A bicycle odometer (which measures distancem5wfh,f +as m4 traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-ifw5+sh, mm 4afnch wheels?

Suppose a disk rotates at constant angular veloct4k,xl8k6cf( 5 dq rvtity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?8q ,5tdc ktrkfvx46(l

Could a nonrigid body be described bzn1h;2l g5 .u.3cv7uffzh xlhy a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than 7jlc 7fvxk p/8a 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 fkj ahg)d)v) 6your head than when your arms are stretched out in fragf)vdhk )j6)ont of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the pedalp6naik4*-x:deshqf -u ;-lpqz q )ef 4 cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which ge6xn-44*e pkfs-uz f h adpl):- q;qqeiar is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on bsu x ;:oqic6 x.rdryi;6b :c9feing able to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distriu6xrcdf; i o96:isy.:bqrc;x bution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, 9 3zrtgqlb h+ 7d.rmqcr99t,snarrow beam?

If the net force on a system is zero, is the net torquvlnmy /p4 qk4/ 9ng8klz+ f*yle also zero? If the net torque on a system is zero, is the net force zerol+ 9k/klmqv4 ypgf*z/4 l8 nny?

Two inclines have the same height but make 4o h+dh 4g28ze(ql6:yoq zwp pdifferent angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the spedqe :ohzh+py (68pzwoqlg424ed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from restuutu j-khw3v6)ot3 8a ) down an incline. One sphere has tw63uh )tuu3 tvo-j 8kawice 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 energy at the bottom?

A sphere and a cylinder have tczj2be * bpk5dsp16 53 gpet5uhe same radius 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 at the bottom? Which has the ggpbc*6jd1b5e3zst pek 5pu52 reater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most mo7/)2kpodw7xspd vh 7cving or rotating objects e/dh vp7cw7xs2kpod 7) ventually slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equator, howy.3fkdlz .c8w .7q 6eylxjb.hz+ g t(s would txk .6. cll 7y.h(zgf.z dy3sb teqjw+8his affect the length of the day?

Can the diver of Fig. 8–29 do a sb eq8 x,bi;(uuomersault without having any initial rotation when she leaves the boaruqx,bb(e; i8u d?

The moment of inertia of a rotating 4yr1w*g+kb .oj dqlgr6vo(jpx *t5 q+solid disk about an axis through its center of qgr*.po1b+y gwld4j j+ *kv6q(5r xtomass 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 2n0z:uj0s 56)mt (kkg hrsd6,ajhu qv)-kg mass in each outstretched hand. If you suddenly dk:6gszndj m )t(6ahuuh)k ,srv5j 0 0qrop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identi xj-l:gq5kgj ,cal and have the same mass. However, one is hollow and the other is solid. Describe an experiment to determine which is wx ,q :g5j-ljgkhich.

The angular velocity of a wheel rotating on a horizontal axle poin/9s :aqfivw iqqfpimj*f442d w0()zots west. In what direction is the linear velocity of a point on the top of/p zi4 (jaw:q9fqq2md oi f 4*)fwvsi0 the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freelyw nonsrkx+g2d x93(t + ,+qjfn rotating turntable. What happens if you walk toward the w sj d nf(9++trxg3k2qon+ ,nxcenter?

A shortstop may leap into the air to catch kbapg9/re,++cvbjjm opl.2 1 a ball and throw it quickly. As he throws tblv/9ra,c pepjjk++o.21 gbm he ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law e1w7mn2sl3r cof conservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operaslwnm31 7ce 2rte to keep the helicopter stable.

  Express the following angles v xogd pk e3q;i(0t)rc5oc5e s/;ozp2in 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 because of an amazing coincidenc 0gtr+xhy pbgf +ka4fqy++ e;-8ms.kke. Calculate, using the information inside the Front Cover, the8+sfexta0b;. y4pkgm k y+qkh+ -frg+ angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directe2t 2 qe;ktku/hd at the Moon, 380,000 km from Earth. The beam diverges at e t2tq;/h uk2kan 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 blende9l3tz nf.qsjok)n j, 7r rotate at a rate of 6500 rpm. When the motor is turned off during ookq9zsjl jft). n 3n7,peration, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to an:g,tecma6 ;st z ufgychm6)x3 ./1mgtok 1m; mother child. If the ball makes 15.0 revolutiogmgu 6o mk/tcfs1mx.m 1 3za;6,m):ghct; y tens, what is its diameter?    m

