A bicycle odometer (whiypye+ (su59god(vc o4uu ko6:h 7vm:ach measures distance 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-inch whe 4 5oduu 6c+9:pv7:oyogm yaus( hk(veels?

Suppose a disk rotates at constant angular velocity. Does a 44c q 1itf7b gk3ndr+rpoint 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 b4kg f idq+rn71t 4rc3cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be cx 5g eydbav(d5om-)* described by a single value of the angular velocity $\omega$ Explain.

Can a small force ever,3 h)b fca3agw exert a greater torque than a larger force? Explain.

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

Why is it more difficult to do a sit-up with your hands behind your h3a3 wk;lwoa, ; ay:alkead than when your arms are stretched out in front of you? A diagra3 awlk,k a;:ay;3olw am may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the peda fmpn3k0s,s4le6+ ox ll cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder tl+0oen fm3,ssx6 l 4kpo pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender mw6tny*dcar0k. f ta pt6 al-lqk891an26n ;v lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why tha2 n.*ttvaynmn1cq6d ak0-6 6l rfp9k;8w latis distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow bea(df0 kxboo0 a4i6z0g qm?

If the net force on a system is zero, is theo / ,a(fdced5t net torque also zero? If the net torque on a system is zerd e 5odc(f/ta,o, is the net force zero?

Two inclines have the same hei41y6z3l*nbex tn cfe0ght but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of tx6e 4tyef0cl*1 3n nzbhe ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down adfsl 26o,l+vuao5p2s n incline. One sphere has twice the lfa vloosu 2d5p62+, sradius 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 the same radius and the same mass. Th /qc/:e /yg+/zrggopqey start from rest at the egqogq+ y: /zpr/g/c/top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentu8u:+tm+ xiozp m and angular momentum are conserved. Yet most moving or rotating objects eventually slow dox 8z+u+:motpi wn and stop. Explain.

If there were a great migr+ph8hovh m6l* ation of people toward the Earth’s equator, how would this affect the lenh+hv8 holm*6p gth of the day?

Can the diver of Fig. 8–29 do a somersaulwmccf 4/21*pewp ge4zt without having any initial rotation when s cz e4wwcpp 24efg1*m/he leaves the board?

The moment of inertia f 3vwqx(w-v8swii+;i of a rotating solid disk about an axis through its center wq;v3 +x i(w8wsviif -of mass 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 eacmo4 8;ix3kozbh outstretched hand. If you suddexoo;zk8ib m34nly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, onep 1ynm0 k8d* gi/3gkgbyx+:e c is hollow and the other is solid nmg/ 0pbye+gg81k: kc3dixy* . Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle pointg) zv481yqlozn.f 0p3zv bk8e s west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular spe z4zygn)bv l 3k8qf1e vz0po8.ed increasing or decreasing?

Suppose you are standing on the edge of a large freely h;h13u ok1 ivbrotating turntable. What happens if you wa v;h1b uok3ih1lk toward the center?

A shortstop may leap into the air to catch a ball and throw it quickly. As he-htd nwk/o :c. throws the ball, the upper part of his bodywd:/k -t.noch 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 of xe(cg6ihyi46 conservation of angular momentum, discuss why a helicopter must have more than one rotor (or pycghi 6 ie6x(4ropeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: zjcoo4t7dkb;b8l 8 k-(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 amay*bt fq(uz/+z zing coincidence. Calculate, using the information inside the Front Cover, the anguz(yt qz*f u/+blar diameters (in radians) 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 fro ixh26 qq47lbvm Earth. The beam diverges at an anghxqb2 iqv4 67lle $\theta$ (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 turnl h9a,*p fco6ped off during operation, the blades slow t6c,ha* l ppfo9o 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 another child. If the ball j)q r/ hgg5pj3ajo0.imakes 15.0 revolutions, what is its di.)higg5 jrpjo0j 3/q aameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutitovnap b)9-)ge n 5hfyd((pp -ons do the whe d ya5onf)n)-(ehg t (-9vppbpels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calcull4 rf zfs 33j0;nb7:bgohdi8wate its angular velocity in7z f:jn;lhisd g o0bbf83w4r3 $rad/s$ $\omega$ =    $rad/sec$
(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 make*;iyvsu)vvy 4g8w fhrn.;g ;ks one complete revolution in 4.0 s (Fig. 8–38). (a) Whatrif sw hu;y)*4 vng;vyk;8. gv is the linear speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbt4r2z( ;hao as ylmqr*j-d 92sit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a vtn:m0yz(i t-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 v7lr,6,n0is:venq1 pw g fgq) rotate if a particle 7.0 cm from the axis of rotation is to experience an acceleration of 100,07rq li6fv:ggn )qws1np, 0,e v00 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly abo r2qvk4-tk4c3dkyo.rh n4q5 lut its center from 130 rpm to 280 rpm in -hqoydr5k 44vk 2tr4nk3 q lc.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 radiu5x;;hjooo g68xo zp9v s $R_1$ 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, astronaut p m-w- /cf037/naevueq6gyy xs aboard the Apollo spacecraft put themselves ey7vn/g0u/c 63p -ya-x wqmf einto a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 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 rmdev8k6h m 6*gpm in 220 s. Through how manv8hgm* d6emk6 y revolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calceu bkm*7. r3c pgno)0culate
(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 stress 85j1q; td rcptu; i,zw5gx(joes of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complu;55cd j xt rwtjo; 8pqg1i(z,ete 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 accel;kbv)ycbc)+ jg- )sxi erates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the whe;)) ickvcjxy +s-gb)bel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It o,oxrnm n/e99c bv ;p;turns 1500 revolutions before it comes to a stop. nnvo99c r;;/ xo,mpbe
(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 51 vtvdm)u sk5 h guy/no437rgits speed uniformly from 95km/h to 45km/h The tires have a diame vg51t)gs7/vou4 rh5nk3dmu yter 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 apb l7j4dl5ux8s the car reduces its speed uniformly from 95k4p78ldxbj l5 um/h to 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

