A bicycle odometer (which measures distance traveled) is attached near thei(wv29axax9 y b9ufn s6+gnjl, :mt :i wheel hub and is designed for 27-inch wuwt9:j6a:sb yx,mfl na x igi9+nv 2(9heels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angny(rvh :p)tj*i0 tjr ko0+o ,iular velocity. Does a point on the rim have radial andy+jik, ) trj (rhon*t:ovp 00i/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?

Could a nonrigid body be described by a single value of the angular x2 v w(ap1tx:tvelocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger for* 0 fbkqoxw4f/ce? 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 123nw*5 uuap k-fxeuqdo a sit-up with your hands behind your head than when your arms are stretched out in front of you? A diagram may help you to a uewn5 x-1kf2*uua pq3nswer this.

A 21-speed bicycle has seven sprockets at the rwj424ow2 wfc.4axso f ear wheel and three at the pedal cranks. In whichw2.cfs44 o4xawo w2jf gear is it harder to pedal, a 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 abto wk9un 0)lvje+c/;1er vl/m le to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34).t+le/ke/cwv9v 1jl) ou0 m;rn On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow bea882s.a14j zsl 8nqs v jvzeu2sm?

If the net force on a system is zero, is the net tojd+;jrre0u 8lz+t)t h 2s-dbyrque also zero? If the net torque on+dtrjbz uj2ty;8+ 0e hrl)sd - a system is zero, is the net force zero?

Two inclines have the same +qy)d s-tb(k /l m*krpheight but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explai l*kk yq)m+tdpr(- /sbn.

Two solid spheres simultaneously start rolling ( vt+ zcp fxi5*b+ut;)g6xha 2lfrom rest) down an incline. One sphere has twice the radius and twice the mass of the othb+f;xtgc )a* 2 6uv zxph+5ilter. 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 sam9prldc 3fnpbq sn:5,.j2 z6xl e 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 gqxlsn :jbnc5dp396z .,lr 2fpreater rotational KE?

We claim that momentum and ang7 azhgptvtez u s92)p-x 78j0zular momentum are conserved. Yet most moving or rotating objects e7px 8 zzz29vh-t pe uj7ats0g)ventually slow down and stop. Explain.

If there were a great migration of people toward the Earthoj 1axk4r2 . /mwj+jj.laff6t’s equator, how would this affect the length of the 2awfj+.x4j 16tml/ k fj.joraday?

Can the diver of Fig. 8–29 do a somersault without having any initial rotation w. hb-sms 9(x*(upz7ev6aelr p/;nyedhen she leavesms/zldpeb7yvshep96 .*; r( ( n-a eux the board?

The moment of inertia of a rotating solid disk about an axis through g(h :/jc kmf9xu1 qzz;its center of mas:kcqz;u 1xf z9/hgm(js 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 holdin,q4y - caup7pag a 2-kg mass in each outstretched hand. If ay7 uc4 p,q-apyou suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one ist q7c- (v l99lo1qae-.4t8dn6hxi t l xsp(buk hollow and the other is solid. D 84o6vnt sl- 9x leaqdx(c1bkuh9ltiq p. -7t(escribe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points westeeb6awz m o )ki) 5b-n9jzsy;s-:ysz2. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration poiwyi szsz2;96bkzanj :mo -e5) sy)-be nts 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 freely rota0sg14k foi )pfting turntable. What happens if pg1fk4 o)0isfyou walk toward the center?

