A bicycle odometer (which measures distance traveled) isrg wjv f/c,e09 attached near the wheel hub and is de/fr,w vj0 g9cesigned 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. D0- s:q viogh)xoes a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the q0v-i:x) oshg point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body 9r7ii t;q at3qm e*y,ybe described by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exer. ap.1 3p2nm fvmdydwd6k(m/ ht 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 qv : o15sdmbe0to do a sit-up with your hands behind your head than when your arms are stretched out in front of you? A diagram may help yous1 eob5mvq0d: to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three(x5wf6ai/me- mu, bw s 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 x-fbs /mww 5(ei6m,u ait harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on beijbok un u (2xnd0l-i)+yso0x8ng able to run fast have slender lower legs with flesh and muscle concentrated high, close to j x28i yo)ns un(xol-k0+0dubthe body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. c:e7vavmje,g ij+( 8m8–35) carry a long, narrow beam?

If the net force on a syste+q.mepx gy7qqoa- 2)d m is zero, is the net torque also zero? If the net torque on aagdq e+p2q-mo .x)7yq system is zero, is the net force zero?

Two inclines have the same height but xv +gs cy)*29etuuz6guj c83 qhgrt21 make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of tc1 ) 6stu 82gg+c9hjuz2 *ug vqxrt3eyhe ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an incline. Onpq nwi8nc76- n mvgms:b2 (o2ve sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy7n i: p2 2g(nvb8-swo6qmvnmc at the bottom?

A sphere and a cylinder have the same radius and the same mass. They start frok-2kjom q- /o7jy)g ctm rest at the top of k7ckjqt2o-- og) m/jyan 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. *dxg2n kyy3zg7ls1k25piwhu9r g( 4wq j*i9rYet most moving or rotating objectyk 2nz*9u4s5* r gy2 qkwd(p g3j9irxg17wlhis eventually slow down and stop. Explain.

If there were a great migration of people toward the Ea6 bpt7 f -e05rgurt27d 0 7lnhkaa2lzmrth’s equator, how would this el dk0gta7mb -hf7 27z6ant25 up0l rraffect the length of the day?

Can the diver of Fig. 8–29 do a somersault without having any initial ru()fo gav :xf- ouv;,lotation when she leaves the boarfov of-guxa l:;v,u )(d?

The moment of inertia of a rotating solid disk about an ax,xk ;gasba(3vd1 e -puis through its center of med;g,-aupv x 3b (aks1ass 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 j7an7oak7nr tu i 5;0.k rz;tron a rotating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? k ;r77a;ra0njzi .o57ttnkr u Explain.

Two spheres look identical and have the same mass. Howev90h+ gfie0wv :i bo3cner, one is hollow and the other is solid. Describe an experiment to dc b+ nge:09ho0i3 fvwietermine which is which.

The angular velocity of a wheel rotating on a horizoigcru4pfni (c(f/m td(+ r-7 bntal 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. It(ip4+irb (cc mf7/undrf- g( s the angular speed increasing or decreasing?

Suppose you are standing on the edge pz(e*0 3k ccjwu, yol:of a large freely rotating turntable. What happe,ouc3y(0 :w*zlke jpcns if you walk toward the center?

A shortstop may leap into the air to catch a ball and throw it quqibe)50y (lh ;fs z.dtickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice thay fhtds bqzi5();.le0t his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of anajyw ;p( z4 ht*w7mj6egular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to k y 6;am(z4et7jw*hjwpeep the helicopter stable.

  Express the following angles in rt zci(x4e1 732(nfzp owrf:g gadians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an ama)vwxk6l 0j1umzing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians )muw6 lx1jvk0) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The ba: e6p+s3lvx( k+ 6wldfy y7je3eb9e beam diverges at an angle9a7ldfj ypyse3xw+6 l b:beve 6k3+( e $\theta$ (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 turned onoa,ez5g/ g.7. :ctfkgqv1 lyff during operation, the blades slow to rest in7:g,kavt fz e.c ol5/n .gygq1 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 t(;eo *e wk jz j+f3ku2)u 1 u9h+pchvmy(jsh,zo another child. If the ball makes 15.0 revolutions, what is its diamet1uueh;jvmoz2*ye +sp3 c(z 9 jf)w+kh ,uh (kjer?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many rboh( zertmxpte +6 5))ijn d.9evolutions do the wheels make?)n5( xdmtpitbor +e 9e.6j )hz    $rev$

