A bicycle odometer (which measures distance traveled) is attache,le dmnn-9-xczoz/7u d near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24z7o9cdn -m,xl /en u-z-inch wheels?

Suppose a disk rotates at- 3ghxyx)o6og constant angular velocity. Does a point on the rim have radial and/or tangential acceleration?6gox-hgy 3)xo If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the angular velv/ )+rjcvomry+.hyx: i tp1j*ocity $\omega$ Explain.

Can a small force ever exert a gr wdi7ws5 3dh6ml8s t8keater 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 uya4/nu2:j i a bhgh(5with your hands behind your head than when your arms are stretched out in front of you? A diagram may h u ahha25ng4i/:(yjub elp you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel pd8 t4yz6cq) pand three at the pedal cranks. In which gear is it harder to pedal, at6pp4c8 dqzy) small rear sprocket or a large rear sprocket? 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 fas00md0o+)*ej6 liohqc w smqfxqh 011wt have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why thimhxqfsi6 0l *qdqoj1 +1)0hwc w0e 0mos distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow ben *umfbp;pumrl/jl5q3 y -kz8s,7h 0 zam?

If the net force on a system is zero, is the net torque also zero? I/uml b7-s,pof phb nl2q11 gf9f the net torque on a system is zero, is t1bqsmln obg- ,fp7pl/f u2h 19he net force zero?

Two inclines have the same height but make different zhmucq 94 8)v5m4trlvangles with the horizontal. The same steel ball is rolled down each inu5 4z9mmv4 t)qr hlv8ccline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) dna;gtmc-9av*sh0 zs4 own 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 scas 9stzvh- 0g*nm a;4peed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius *liwf ej/)okc/os1saw 01u n,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 kinetic energy at1w 0o1, u *cj/isso wnkl/fae) the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most moving or r/) xfqlijc q36p aey/7otating objects eventually slow down and stop. Explaip/qjxla7/ 3)ce i y6fqn.

If there were a greaxvw;f(vcj wc;22 i0o ht migration of people toward the Earth’s equator, how would this affect the le2io0 fvcx(vwcj2 ;w; hngth of the day?

Can the diver of Fig. 8–29 do a somersault witg6i y-wi d2ljf +-8jp)es:ksvspm00 nhout having any initial rotation when shms ey0 ) -fisp0w+n-dl vjiskj2g8:6pe leaves the board?

The moment of inertia of a rotating solid disk about an axis5 cke(*hxo vj5j 0y-as through its center of macxoesyj j -h(0avk* 55ss 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 in each ot9bdywi7s f7p;i3co+ bzx: *k utstretched hand. If you suddenly drop the masses, will your ayfwsozi3cxi p7b*b d +t7:;9k ngular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, ov7qxf-f :cax5y , pbu,ne is hollow and the other is solid. Descpfca v-, 5u x7,xybfq:ribe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle poi:qp8urex /+ h :o4rjr od,1yxfnts 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 trudr 1oej:px+qx yrhf8,:o/4 angential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turntable. Whb*7h 2t6nc axugrf* (qsf(cz/ at happens if you walk toward th *( 2ru6*z 7fxgb fs(hcqna/cte center?

A shortstop may leap into the air to catch a ball and throw it ya9d t,ym5xpirap;0 ( quickly. As he throws 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) 0pxr5ay td y,p;a9im(. Explain.

