A bicycle odometer (which measures distance12fzhm jl2z:z a2o6n l traveled) is attached near the wheel hlm6jfhanl1:2 z 2zz2oub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a 9ovo 2-m- 1d8ho5ickpk2 j 7a4vtwole point on the rim have radial and/or tangential acceleration? If the disk’s -vt45 h k odwk oeo22j98ao 7c-vil1pmangular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a s :7do+07v++of ame sthesq s/lingle value of the angular velocity $\omega$ Explain.

Can a small force everkt9vqiv ,4/pr a1o9 gn exert a greater 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 sit-up with your hands behind your head tha0u;zum:zs,x vn when your arms are stretched out in front of you? A diagram may help umus zz ,x:0;vyou to answer this.

A 21-speed bicycle has seven sprockets at the rear whenyn +:e uos1yxh7 /dn+el and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket1y/+s7e+uxonyh n nd: ? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lowe c +3gamzw.in+r legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this dg am.zin ++wc3istribution of mass is advantageous.

Why do tightrope walkers /,n+: (ams+hfh n xzi rney:w2(Fig. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the net tor7 u i0rulx:/fvque also zero? If the net torque on a system is zero, is the net f7uf ilxruv/:0 orce zero?

Two inclines have the same height butcq,meer3(- c,nhtsls +:ps;3scdsm 7 make different angles with the horizontal. The same steel ball is rolled down each incline. On which inehtessc+sl (,cqp c;r7 sm-:ns3, md3cline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously swh v tjph.wlu-*h*/-2nen 2ybtart rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the inc h 2/u.yhlhb*p*twn w ejn-2-vline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius d u0o7b/ zzr5h:q/vmd and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinub r/5: dzq7v/oh 0zmdetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angj 87 o ,(xdt 5u0lu4tnzamds3fp4n9caular momentum are conserved. Yet most moving or rotating objects eventualpdtc0ftz8n j(4ds9a3 5nmu7 ao4ulx, ly slow down and stop. Explain.

If there were a great migration ofym)f3k;omm )5o. -ffms,fuqr people toward the Earth’s equator, how would this affect the length ooqr)u,5 m-fs3f. mym f mf)k;of the day?

Can the diver of Fig. 8–29 do a somersault wiv 2q wiq(3op c:er3am;thout having any initial rotation when she lep2 qarwo qimv:3e;c 3(aves the board?

The moment of inertia of a rotating solid disk about an.f/nvz* w/rcp axis through its center of /zp/rfnc * .vwmass 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 r j o.x9breh/f*- lodw;f9j v62-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity incfjw bx.9h9ol6rfo e-v j*d /r;rease, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hollow and :z 2:aiber.q wth:2rbq .zewi a:e other is solid. Describe an experiment to determine which is which.

The angular velocity of a wqzlr cx-f33. grnana8/l m fb3n0 r1i1heel rotating on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is thea q3-ag1/z cn nrr3x l fbf0.3mn8ril1 angular speed increasing or decreasing?

Suppose you are standing on the edge of+ 9.ew)l nk d-;8bz3snagkee j a large freely rotating turntable. What happens if you walk toward the cente )w ;akk3n slg8. dbn9+jzeee-r?

A shortstop may leap into the ai mmo8x ;n bs1br34lzrr0z8t6dr 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 notbznt3mr s8x dm8r; lz 4bo061rice 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,p jrw-ro(i4g5maw ,a, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate tr omaiwa,5j-rp,(gw 4o keep the helicopter stable.

  Express the following angles in.iev3i6d +;3de earjd 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 bfw,7xu uceif(0/7kl9 fb j a:recause of an amazing coincidence. Calculate, using the information inside the Frlcw i/fa (u7kb ejffr,u0 7:x9ont Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is dire2z ;hsxo/*ju1 nf he6p tc:2qjcted at the Moon, 380,000 km from Earth. The beam diverges at an /x6nuoe1h22sq;tzf j h:pc j *angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. When the motor is turnki*,)afb ncc + /5ospred off during operation, the blades slow to rest in 3.0 s. What is the a+,oa f) k/ircp5csn *bngular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the ball makesaga skry6 6q99 15a rag k9669qys.0 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm vewc,t+xlnogtrsrthf 5+(;j3us:7 ( in diameter travels 8.0 km. How many revolutions do the wh5+vts sfo(,rtlwxr;c7:j(h3+n uetgeels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculatp yy0.zxndc/w(grk9u7 +d * vme its angulac09uykpzd*(dyn/ +mx .7w v grr velocity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round maa vaw1kldjmwhn4d7sh ) 54:-ukes one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m fr5kuah:w )4d n1-4lsm d wj7havom 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 Ec2jel( o 3nit*w5zge,3u as:d arth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear spe rm2 cb/nl)k1bv:xjy- ed of a 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 particle 7.0 cm from the axi.fxuus1n -,yks of rotation is to ex- kuu1xsnyf,. perience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel awq0y 9 /w. j(a i(.6nvtzo9jif; oekwaccelerates uniformly about its center from 130 rpm to 280 rpm in 4.0 s. Detean(w.kowi/iz aw 0;ft qj6 oe( 9j.v9yrmine
(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 radiu jb,jervy .oj u5(b4,xs $R_1$ 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 spacecr+uh-ji ael67 iaft 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 diameter of 8.5h+l6 uj7i-a ie 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 15fq3dqenn p864,000 rpm in 220 s. Through how many revolutions did it turn in this timeqnf3n6 e8 p4dq?    $rev$