A shortstop may leap into the air a5,j +ep*8urofs-yb vto catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you wibj, e*+rayuo s8v5pf-ll notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the lawuk d2.:s zfaz8l ug73b of conservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss bsu3 8ku 2.lagdfz:z7one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following anglgfz)fprqh j5w+ ,7( a3-mfwxd es in 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 becauzzi(y 66c(r bzse of an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diametersy6 z( z(zic6rb (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, n au5;z*l*ucn 380,000 km from Earth. The beam diverges azcl*a* nnu5 ;ut an angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotaur5om oj, 365: ggozd) s4cpwlte at a rate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades ,5)jo4zo : ggd6u3powmc5rl s slow down?    $rad/s^2$

  A child rolls a ball on a lhucy; e u;x ++g szb-t5d*2cbcevel floor 3.5 m to another child. If the ball makes 15.0 revo dhbu+etc;bz- c5g 2y;xsu* +clutions, what is its diameter?    m

  A bicycle with tires 6+; ch5(oiqhc ps0ofq4 8 cm in diameter travels 8.0 km. How many revolutions do the wheelpqs;foc i 0c+h5oh(4qs make?    $rev$

  (a) A grinding wheel 0.35 m in diameter who-g y 5fj(fu,-a6t brotates at 2500 rpm. Calculate its angular velociybt j5a gf(u- -fw6ho,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 4.0 s (Fig. 8–38 3is 9p 88*vt5g*a;d*acclteghnp1k y). (a) What is the linear speed of a child seated 1.2 m from theeha1;altd85** tgvcic38n spp9 *yg k center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity o2y t8ufn a,( t-ethgxv85f jyt0j9qq;f 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 sm8/75phl ksacpeed of a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a 9fj8ne.a m bu8xr-b;a mjd 5a2particle 7.0 cm from the axis of rotation is to experience an acceleration of 100,0rde5 m x.8abm-f9;aubn j a8j200 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its c4wbbpv7:q5ns 0irj u8b .0qt k,k6aimenter from 130 rpm to 280 rpm in 4.0 s. Determin0tnr: .5 vq0 kquw4bi p6jbisk8ab ,m7e
(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 radiusibsuf01d:ah 3xo5-nx $R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard the Apollo spacem 1g:x1eb-vmni,f /2hjqxdyb e1e s 8,craft put themselves into a s 1e1b:/,e,2s fjvdxxemmhg i y1-bnq8low 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 rpm in 220 s. Through how *;k tifza sdn247gxm2 many revolutions did it turn ini4tza mf;2kx sn7gd2* this time?    $rev$