  (a) A grinding wheel 0.35 m in diameter rotatq2d c6sdgvlo6 s (n45qes at 2500 rpm. Calculate its angular velq(65sq2 dlnd g6s4cv oocity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.en5jvcevyia c m+ ulz7f)1+ ./0 s (Fig. 8–38). (a) What is the linear speed of a chiel ) .mccv fnza/e i71v+5uyj+ld 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 velocita/-04v7/*d nnpi+qq0n r yowyr(diqe y of the Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a pkt8sp*j+3:y7m( om:k zzxtcj oint
(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 fro lxk 7:l*6m8gh tt,nnem the axis of rotation is to experience an n 6g7xhl8t:ke,tn*m lacceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates unipl 7g-s z5ol4qpv(/dwformly about its center from 130 rpm to 280 rpm in / (l-wzpovgq p ds745l4.0 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 radiu p1zlt f 8urld+91aig7s $R_1$ 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, astron;ub,frdc cw)a6af 75oauts aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their aubc6o ;d)5f,7ac rfw 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 uniformly from rest to 15,000 rpm ia 79h:etrmw1j* b rvs:n 220 s. Through how many revolutions did it turn in thi rh*wr7e:sm19a: bvjts time?    $rev$

  An automobile engine slows down froekxji3p-x(kw z2s3h* m 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 highspeed jets in+iz0i/mzniu(ul5: gcs9vn)j a whirling “human centrifuge,” which tazcjmiu/5iv0(zn nu sl 9+g):i kes 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 unifox5z w7 rd6gufwezlvdn t,e0v;q;5m2;rmly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled 7 ;m; wr ,lz0uv2dv6fwge5x5tn; ezqdin this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 1500 r iy*x vgr-o6 teu+3xwdun63f/evolutions be+63of/ iuud3wg e*y-t xrv nx6fore it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed uniformly fux4m phmw(: ( ga/we*j2tsfx 4oq5yz -rom 95km/h to( x fwowa45hqy(g x:j2m/t 4*zesump- 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 6*yrbnx kuei8ha1mpz / x7+ rfmu7m 8205 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The y m018zhmk ux 7rex*nm/p+riab827f utires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each pedal whennul -5s xj;xa ul1t93a climbing a hill. The pedals rotate inx1l3lau s;jxa 59nu-t a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cm wide;*g b e,kxm yl5( 5h*ugwsz8we. What is the magnitude of the torqk ew,(g g sx b55*;z*8yhwuelmue if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of3q+y*xth o5iqc s 4e5y the wheel shown in Fig. 8–39. Assume 5tq4q3i hey ox5*c+y sthat a friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a masslesdx4oa krs42 co(7qg 0gk.9 ples rod which pivots as shown in Fig. 8–40. Initially tl4pkk.er2(q0gd7a o gso c 49xhe 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 cylindeyheq88d*ulf w.j0rb4r head of an engine require tightening to a torque of 38 h08 e qyrlj8fu.d*bw4$m \cdot N$ 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 oa, balarme80vvh h863 f a 10.8-kg sphere of radius 0.648 m when the axiae6rahm,a lb h8380 vvs of rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia ozb648qu.ctm *m an ta5f a bicycle wheel 66.7 cm in diameter. The rim and tirb q4am 6namc *5zttu8.e have a combined 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 r) ewza64xums,j-hd-sod is rotated in a horizontal circle of radiusm,h szj)--du xa64w es 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 angula x;)a2k b7wqiszl 4j,sr speed (Fi,lz) ksws;2jqx 47abig. 8–42). The 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 pm/6l*sxq;1 rvmsyb9objects shown in Fig. 8–43 aboum* xysq9smplrvb 6/1 ;t
(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 wnqh96q7shbs 1 v*b 7rehose 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 cyli/ olhue, i8;,-xevb xfng-k *kndrical satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, whatvk k8eg n *f ebu,i-o/x-x;,lh 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 u5kd, tqhi,)zrk7zyo7 niform cylinder with a radius of 8.50 cm and a mass of 0.580 kg. 7 5doty,ki hkzqr,)7zCalculate
(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, accelervkzv8wpf4 d 0k+jb07 rating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merry-go-ro*q75o l)0l b z;:astocjzyk 1iund 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 eaczi o7j: ;oc0z*y5 )1lbktsqlah other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating.0fr;*ojx/4q 0e di ln -bjlir at 10,300 rpm is shut off and is eventually brought uniformly 0lxqifj ;rb*d ljn40ero.i /-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 a9kg680jgase qrht,g 7ccelerates a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the action of the forearm, whlibx2 mly+ r 3xi-id(+ich rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformri2y bl+md+-ilx( 3xily from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as shown in Fiw-ihg, ib :(zag. 8–46.haw g i(-i:,zb
(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 co6 b(euzd8xh/ dnsists of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest within four fulluy+-q z(l1epx5a mt*j9ed;sn turns (revoluti- m d1e;zexql 9p(5yatjn +s*uons) and releases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a mojge a(x :4qca8auz:j :m,w/bcment 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 deve4(buo, fx-puqlops 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 and radius 9.0 cm rolls without slipping down :mwf2/koec o,vzdj x9c:)q*z a lanew ,zdokvzqc 2x 9)*m:o:efj /c at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth 4 kx9afi(8sa iwith respect to the Sun as the sum of two texfk si48a(ai9 rms,