  A 55-kg person riding a bike puts all h)- wydbs)gju 5+kq 5u22utrh yer weight on each pedal when climbing a hill. The pedals rotate in a circle-hqk5ruu ydw))b5j 2u2s+t yg 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 wi; mqt 0(ltiaoqe5+jc .de. What is the magnitude oe5laqq+o( .imttcj 0;f the torque 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 abouj ; /csz5 uba0or(5qimt the axle of the wheel shown in Fig. 8–39. Assume that a frictiorbojs ;5u(qa/ m z0i5cn torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attalm+u33i jffrnbpg+ a5ha4 0u.ched to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the hru 5aai3 +n 4.m0bfg3 pfu+ljhorizontal 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 tightening to a torqueg-yv ww(:s-vef/ +by /h0varq of 38 av/f rw0 yev/h:vsq--b+(gyw $m \cdot N$ 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 0.6:vxbk:h dol9(, h :ges48 m when the axis of rotation is through its center. vg:o d( ,k:ehs:lh9bx   $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameternjb7m9d4wdfu -,7j w hpf( aejdw8h3-. The rim and tire have a combined mass of 1.25 kg. The h 7ph87wn w-wd-a f ef9d,3(juj 4dbmjmass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod w niz61 3jsd0jis rotated in a horizontal circle of radius 1.2 m. Calcdij06 zwn1sj3ulate
(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 rotatiqps0 h1s (i-umng at constant angular speed (Fig. 8–42). The friction force between her hands and the cl0pum-1s is(hq ay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objects shown in dgv u0+tvdzof /y. k*4Fig. 8–434g 0 t/zudv*k.o+vydf 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 consists of two oxygen at ,whsuve5-uhzh vij36 x)o6xgx0c4s8oms 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 ad58 82izw f2fscyxy*wt 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 ra xy82ff dz8w5isc 2y*wdius 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 unifo w)okfm ric*n,hb -f41rm cylinder with a radius of 8.50 cm and a mass of 0.5814* rmn,wfhb) fo-ik c0 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, accelerating it from razw-wt-o5o*r yltt8 sx 2h-+ pest 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 and is ey2mo hat()+(nnvw5 o 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 2ne vwo2(y)tm( nah+o5.5 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 torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shfpf .jmn*t4.d ut off and is eventually brought uniformly to rest by a frictional tofd*.p.jf mt4 nrque 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 acceleratez:ksfm 5p 2g1 lbkh13fs 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 sol8u7w h6se .omif8v8 aoely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, si . v7uw8ooeam f6h88Fig. 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 bladebrh2 r(f.dq qf ql zgs(r2 -sjso(,.q. can be considered a long thin rod, as shown in Fig. 8–4(2.qr(., sq(qgfb df2jh l-srzs.rqo 6.
(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 twopt o03za ldh 30qh)0is8by ;qm 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 h wa :ge3cdd4 4w)l3adqammer from rest within four full turns (revolutionsc:ad)4q4d weda3 lg3w) 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 mozq;/:f butn-q ment 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 /uct4oy5no6*yo:t4,z wtql ldevelops 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 slippmw(9 xzk5;r o8enbn v;ing down a lane atnv8m;bx5(o e 9k r;wzn 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect mrx ih8qhn/:+pt( g l+to the Sun as the sum of two terms, h +(pg nixr 8h/tmq:l+
(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 r:ybi6 2mh 3uf,urkld6adius of 7.50 m. How much net work is required to 6 uu:h kird 6bl2m,3fyaccelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and ro 6v *cbt9gfkc)lls without slippingk)fc6vcg 9t*b 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 balak;v fx-arfo1t0u ; wpos /h,,bnced vertically on its tip. It starts to fall and its lower end does not slip. What will be tfpof a -kxoh,vt,uswr 1;;/b0 he 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 rotatinkm m *(s yhsx;ct56y86m) jrxe+v k;vmg on the end of a thin string in a circle of radius 1.10 m at an angule) +k8xms;y vxct5(mrsj mky *; v66hmar speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a khi,lp jrf:8lw:lr lm5b5v 632.