  An automobile engine slows down fr ;5ldnv otz0lzg/rc./ps kj5*om 4500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying (cb 11vgeu.:rbr/kop highspeed jets in a whirling “human centrifuge,” whbecr1orb/k u:( pv 1.gich 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 uniformly from 240 rpm to 360 rpm in 6uew ykzo(yndj 8d- 6.6.5 s. How far will adozu8 yw 6d ej-ky6(n. point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/mbob( 8dzd7qix3 6ncz 8in It turns 1500 revolutions before it comes to qi bcd x8n6b(zo738dza 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 it5qjw+av1o1*k d7 kwwifs62 1ha6sq lx s speed uniformly from 95km/h to 4517ij5laqxdwo216vsk6 wf1+q k w*has km/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 65 revolutions as poc2k+ 4ky -gg 366mg7nim kqkthe car reduces its speed uniformly from 95km/h to 45km/h The tires have a diamet 2k6mg7k -k+pyo46 k3ig ngqmcer 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 dwlx39jpd n4/ pedal when climbing a hill. The pedals rotate in a circle of radius 17 cm.3w /dpj4 xdnl9
(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 mag cdq* jc 3y2w1nhg2s7d7 0ttwxnitude of the torque if 1wct70st2 d72*hc jwqx dgn 3ythe 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 uz3 cz.7d qzo*the axle of the wheel shown in Fig. 8–39. Assume that a frictcuo.z 3qzdz* 7ion torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a massless rod w qgf34 ae(gpa 6x;,tkxhich 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 thi ,x4qe(3g a;kgfa tx6ps system.

  The bolts on the cylinder head ;d ki;hnrjvxw6w:0 +rof an engine require tightening to a torque of 38 ;j+ ixvdh w0:kw;nrr6 $m \cdot N$ 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 momentzbo:v8 0yvsr4 of inertia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is th sv08z:vo 4rbyrough its center.    $kg \cdot m^2$