  An automobile engine slows down from 4500:a ttzy5s fov0 n :li-o64rz)k rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flyi1egq6ev:d mvx *5bvv 8ng highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to tur qgv6 dxm:ebvvev185*n through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in dia q 4i9o4nw 0la0htxw ;xy.9ehqmeter accelerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far wqwa0eqnx4xh t.w90i;4oyl h9ill a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is ros:fa* v6ut 0s mjcca4)4gi/i unning at 850rev/min It turns 1500 revolutions before it comes to a stop.sgcam4 taiiv*s:0f/6)juc o4
(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 speedy::(5e 9bed0hlm ieme uniformly from 95km/h to 45km/h The tires have a dm: y l5eb (ieeh9m0e: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 speed uwf3nlc/ qqtf:zh +,r) niformly frr ctzl /)ff:h nw3,q+qom 95km/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 her weight ,/.gm bmtndg/urz 6jv(x5 h,lon each pedal when climbing a hi,tumjg.g /v, bh(/dxl 6rmnz 5ll. 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 force of 55 N on the end of a doo sl48xh q0li8qr 74 cm wide. What is the magnitude oqs8l0i hx48 lqf 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 about the axle of the wheel a,uz/g)h-(k-x)t -fwo;rnt se tw,zrshown in Fig. 8–39. Assume t- xfr/)t) ozwghk-usta;,-etz r(w , nhat a friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attache,nuyr--bcy gqzm 33k4 d to the ends of a massless rod which pivots as shown in Fig. 8–40. 3gry4qc 3-nbu-m yz k,Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder headd x19 catna,a 3x1qis c6pzu42)vhfa, of an engine require tightening to a torque of 38qc9)icn2avhad43af1x s 1u6, ap,zt x $m \cdot N$ 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 0.64kvmiu1 (v:)se 8 m when the axis of rotation is through its cent1(v i)k:vsmu eer.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameteso n;rhi*-hx3r. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignored (hhr3-n;os ix* why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is roo8 c9vn7)9 nwy-*jgt la(ju:5z xhbwbtated in a horizontal circle of rady9job5tjxu(*c-n89w)baw:gzh 7 ln vius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating at +b5s5*h4j g5ge/cic,wv fb z cl i25agconstant angular speed (Fig. 8–42). The friction force between her hands and the clay is a4g f jc*5e25g cc5sbi5/vh,bwglzi+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 uee11op4v8m ; buyew+:6ti tuin Fig. 8–43 abouiwytt+ 4 ou81b 6 1epvuee:;mut
(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 gwelk* 7.if wor.4 .p57z6+xwap n rrhtwo 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 cylindricabj75twpfyd)8 sx/e x0 u(zkp+l satellite spinning at the correct rate, engineers fire four tangential rocket7+5tx 8(x dp/yzsep 0)fjkbuw s as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 cm and a m4e0va-cdm8 t9 rqir,d f oh//gass of 0.58,cd-mvr a8 /0g9 ethqrd4i f/o0 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 batk enn ,5g+ttc3, 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 small hand-driven menrq ) k-b3f78povngsif 8z)p4rry-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 83-z p gvbf o)r nn748q)pkfisis 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 torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at ;4u,,nxtydo/nv gs4o 10,300 rpm is shut off and is eventually brought uniformlyts,odn ;/uy,gnx4 ov4 to rest by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a 3.6-kg bq:ia1pta. afm+ d 5,qy8ud2bwall 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,+oz)ye 8gm * p6xqkxuewqa 6*v-l x6v is thrown solely by the action of the forearm, which rotates about the e6vzg6xke my*)wxl p,+q6oaeq*-xv8 ulbow joint 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 considered a long vu a/f c+,xyvp( p 4e5unjxy+w09on h:thin rod, as shown in Fig. 8–46.v f 9u exw5(,oc una:j4vypy0++/p xhn
(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 co,r8o )e /gfzkknsists of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer f:73r.u;mbfe; ix*sjw. gxtb nrom rest within four full turns (revolutions) and releases it at a speed of fjg3; mtbrewx :.sb 7.; x*iun28 $m/s$ 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 hc p7x zq+p y2xkk0v80m fv5k:fas 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 develzh7 gg5t8(abj ntd* jg a7 ia2;zg(1thops 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 withouaqbedg.15/j(m j/1ew ck 4lkw t slipping down a lane akm jqw a1b/./w(kdcje 5le14gt 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respectbxe)mpbn t+54a+ 3zh k to the Sun as the sum of two m b5 k+ +hntpbaex34)zterms,
(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-,v;jc6f p)tovwtpu(g b5i j9itxb:0t86 uaof 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revji btp8f9 6bx:t 5;ott0, iv-o) (vucjaug6wpolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg eluq):a 9.f0 ci-rnrwstarts from rest and rolls without slipping down a 30.0lun-c9:a)e iqfr w.r 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 vertically1nd 3c. xekr;b on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits 31kr edn;.xcb the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end of a thin) l+g6 g(n-y;hujk3h yim)x.akr7qzb string in a circle of radius 1.10 m atlg h xr7hbiq m-(k;6n+u)jyyagz3) k. an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum o s zsj5p23r+pdf a 2.8-kg uniform cylindrical grinding wheel of radi3dsz5pj +p rs2us 18 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at *yxu;wjejb5wi o:wu0gm;.m zz01u l/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, the speed of rotaty1;leo j/jmg w0uzbux;mu0 z wi:w. *5ion 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 he(8no rsq(lbqw2e- 9bmr 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 pb mo9(2qs-qrwbl8( ne osition, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from z)gb s5h+ +echv1zc9pan initial rate of 1.0 pce+95+ ) vs hzbh1cgzrev 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 arouly (odq 3552bkobunz- eq1m o/nd 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 (1n 5/o3eb-y2o zo5bdm qklqudisk 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 momen;3byujqyslc5b9a jm ,0wri93 tum 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 moment()ub xexjo)(+dvv2 tp)u tzu+f z1zx7um 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 iner2g3o8 zv v fzsk6wu16ytia I is dropped onto an identical dkyu3 fvw o668s2gz zv1isk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns ap5bl7*sw5 o1 gtkr uw0t 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 9+yoyhhc/ oys 3) tsuh83a c a.k.w5khstands at the center of a rotating merry-go-round platform of radius 3.0 m anh+o)3y3s c.a huoh9cyk./ w5ya hts8kd 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 rvppjae ac 9/ a aoji9q(6hv0j) bx3g;;otating freely with an angular velocity of 0.8 g03 /ovpp;c;e9q h(xa ja6aiaj )vbj9$rad/s$ 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 collapses1ilcgk v 6s r23 +/jwzax3eei- 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 mass carries away no angular momentum, what would th 2gki-6esr+/a33cv jlx1iw eze 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 i8jlk68ufmjxpvy/6d 2 k ue*g 1n 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 v4zqoz l5s (6v$ 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 can70u brrf;1vvpov;/tk rotate freely without friction. The moment of0 rvtrb7up 1k;f o;/vv 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 merry-go-round t amatk6(j:*b ep*-xsf urntable that is mounted on frictionless bearings and has a moment of inertiab eafapk-j*:smx(t *6 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 lying on t ugq.:/rdkalmkpk83;o o ps- kb/ 6(vyhe top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s 8vpdyu.og/p6 3 ;l rb(/ka k-m:ok sqkcenter of mass move?