On the basis of the law of conservation of angular momentua2+y6h 3ff w7z kci)jdm, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways ty3hdjaf)z2 c7kwf6i+he second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (a) 303 ll8jsy5j+vjodm52+ .l ausa $^{\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 amas:ok( 4m 4jqzozing coincidence. Calculate, using the information inside the Front Cover, the angular diameters ( km4o4ozs:(qjin radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at th388)cwotnsuf x2b yz* e Moon, 380,000 km from Earth. The beam diverges at an angley83) tnz bxo*c82 wfus $\theta$ (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 turned5p05vxc//adpa 8m u ft off during operation, the blades slow to rest in 3.0 s. What is the angularcxp au8mv/p5aft05/d 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 ,; qcx agg9*mztq*j yla,0q *pmakes 15.0 revolutions, what is its diameter?9y0**c agm ,glq,qa*;xpj z tq    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolution;w (0(u ebj/o,r3ctxb1ydg hds do the0eb;o3jhby /( t(x1d u rdgcw, wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate is(il14(-e 0hosex ley ts angular h4l 1y((e- eios elsx0velocity 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 maket*dco/0ml6 (es nggh;s one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child sel/6( *nohd 0 cgmtesg;ated 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)lxy+ )vf0;hq,jn2 smk in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed o*via*s; e*yhs5ok1q hf 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 cmn o5n5 +veg- v;smn:nm from the axis ofse5: m-mgnnov n; vn+5 rotation is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 dtco8n.6j5v3c 1 ixne rpm to 28v t5jond1x36 c8.ceni0 rpm in 4.0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radiud ,)lv fjrhrzbxq0mj b835g7: ef0+-mgn: kq js $R_1$ 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,di/o60* inh 50ss th3how .z 00ldgjyf astronauts aboard the Apollo spacecraft put themselves into a slow rotatij0li53* .s0ds ihwn/gd0o 6hf zy 0ohton 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 8.5 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 in 220 s. T5p2(6wbqubaou, du s* hrough how many revolutions did it turn in this ti5 u *bb2suwa6udqop(,me?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5qybgtqypb a w5;-.e/ 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 stressesj u2rw9yp7lb/a2,oo,nep ud+ of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching it ,l,9upj u 2bd2waron /e+oy7ps 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 acceledt6ex utv5+grvv.l d zs ed2:.qg4;:n rates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the w t2tlr46 +dngu devvdx;zv .q:s.5e g:heel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turmjy 0a yv3e*+totipq6/- 3phnns 1500 revolutions before it03+tjo6*3 h- itnyamevppq /y 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 uwyp; rh 6bbqkf2t chpqh7,931niformly from 95km/h to 45km/h 23p h;ckp9h1 rtb q6b 7,qwyfhThe 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 the car reduces its speedybhrxee x)3c:8n 1 qx: uniformly from 95km/h to 1re: q8hxxxbe3n )y:c45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weighlj hlj1qjiw+s, -:/r yt on each pedal when climbing a hywsj +:ji1 , h/qlrlj-ill. The pedals rotate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a forcaxqiumru0x v9n8: c .*e of 55 N on the end of a door 74 cm wide. What is the magnitude of the torqu.xu8qc: xa09*m uirnve 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 gmxhr,s kyk du/05.z,of the wheel shown in Fig. 8–39. Assume td5 ,g hmk/kuzsx,.y0rhat 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, a:6 w+ni(a(a6 rkleboh hmj )s7re attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude a)ea j6(n lsa(:k7h o6mhi rw+bnd direction of the net torque on this system.

