f yahj(2 alsrod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on thik(2svyuj ahq;a 7l-* fs system.

  The bolts on the cylinder head of an engine require tightening to a tor+x sq nctzp.;;*otbm- +8 tqutque of 38 -.ut*tson;t+ztm qb8;p +xqc $m \cdot N$ 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 1 1iltdeq .i0t90.8-kg sphere of radius 0.648 m when the axis of rotation is througi1tei09 .q dlth its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm gi peuis8vkuk38 8+91f sz7 t8cpq/iqin diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub 8i8 su i cv3upge8zi7ktsk/f8 qp19+qcan be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in :xo;w5wiw 6c1y-d ihp a horizontal circle of rad ; : wx1icpw6-woyihd5ius 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 rotating at constant q/zmaf:2 iqe2 26zsgpangular speed (Fig. 8–42). The friction force between her hands and the qz2z2ps6g/2 a em qf:iclay 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 o,:dx5 tpp:(mtcsw, tn. km (/ tz/ucyrbjects shown in Fig. 8–43 abokw dpxc5my, :,t .t/mt(s:z ntpu/( crut
(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 atv kn7 ain6a+x.d ub2g6oms 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 corre8jkfwf)gh hmfy(f()s m 5bl32 ct 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 mfmyflk)hsfjf)(hgw3 2 b m (85, 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 wits4f 4u53sp5pw+ os ecuh a radius of 8.50 cm and a mass of 0.580 kg.pps4wuo+s5 ues 3fc4 5 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 from rest q.eybxo l:m;jf eajs;a5 -3jmha1rkoog 8 ,-8to 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 handxgzc.u4izz* 3kj,x h:-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, n cxu4*zg3,.jhikx:zz eglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut offv;bsy qa j7054b e7jzh and is eventually brought uniformly to rest by a frictio7 jbya sve47 hjb05zq;nal 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. dt)w )pdl2mrk,*ddk 5j*z-cj,jjc s;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 solelyji*a qps- q1ny4p .b;:bwya1t by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig.;waq1 :i41bytp aq-by. ps *nj 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 consiwo lq1: n/ouli(m 4v0b xg8 ;pb0haf+kdered a long thin rod, as shown in Fig. 8–46.1gvo;w 00l uxqkn bab8 4 (ipmlof:/h+
(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 nz+-h1ujm b*bd6z)j+yekegw nf3o+ (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 full t-ne,(l p 4oaxnqb 8z7lurns (revolutions) and releases it at a x oeq,z8- bn4(n7l palspeed 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 og8 thvm c*3cl3f 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 85 i ujv)(kpmfi dj;,(g;kxcz 6bug(o280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and ra2dt2czz9e (y zdius 9.0 cm rolls without slipping down a lane t z22ezzyd(c9 at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the sum,uc58i t4c.i trfjrk5 of two terms,54r.c t5fc iik8ut,jr
(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 k:ws(ly;vkkae// i 8 ncg and a radius of 7.50 m. How much net work is ry vnw(ic:/ls 8eakk/;equired 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 from6ej+(8* gt ifz.upjmj rest and rolls without slipping e.+jf p*tjijug86( zmdown 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 start. ,v -smaazj58 nblb5f rfvu77s to fall and its lower end does not slip. What will be the speed of the upper end of the pole just b.baba v,8us7l7 fm -fvrj5n5zefore it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball ronmcn*r 5 ;/bcwn4lp w4tating on the end of a thin string in a circle of radius 1.10 m at an angular n44cbnlcw/n; *m5rwp speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angula ahmdrm hg58m3:p 9:vur momentum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotating at 1500 u8h::dah p rmv9m3g 5mrpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, okkirdav9z0 y hd5 (vv*z (1*uwn a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fi 5v ad09vh(kz*k(dwvi1r u z*yg. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce her moment of iny .e2wo x+wa3certia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rowoxea2+.c wy 3tations 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;o(6h en6bl ycu 2 5rs7pvsf9cj j2-de her spin rotation rate from an initial rate of 1.0 rev every 2.0 s to a final rate ehcs92n;f6dbju y27es v5-clr6p(oj of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis through wk)gw89maiy8 its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk oaw)8kg8im 9 wyf 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 spwn u09s;hiv2r42;lyfa ocfy 1inning 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 of1:ilv cs *xj4u 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 o2dy -y z5byt3g9itsl 7v.m7nd9qjp2 af inertia I is dropped onto an identical dis7ydv2pzg3stnd l9 297q ya 5ty-jmib. k rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4kt4e87 pie0xajd)( h/tj 1ofv $rev/s$ 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-roun;o ia9gl*t un)d platform of radius 3.0 m and moment of inerti 9au *nto);gila 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 abybi 2r)s6 *inawa/)7coifu, n angular velocity of 0.67)bc,u)awif /niab 2*ryois8 $rad/s$ 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 colehx s h)4nkl*-lapses into a white dwarf, losing about half its mass in the *hls)he-4xkn 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?(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 120uuzu d2: v8q9us;*)wefsucs*3y 5 tiu $km/h$ 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 :sjm,pyi izisf fyo3c+-5a(-$ 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 platformwaej i, l;8gre:2p l o.0aat/b, initially at rest, that can rotate freely without friction. The moment of inertr.2eio l,;e8w ag ajaltb0/p:ia 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 6.5hu /+d2 j.l)qmxeeq(70uzqsi:qtd 8g+ i ,twh-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia of dh,uu/:xdlesi(qq)7. 28jqtw m +iqht0ezg + 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of th4e *k+2 a+l +bv/ bve9ispco;3 zendhee rope lying on the top edge of the spool. A person grabs th dle2+eb+np9+;vzvs /eb i3*e4ahcoke 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 such that the same side always faces+rinw)xv9. v sadxxjm v62c; 0 the Earth. Determine the ratio of the Moon’s spin j9sdrmw6 i.+v xc2a;xvnx 0v )angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates ff/) m,nz 1cycgc6j0ivrom rest at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder ;u/fwax3iz4 vr w.qt.of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculatewxw/qru 3a..i; vt zf4 the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two a tz4g;fukgg1 l8 fu10solid 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 1kal8ugfg 1 tug; f04zdiameter 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, h95)wtt r 1mcl*f5ffpqjz:je- ow is the 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 Sun,1dt6t8uh;l7(ofudqh.xu v+u but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, f6huu(7 o81huqvtd+x l;d u. tand 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 ene rlpzu/xex-nx8j.y5 7 rgy stored in a heavy rotating flywheel. Suppose such a cu5-e xp7y rxz.jnl/8xar has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without 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 illustrates2y t +hpokf:yd44r v6h 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 onpt5c;lr q lbjw9 y/m57m/nu1 x 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 pivot freely (i.e., we ignore frictid uz+4vll:n;k w+n y) e1jxfy1on) 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 graviyzn4n+f +d1;w1k):lx ulyv je ty 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 vertically on the floor, k (.myfpab1)nddn a7k8 f5r,hand we want to exert a horizontal force F at its axle so that it willff,k m. y)8k(a n dphnab751dr 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 makingzwcto/ e8k 4cz9 mw -z;pe-jldo9*k7a a turn with a radius The forces acting on ok8dwe--e9zcz jz* t; 9plo/wc4a7mkthe 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 whir2 74-:y 7 bcqxmmvkmdaling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque requi2vqm -47ab7myk: xdcm red to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s bodyh0.y * tn ta;:g4t)agkm e3g ffdflj52 as a solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of tenfj;.4 g2ka*5g 0y) 3ftgtml tah df:he angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly w/ hu nnnsx3hj ree;s+-hci )26hich consists of two cylindrical plates, of hehs+ ;rc h2jxenisnn-3/)6u mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 r1.xn uki/ix,mc; (vlrolls along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marb i;1n.lki/v xu( crxm,le 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, butgkkcmmfu,9q01, ; c qj bk+19tx xndl; do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolution5w82amxfk; ,b3i,-q z 8fojj fpjp4bs s as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was 3 jjqb;,k4z-p p wfb88 s2f,amixo5fjthe 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