  A bicycle with tires 68 cm in diameter tlxap 3hfo9)/8s gv sv5ravels 8.0 km. How many revolutions do the wheels makepgsho8a 9)/fl5 vxs3 v?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calcoetv-x d3k5 ,1c) ztokulate its angular velocitokt,k dvc5 )x1 3eoz-ty 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 sw d 9qf;+bboe, 16fn:iscw0w.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m frw1:eq d ,+6ofisf bnw90bc;w som the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular veet ,p4u)*kgyk)wa f6ul(ord . locity 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 offq(u,4ul ;tck 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 r.z9bg/vg7tc* g( kjlrwmke + 3uy2rq, otate if a particle 7.0 cm from the axis of rotation is tolm 7gkj9 rb.gtg,ezr/ *y2ukv3(w+cq experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about j r+fc2 rcjd/r hg/d7*its center from 130 rpm to 280 rpm in 4.0 s. Determh rcrg*c7jr2d j//fd +ine
(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 radiu6oxr sj+e(jt/ibv.7 zs $R_1$ 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 Mc3af ;)7xcq4font3 :uan .uas oon, astronauts aboard the Apollo spacecraft put themselves into a slow rotatio7at:f)u3; snacounaf. 4x q3cn 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. Throuumxfq(g/go:gt8 6 .s5p8t gdygh how many revolutions 8y5 gpmtgx :gqo /(68.fudtsgdid it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Cal0c da a07 gu7pci* 0 glt9esmis(xh1u*culate
(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 61/r h0kahnjpc kyx5, in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions befor/ p1rkj ,any c06xkhh5e 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 frodf+vpd :)hrv2 m 240 rpm to 360 rpm in 6.5 s. How far will a point on thf2 p)vdh v+:dre edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 1500 rmm5qvj8tv2;9bi ikb1 evolutions before it comes to a stop.qtmm81b i 2kiv5b9 vj;
(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 s7g+rhbvgd/ ) w5j ue;zpeed uniformly from 95km/h to 45km/h The tire/wv5g;7)dr+ jzb ehgu s have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions ajj1 rt6t. x8 k2be4sysupyye9 /jwa*.s the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.80 serjt4xk b 92w6s /etyjy*pyau. j18. 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 g p ln,c9h x;o1a(kb3iher weight on each pedal when climbing a hill. The pedals rotate in a circle ofa hnok,c3(; x9g1bi lp 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 azqx :4od1 cup+w6ii4l door 74 cm wide. What is the magnitude of the torque if the fo64z:iqipw1 lc u+ dx4orce 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 ;ieb4sl5m8 f c*(0zm 3awzm b axa1c1raxle of the wheel shown in Fig. 8–39. Assume that a friction torque of 0. aza3maz;fiw1m l 5es0ccxb*4br(8m1 4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, ah a7rizvxqs1e: y-u,7re attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal qha7 yis-e ,:vru7xz1 position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require ti2g ymak- gkwn6xf1e0 ,rjeom5 j ()0vyghtening to a torque of 381ya v6g ym5f)0xjrnegk,-ek2wj( m o 0 $m \cdot N$ 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 m; zeu57qwe;pkr 5n+g2n97o qzk) toz4iekc 4 v when the axis of rotation is through its k5z+kc 95 k)eown4vz pti;ugq2r;7 7oeen4q zcenter.    $kg \cdot m^2$