  The bolts on the cylinder head of an engine requ1n mpy k1)ais5ire tightening to a torque of 38ms5y1 ip1n)ak $m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radius 0.648 1y/zzj gf 4hn+m when the axis of rotation gz h/j1yz 4nf+is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bpu)4 +k8) ejd.j7djd ;t1pq ncy 3rrzcicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. T ek.)c3dpy+7jt)r dzq jr 8p;undc1j4he mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end o0tf2 e 2t,lbk0pd)v j /i ycim3ao5sw0f a thin, light rod is rotated in a horizontal circle of radius 02 5cv3p0eti2sawy0kjlmt bod)f ,i/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 rot1k ow7pyiqy6.q 45ul1 kzaf3.z-fowqating at constant angular speed (Fig. 8–42). The friction force between her hands and the clay is 15o3aq zf17k. -k.of4iy16 qyzwu pwlq .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 objectsi w(py w5cxctp*.tg a4*q ae* j+;wty4 shown in Fig. 8–43 g;(*j yq caix5c t4awpyp+tw.w4e**t about
(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 consis9g4md *5zd /6 lhzcwwrts of two oxygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellit:mop3c e9 - t *,ds njb;ksa96iqh+miqe spinning at the correct rate, engineers fire four tangentin hmm q9 sbk+di9*p ;qcojei ta:6-s3,al 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 cr*(d/ qh tp+e8 hs )e+uf(wmprm and a mass of 0.580 kg. Calq8*u)e/pmr +( hhtpsdferw +(culate
(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+ boze; 7cca9sr 3y xzsfrjqaa,4 /z4., accelerating 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 smsp:c hu 2)jkp*all 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. Calculasuch kj*)2:p pte the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut of/ k7. nz750euzfddbopf and is eventually brought uniformly to rest byzu d.k7/dzo n5fb7e0p 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. 83tvcsrjbjxw 54mjc2)vsw r :*v jd+4*–45 accelerates 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 thrownx cgiaxt 76nz 5 mcd.-l+b*4ehyxn,:a solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly gx x5 :elyi.t6hcx4md-ban+*ca7,znfrom 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 rwiky- 9tah*/bod, as shown in Fig. 8–46.yw k-/atibh9*
(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 masses,h 4:,jyjsxvx,q c9m yjluopdh/ 3z:v ++gov )* $m_1$ 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 fot17wi6z/h*qxee g ,k( n4gl urom rest within four full turns (revolutions) and releases it 64i,kzu ntgelh /7 (xg1wqo*e 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 mow)rdl; he /h)ohs2xft ul8ji/-0r ,ekment 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 develo0r*vwgk:(c r xps 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 9vsj,i 8 x2e1mvpc78 mbglo1 l/.0 cm rolls without slipping down a lane alv iso m8/17bj8l ec2, gm1xvpt 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energq kvwsh;wxutl (yj.2f 4s+ j:;y of the Earth with respect to the Sun as the sum of two tetjuf2s:+.v;xl;h(y k ww sjq 4rms,
(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 1640dfwjk5 . o-f1p 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 s? Assume it is jw-5o f1dkf p.a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and 8 p+)dzbkro7klm,8a prolls without slipping down a 3b kzkp)+r7lpa88,mo d0.0 $^{\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 , sym iquf+d-*:hqu)z 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 befd+)qh umzf : u*,-syiqore it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of aofg) (1v qzr(n 0.210-kg ball rotating on the end of a thin string in a circle of radius rz gnqfv(o(1)1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grib( r0o8m(qr ( 7jxau0h *vsvainding wheel of radius 18 cm 0q(rr8b(s0m(*oivuv7 xha aj 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 u,qp;4eu1 si ycn), oiside, on a platform that is rotating at a rate of 1.3rc ,u;s4,e iy)pq1ouniev/s If he raises 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 hem( 7 6wf ensl )il0x3/b 4sjlnxpp9kn*r moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angul4ppw *f il 3be(l/n 7s9xx6n) mskj0nlar speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an7,udeb svl 6; l56hj5axdph z9 initial rate of 1.0 rev5s6h9;ub l5 7zxpja6lhd, ved every 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 a v0pn3 va0q+w9ndg 2bak bl uy+,ertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 wba239+b n+ ng0d0l apy,kuvq kg and 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 spin( fll9/d.y wljning 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 mome ;30nblcpf1,kla5 hx wntum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia o2yk);y rlagti :kp5p7 jg85r I is dropped onto an identical disk rotating a: a rr j)kyo;y2gt5ikp7l p5g8t angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns ato7glo q7mwx6y o5y 6)y 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotating merry-go-round plsf:sn21xu l a) swnd.*atform of radi sfs* )nasx.:uldnw 21us 3.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angular velocity of oiomhtw; /n jc(r;5fb +)h.bo svx)f(0(hfh im)cb/ow r oos(tj ).+fx;bv5n;.8 $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into8) /f ps8xzqe0ug.cg q a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no / qesz88q 0fx.gpu gc)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 z i feq) 0j9ezee:5 twq*),i(9lfur caof 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 ki q.:xq 3ig12vxrp*n pgz2w 5$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, that can rotate freely withoukfbk.euvt,vu zi+2: x g,1x3pt fri u3zkiefu2 v ,x.tv,bpg+kx 1:ction. The moment of inertia 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 6.5-m diameter me*rk1ugxi+ a 80va ev5krry-go-round turntable that is mounted on frictionless bearings and has a moment of i*a xk5vru0 aivg8+ 1eknertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of thew0mjs x-;cuxgh74 xi 3 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 le -i 7x;s4xw j3h0gucmxngth of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that thbie 7 7i ahjk1q5wm.c-e same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (abou7 i5c-h7bwek.j1m aqit 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 from rest at a rate of 1 m/esv l21b0 yr()b;fwnga lg02k$s^2$ 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 t). iynmo o1x)m 7rc )wvw,*ooshe shape of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor mo morco.v)i,xw 7nw*oy1s)).    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, c+,omdl o 01vyd y.4btkv;s7w 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.sy 4d,+cvbm0 ;d ykotwlvo.170-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular gg5,/ada2(vgurxz/f 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 of0xg6 4mm u2mtsbx/zb vxh045 d our Sun, but with mass 8.0 times as great, were rotating at a speed of 1.0 re mxzb m0vdms24 /t4b6x0g5xhuvolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low4yjmv8 t1d5 u)pvdy f(-pollution automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass4 v mydftj()v851 dyup 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 illustra gla ifl;wb 5ouokcz*06c8pm )7cd ;n)tes 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 on a horizon1 w)gkyt(guald0fp,dig i0 *6tal 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 c h )((v1chqgbban pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. qb(hgh(bv1 )c8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It wbp+24q5 pvfu is standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climbq +fv4 wup52pb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist travelingu):gc+is;jc v with speed v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist g sv ijcc)+;u: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-h*a+ :.+.h*jkp dgq9u cg8jtel nq3 azo:e4qgkg rock into a sling of length 1.5 m 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 s. What is the to*qz.a njgg+k* 9c+gpe :dh4: a ueq oq.h83ltjrque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skate4f q*1 yj0- lloqef9+nx1m svrr’s body as a solid cylinder and her arms as thin rods, making reasonable estimates m-lsy4jx10nqvf ql r+19*f eo for the dimensions. Then 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 cylinx9rmp zyj+o uqb(i84pm0 m45gdrical plates, of mmj(oxbp04+u4 i rymp8 z59g qmass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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/,d;trb3 g x +*bzsarx track of Fig. 8–58. What is the minimum value of +a*dsrbx/bz;, 3x tgr the vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not ag 17 czgj0wytxw9 :x;i j2rlm:ssume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed un;v ugw xi//bd 2dh0-ntiformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the a0n;dxitw 2/bd -g uv/hngular 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