  The Moon orbits the Earthbct9ju* aq5qw q*s-/h such that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Ear*5c-9h uqa/*s qwqbjtth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a ra2 vk1m1p*1si sjxdn5rte of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of radius 0.20 m uk:-z 0e-uigi acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the0z i ugu-:-kei motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 kgxkf r*1:gqbdgy k)p: * and diameter 0.075 m, joined by a (concentric) thin rg:g qf*1yp: )xb*kdksolid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is6 vu*.v9foez5z t8aos the angular speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mwfn yu 4p/ti/7c po2*qass 8.0 times as great, were rotatin*iqcp np/t2y/u4f7owg at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the 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-polluaw)0p6g l-epption automobile is for it to use energy stored in a heavy rotating flywheel. 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 356-lpw p)e0ap g0 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates a0)zdh, 4ai8ewsmjvd* lu yxwu -1w:z)n $H_2O$ 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 nik 1olx:+;9h fjck6e rolling on a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we ignore frictim 3lxigg).09q osvp, mon) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release0spimvg) gxl3mq 9. o,, 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 v 69 ptq pidc qd7kn9/wm8)2r,-kseyv kertically on the floor, and we want to exer p/nq-t9skkvcm )d7yp,q6 2rki de8 9wt a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist travelinzij 8bxr,6a+2eg3y2i o yt1 elg with speed v=4.2m/s on a flat road is making a turn with ab281egl e rjxy23oiza yt,6+i radius 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 lengt) :y( w83gr 5wt.qb*f c 0ozpqqwgqi8qh 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 rygc r b( ztg*oq3qiw8:.p8 qq05w)wqfpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as t9b5 cou,t-qip hin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, tiuc q,b 5o9p-and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindrical plal:fphb si cz+xv-2/0mtes, of masz2f bsl-+ x/p0chmi:v s $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 rouaeljd .5r(rl9 gh 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 leaving the track? Assua (r.dl59r eljme $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but h-e6 ,i7hcdlt do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car redu g5opy4vyp3w,m2 rg :+6uezhn ces its speed uniformly fro gu24h+yery, 3vopg w5 n:m6zpm 90km/h to 60km/h The tires have a diameter of 0.90 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