  Calculate the moment of inertia o(;c koo)3 zpz7mlh(y uf a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub cu(;z)ho 7 ypl (zcm3okan be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in a hor 8ubzqy c 0eptp5zl4w:8:o9kgizontal circle of raditocg: 09keb5zz y8ulq :48wpp us 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 aa+yykjyu c24 fji7/, c potter’s wheel rotating at constant angular speed (Fig. 8–42).jkc yyyj 2 u/ac7f+i4, 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 objects i 3r) a8 gfw04on9yoclshown in Fig. 8–43 about8rgfoawi3 l9 nc4)o 0y
(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 atgo5n -*ttx/p(tvm2bz oms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the corre /siug9 hxm11jct 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, what is the required steih1sx/j1m 9 guady 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 az2 /5,duum8tfn5 ydxfnd a mass of 0.580 u 5du/8fmynxt,f 5zd2 kg. Calculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, acceleratinv 0q wd ;v1atu+ov(-9ajmjeo*g 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-round an8 ifwlu7(iaqad6* vu 3acb9k4 d is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torq kd i67ulfa cv93(8wqbu4ia a*ue. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotatinf0-g,uur7p d0+ bcr b fb.p71fp a(vfag at 10,300 rpm is shut off and is eventually brought uniformly to rest by a fr7fac0f -r 0,pfp(rvufud+ g1bp.b 7 baictional 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 6 qe*rgm bmm-7d6y 2usball 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 bgr(+ r 0iovs4of the forearm, which rotates about the elbow jo0ios4r +(vgrb int under the action of the triceps 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 consid+t. k j.6 b:i+pqttiaxered a long thin rod, as shown in Fig. 8–46t tqtbjx:i+ip.+k. 6a.
(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 masse7t/zb5:kmgv 76 t fxxws, $m_1$ 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 accel1k+03pj1 c6fpgkki kx erates the hammer from rest within four full turns (revolutions) and releases it at a speed of fj6pp k31xc+kkgi0k 1 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 ha/e//nbk k +mihs a 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 developsvll2lr b/kpqb8a)+v,)jtak 5 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 r4s u x8 tj vr*qry/ly7l65a8bbolls without slipping down a lane at 3.vsbb 8tr4/ 586 al*qylr7xju y3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth wit l,ezc )8at)gih respect to the Sun as the sum of tw8)g)iza te lc,o terms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius o a,;c, ce:b15xpiqja kf 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 a c:abi kap q 1,,x5ec;jsolid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from restel v/z d9dwp7iu1 u-d( and rolls without slipping down a 30.0 (p7-wd/ze9u d1lv d ui$^{\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 9,a0l w nknqawg) nq,kj8;).ltws0i op3ot7a balanced vertically on its tip. It starts to fall and its lower e 0 sq8o97)ng;)w3qk,o an twtpwl ,ak0l.a ijnnd does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the endqo1uckf4 u (uvyv8 l:3p9qh 5d of a thin string in a cirq 1 fpyvu v:4ouc93d8h(k5lqucle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum3+tasy z1nr/kn9t tf; of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotating at 1500 rpftzk sa;+39 r1/nttn ym?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands atx1 5h;kb y5dpv his side, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, v;by1dxk 5h 5pthe 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) can3n1 jyc:4sns 1u gx3/d adlo3s/h kxf: reduce her moment of inertia by aoc/dg :4xj s:3xl ua 13snn13/h kdsyf 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 angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase hercmpr8-)jmig sv2a,3)ij kc6izs,)m x spin rotation rate from an initial rate of 1.0 rev every 2.0 s to a final r3 xm smc grjczi ,jvsa )i68p,m-2))ikate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis through its center up3gg 9x.k;ih at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the whe3gp; i9h ugx.kel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentunot a5b0(i+mz m of a figure skater 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 momentum of theorb* y xlln,r*ys3 sig-h1 +d+ 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 diskc6.arui3*wce ,gjsrg4w * , e3zv bmb9 of moment of inertia I is dropped onto an identical disk rotating at an z.i *srm*bg 943 rvce6e bw,gau3jc,wgular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at py ;y. v9, uitj),aeic2.4 $rev/s$ 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 stancl 2uoggt*1.1 * yzjeeds at the center of a rotating merry-go-round platform of radius 3.0 m*cz1j g1l2ge ot*ey .u 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 w 2pz 3ipe0p/ocith an angular velocity of 0.8p ioe/p02pz3 c $rad/s$ 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 eventualvx yzff23f (c*ly collapses 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 the lost masxyfc( z2f3*fvs 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 e( ,)o*tq ) c 2djgmvrr.qvto.axcess 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 n-gss :-i6kvzk 1ownx95ys 9qgm3 f/ f$ 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 5ska/2) yhoirk 9g;2xkiw a/eat rest, that can rotate freely without friction. The moment of inertia of the persoe r/h522i9oia/gk;k)x ya ks wn 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 edgu hi8fc2xur zw /c5*6 zssg/9se of a 6.5-m diameter merry-go-round turntable that is mounted on fs/z x iwh95ru8/s*62 cfz gcsurictionless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of the rope l r k/*.q0ldzi9xcr8-8kcpivx ying on the top edge of the spool. A person grabs the e/ v90kp.cxq l 8irz-rck*8 xdind of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always f02 2s(huvzwr aaces the Earth. Determine the sh2avr2wz u(0ratio 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 orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rajy:jq)q,6 ixo,cd3igdv j lw3 xu20 u3te 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 ;s)fw -r hf it 9ocao7o 0yx402vrth7ua uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleratvrafft-to r4y2)9ohh;7x u w0oi70c s ion. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cyl:zr8p w3d6z(fnaj 8qwindrical 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 the end of its 1.0d8jzqwa (8pw6 r 3:fzn-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 speedgw0oo wru8a/ld 3ja i 215xqq1 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 times as great, wer(vj850gw0+z latkga5 e d 7hnce 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 i50cvj hw a8g(e7t dk5lnga+0 zts 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 autoxh:xg ;kpm,:omobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, gxm:;: k o,xphuses 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 illustrat9o6fdr/ x .lcies 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 /,t5hyy nxn.p5d2lforolling 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 am-6g c4d2xsh fnd length 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 released. At the moment of release, determine (a) the angulah gf 62mdcx4s-r 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/k p stkm0*piz e.4wo8 vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step h0is * kpwmo48ezp/. tkas height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making7 ,8d yrz/jumh a turn with a radiu8mrhj7d/y,zu s The forces 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 lfxop*3hr;vky;-4y sbmqp(a m((ef 3)bsdm: 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 torque required to achieve thfs(yq-erfpoh :ydm a3 l)( ;*3p(mx4 ;sbbvmkis feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and herkf+iit5sa7 3t 1g* faz arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against ffs +t*g k azt175iia3her body.   

  You are designing a cl l2 vg zcvx2f uf7f3+dup;zz9/5ucvt-utch assembly which consists of two cylindrical plates, of mass fpv5 2vtc2 ;97fu xu dzzf-lcg/vu+3z$M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 loopedconl from,s 7+dx . c-u46kel0 rough track of Fig. 8–58. What is tf-rlcuo ls 6e .mcdn4+7,x0 kohe minimum value of 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 assumelgavx bp 6)x;twan gk 0866;hg $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions a:u 5+mkda .tbm4sfl 8mg g.l*ek5 r4qys the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What ls+. 5 ke:d8m*km5l4 4a uf gg.trmbqywas 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