  Calculate the moment of inertia -9fxy/vxvv 7byb/b+p of a bicycle wheel 66.7 cm in diameter. The rim and tire have a c-/b/yvx v yp79bf vx+bombined mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod q*vmidr o:b8+yj q8h.is rotated in a horizontal circle of radius 1.2 jb od:8mvyi+ qqr8 .*hm. 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 angular r5ievu v9olm66 qe g13speed (Fig. 8–42). T l vor91m g6ueiv3e5q6he friction force between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objects shown in F7cz4jgc4ovf) m*;75-v;bgrykicmr f ig. 8–43 accgg z 4k*orri) ffvb;vj-5my7c 4m7;bout
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen atoms whos,awwqb5zx/e (dt/ wf;3r. p gue total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the correct ru*yk6w22ilh pdxr0 u)yf 8t q*ate, engineers firp0 f*2lxqy 2u ywkhir)t *86ude 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 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 oc 3osgf-g j)o*vwq (h+f 0.580 kg. Calcula)foqgs*+cvo hwg(j 3 -te
(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 frat/ xz4 lac;v3om 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 smali8 +p tnl+bn*xw:j ay).j4rpll hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque re:n+wp 8r4*pb jtn )j.l+a ilxyquired 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 bx va;,bvk3w03h /khirpm is shut off and is eventually brought uniformly3vwvkbhahi,x 3/ bk;0 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 -o-f/ rwu l9lwtvz c(q+c(rh9at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the action of the forearm,:: ln2nz,e xhr which rotates about the elbow joint under the action of the tric:nrn:z xhel,2eps 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 ca e 73id7j4+j duz ;-qsi hoch+uc2,znonsidered a long thin rod, as shown in Fig. 8–46 ahdcji,se3 i4+77n qz-z;jd2 ouu+hc.
(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 massen 5 od0,-olmroa/d)qes, $m_1$ 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 acc yn;oeav1l)ygo.* 4eto*(p rm elerates the hammer from rest within four full turns (revoovy4le mo(g.tye* )*pro1a ;nlutions) 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 gm0q7hp (p rn.px,lf: gbo0 ;ma moment of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine developsb+9qr won +qeflnp3:jn r(e.m..r k5 j a torque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg aqmhg; e1 71zxx i6qf0rnd radius 9.0 cm rolls without slipping down a lane1m7g6xhq1; if0exz qr 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 ps-rswvfyr .g0u)wq;m :6i1 f pxcc8*.lpp1 b of two t;fmicr wf c61bv .l08pg) p:r.1 *upqpwssyx-erms,
(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 1640i-zqy jz;jvy, w*9h,u kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 su *w-i ,,qvj9;yjzhzy? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls k/upt9y3p 4xr7*qxmt without slipping down4uytkp*39 mxx7tpq/ r 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 sanz d98-f0s fktarts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just beffza8 ks9d0-fnore it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg bgahz *1znpmh8p4q* k :wmprg9x;p ;( 4 vwgq-hall rotating on the end of a thin string in a circle of radius 1.10 m at an apmaxhz wwmv14qg(z ng :; r8 hppqg4h; 9-k**pngular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindripcl. g-kl)q2 ya:xs5mcal grinding wheel k) :lxgp.m5y sq2c-alof radius 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 platforz/9jwh ofg;v *9zhks0bl j/ ,ym that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases toyg9sbl0,;oj k w//zf hzv9j*h 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 trcrjrg f8n8aq9a7-zjy9 9o; reduce her moment of inertia by a factor of about 3.5 when changr jf989z7;9raqt8 cagr-n yojing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from angw 0rh ri69y55 ywgke( +nr7in initial rate of 1.0 rev every 2.0 s 0y57ig+r r(ww5ng6hkeyrin 9to a 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 rotpimykbws ecv khh(5.8d 7y3+,ating around 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 an3ed kih 5yb+kh .cpyv ,8(sm7wd diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinning at 3.5j y1rr6tyiz82 $rev/s$ 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 ofloh f:7.f)an 1g*mary 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 droppedip6v +y,w* jexvg1,q wdb:+0aa (gjq k onto an identical di*xw(w,+q vdpa ybqakgi j,1+:6ge 0vjsk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns a(xpu3wbw -3):g ix f(q 2o(kya dhmw )rhplp()t 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 k34 2e) bumxnjb/ og8mat the center of a rotating merry-go-round platform of radius 3.0 m and moment of in43k/ gb)8mu ej2xno mbertia 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 frsu)te9xbv1n ) eely with an angular velocity of 0.8st)n1 vu)eb9 x $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, losing cvhzipom8dua(8)tf3f u6 *es (( * qvlabout half its mass in hmzft 6v**(i8luved8 scfu3p)o ( qa(the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(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 w0nn,h z30/cf1ku o u.d aikil+inds in excess of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass 25hr:+ x oubgn$ 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, :qf 12)wp.ras-;+fjc: zytg.vp x cphinitially at rest, that can rotate freely without friction. The moment of inertia of t :ay z;psf-)pthx 2vcqf j.+1.pgwr:c he 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 persjj m g ,obltxsin/4x 2f;z+f/9on stands at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia of 170 t fnlj92z4f/x,gjs/o+xi mb; 0 $kg \cdot m^2$ 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 the rope lyjr0 xjx ) ,ii8scbj.)ting on the top edge of the spool. A person grabs the end of the rope and walks a distance L.i)jj8 rxbjc)x 0t,si, 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 sides0nnax0ynvg 0 ,1cwe*ltq*x 4 always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momens0tl0 *xew nn4yx a,c* gv10nqtum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates awp64ctcv6 2y from rest at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of radiuof -il.db ; ,2uyhq 5zd8irez6s 0.20 m acquires a rotational rate o5dd6u h2ieql;oz .ri-f,y 8 zbf from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disksp;h1;ff svu2tlp(n+b , 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 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 fropu 2nbpth;+ s1f;( lvfm rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular a5:gv -pnw d,qspeed 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 tivc hdt6;j- 4xumes 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 s6utc4 hdv x;j-tar is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use energy stzq) 44ei onnotk59wr1ax*nv */hro t(ored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of 4/nehz5wx)q( *n r9ia oo4otvrt*n1kdiameter 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 8*s,ta9gx 2mv 8ax.ewz2 hdmpwl9pl 9an $H_2O$ 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 surface at speed 3z zvng*x3frc(m904oeik8c q 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 leng4 e62nf0zjps xth L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then releap20e6snf z4 xjsed. 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 vertically on thxxawb+49 raar(aji4l0g n6)he 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, w wi)r a(ab9ajnlxra064 x4g+ hhere h

  A bicyclist traveling with speed vyfu :yx7j rd+)=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist and cycle arf j: rx+udy)y7e 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 beginss1gl.t43a o ea whirling 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 torque4 tl1os3a. ega required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thinki8h-cf5k kdf3o3 of+ueb65j1u be l* rods, making reasonable estimates for the dimensions. +u k*f-5o368fkfukbj c5oeid 3elb h1Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two b hb(qp9zi/7 xcylindrical plates, of97hxzb(/ p qib 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 rolls along the looped rough tq:f3 lfi3xrr 1rack 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 highest point of the loop3r3 i1rqf :fxl without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, butc/36cycenaz-, xnu9hv9rp 6l do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed unifodb x)5(c ;m8b+yzggtmrmly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) W +mb8()xz cmgyd;5 tgbhat 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