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotating at 1500 rpm?:rm6lwivr 5fh8bl ,l3kp:5j l    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at hi-djdxee5 +c+/ f( lxqcq;we /xs side, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a ho-+wdfxc/ql /ceqd(e+ 5xj xe;rizontal 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 moment of inexhqqrn)(z d y)b55 (tsrtia 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 angular speed ((y(q 5sdr bxt5zq hn))$rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her sp(6l yagfwtsb94;o0f fs y; 2zxin rotation rate from an initial rate of 1.0 rext;9af;sg (yyfws 642 zobf0 lv 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 kip3su( h89qjou 7rmel( 58 vprotating around a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg 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 centersu3ek qpv(8j 9(pl5horumi7 8 of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum9y) roft p/1+h+- jz pthifa.o 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 mome7 roa/ek ikh19t 4g(qmntum 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 ineuqd xb0fl(- (artia I is dropped onto an identical disk rotating at angular uq (xd( f-bl0aspeed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns 1a)o6u imu;fo at 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 ce9ll1+8jccu)n dlcu 5enter of a rotating merry-go-round platform of radius 3.0 m 8 ul jc1)ll5+ud9ccne 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 vef7:zgs khj- vjk+a 7;plocity of 0.k7hjz7;asj gv :f+- pk8 $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwa(+fvoxk j g(drwa6/ q6rf, 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 angular momentum, what would the Sun’s new rotation rate be?(round to the nearog /xra wfq(+dk(j6v6 est 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 involu b a/+vkif*5bve winds in excess of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass mh+3ddmilf v3 /wyapcds2. 4($ 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, initial9br2wi 0t6e xpqd )fd7qug ( gqo;za7(ly at rest, that can rotate freely without friction. The moment of inertia of the person plus the0rg ego()qubdai(9zd ;6 2qx7p 7qtw f 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 stankl9, gzex6 w(sds at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia of 170k9 xg(s6z,le w0 $kg \cdot m^2$ 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 withnyz rsra/-twg:qt8l2yd(*1 c 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. Whas2nqyrtdr(yl a 1tc8zw :-/g *t length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Ea;b-/ aflzs/ imrth such that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to ;- /lziamfsb/ its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from resy gu.zczvnc-/ 1,vyb8 o8v;qbt at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder t y52t(wr :og,l0btnfof radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration.b0 gr ty,t:n(owlf2t5 Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two :0gd/ wab-guj 3u.bn esolid cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is releaudgg./n3: a- 0jbub wesed from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of the 5c9+rh5uie-l yw(nc brear 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 7 4;4om kdsaij( j sb7yniyjk-s+5(avof 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 neutron star nyjo775jsmyb(a4kks-s v d i;a4ij(+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 i;o; 3;u0ad,nnas qsk ct to use energy stored in a heavy rotating flyo qnna;,;ua 0kcd s3;swheel. Suppose such 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 illustratesusymr :h.b.o:r-u (0i/mh juoq ; 2dxh 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 horizoni7 ex* zr,m9 lkd2lfm1tal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e1no ,yoz*b 8ib., 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 releasob*b 8y 1o,inze, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing verticifs86 lks0thbblk6fr:s 4 l96ally on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against whll 06k bsrkb68 iff:9hs st46lich it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with spefmhzms+, br/84/h jhued v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist andhz hr hm8fb+s j/,u4m/ 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 0.50ubv tk1q*md +0-kg 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 torque required to achieve this feat, and where does*1+0t qmbukvd the torque come from?    $m \cdot N$

  Model a figure skater’s body as a so- wanu v8h:wrmf / ;y9aflmbhn0,d9e *lid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular sp hfb*mawh:af n; lu9,vmnd8 we9/y-r0eeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a +edutb )g*l(; 0cez 4vjrfj:,xrw4 dvclutch assembly which consists of two cylindrical plates, of m zdf(,jdrtj+v b4x*e: cug lwve40;r )ass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 2jf)hc xet3a4 o5pz/rj4 b8wqrough 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 loop without l544)xe/3z28rcp wha fj otjqbeaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not as 2kn;gz)kdta.hys k69sobpbp5 cfkl17 -5 ; ubsume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed unyf fgz,sya3 er*+ au9;iformly from 90km/h to 60km/h The tires have a diameter of 0.z3+r;gy9aue, a *y fsf90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s