(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 7.5 leef9cezei./: hts4,-tv sv *0 m. How much net work is required to accelerate it from rest to a rotation rateetz,*h4lcv sfe-v:t.9s e ei / of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls withv8yk4wx4iv9 aout slip94 vaxwk iv4y8ping 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 it:/f5ydrow avq-4z;f ts tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Usaf5 vorq-;w:y/ fz4 tde conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ba8x wt9j/k svw0t;h9posgfriq 6e1 04all rotating on the end of a thin string in a circle of 01vf 89 eqt; p4tsswhgx6w /09i jkaorradius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a /s rdx o4)lxmbrh s0712.8-kg uniform cylindrical grinding wheel of radius dxrr m7/lhosx1)0b4s 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 hisji87ko l953zfp3w1s wp elg 8u side, on a platform that is rotating at a rate of 1.3rev/s If he raiw 5kz o9efu 8 gi83lpp1j37wlsses his arms 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 Fig. 8–29) can reduce her momp,9, +kigb(ng q2 nqtsent of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is hernqpgt9sgi,qnb 2+ (k, angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotationyse 3cg pbanjb/; 2;s2 rate from an initial rate of 1.0 rev ev;;j/ 3bges 2ynbsacp2ery 2.0 s to 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 rotating around 1t0m sbhze 7i3xaz x2; qn/,oqi5a3l ca 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 andx;0maa7l1zis5 etbxq q2n/hic oz 3,3 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 ska u0j),olrnr) oo+ 6sjbzh0pr 4ter spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentume 8,zk2 ssvzhe6ff7+v 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 cylindricnsio0/0x3ev s al disk of moment of inertia I is dropped onto an identical disk rotating at angular s v03oxse0ns /ipeed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4u,-fk hp6,m,rzz;du;pel7* a s//bxz 0p fyx y $rev/s$ 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 8medd k-sjs.n0elmc7q;-* 8d xox5or-go-round platform of radius 3.0 m and moment of inerti8cd 7nsroekd- oe;5sqd0xm. -8lxjm* a 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 velocitybo /a4n-os5j6lud.pcb5 cbj: of 0. / d6bucpa:so.nblc-4bj oj558 $rad/s$ 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 cood0a2 )4nk dji6jz -gillapses into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming thedi4g 6o)z-i akj2n dj0 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 1 qhj4ww7q;v;o 20 $km/h$ 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 g6d8 7ff bwrj c/8qz;b$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, i .kjfn,5)xhxlr *uw+:mcz m:nnitially at rest, that can rotate freely without friction. The moment of inertia of the person plus the plcj:l: .nw km*+ x)f,z5r nxmuhatform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diamet68 h9mt3rle(ifgtuz(d b4 :nw er merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia dtb9min43fweztu hl g 8r:(( 6of 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 th3*t levvlhtvj6.+pe6e ground with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope un6t3pv. vle+h* t vlej6winds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same sidbq 6tl(t b9y-wx q,o:je always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbitingw6ybqqt,ojx- l(tb: 9 the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 1 myjq *kaca) mos;7d8.o/$s^2$ 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 uryic 2: d zjbs9k4mde2;cc; ozl-f v41niform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval aez:zcy4l ;9i4v- co;jkmdr22bc1dfs t constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cyl0 jo65 wds uv7fzu*l+kindrical 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 diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches thekl 5uw6o+uzd0 s *f7jv 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 isdwh;dv)9eq+z 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 siv40pebgol0 (ux4g p 6xze 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 neut ( pglxogubx60 ev44p0ron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use enelg 6+ qyc/+ lvr6h(jysrgy stored in a heavy rotating flywheel. Suppose sur l/q+jly+(yg6c6hs vch a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to 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 an f(kt0qko5fn4t p5 r/,3ya jqr$H_2O$ 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 roll9+ eb1p wee:wmtp7 bp1ing 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 r7 czkr b9nxfm78+2my6tvzo) 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 released. At the moment of release, determine (a) the angular accel6x t+)rm7fv o2 yczk mnb9z87reration 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 the floovg)rj91emz f)+ wi ig3r, and we want to exert a horizontal force F at its axle so that it will cli v+)9)rw f3ei1mzg ijgmb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a turn w:ro4u/zsbi x:a /cm2okk -f2 mith a radius The forceam 2o2ck/: m sobifk/u r-z4x:s acting on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5 m5iah2-d1e -okpa 0 b:nn;z uqi and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 ih2d np-1e0;bq : ozn5-ai ukas. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thin rod0x )r ql;v*ftt) co7ngse4hh1 s, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outch1 v4flx;)*tnr 7eqgth)s 0o stretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembwl* v; zb7x:hd*ajfn- ly which consists of two cylindrical plates, of m l-hwfnj:x*badz7 *; vass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 loopeita(unlq 9 3/uss9i3zd rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point of the a3/uiqusl( 3n9zs it9loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assume 1r ;ldkq;a3 vwlhu7j(i.hf ,z $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutinxrid+ 96yq5 yons 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 the anrdn5i yyxq+9 6gular 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