A bicycle odometer (which measures distance travel3lx5;xw:m0w e1yk snp ed) 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 whewk ymns :x0e5; x13lwpels?

Suppose a disk rotates at cod i8/nx: e+lnr7p9r ytnstant angular velocity. 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 acceleratipitn r:e9n7x /+lrdy8 on? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be describe6e-yhe uw8 ix3d by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert .2 pwy;mbvddb657ck c1pu 6zaa 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-upr-q c1w51jar 1 d6xyyjo5lg(s with your hands behind your head than when your arms are stretched rxq-yl61j1 jcgs5r wdo a(y51out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at thf2o3z v obyh1 -+pyu:m:aaz84(wfvyo e rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprovwohp348:b z+ya-y y1zoof v(2mauf:cket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have sleng o,vi k)70ixkder lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageo xkik7giv, )o0us.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow beam?u+0l+ u08 )2fqdc6ie6gn(n bputhls k

If the net force on a systemcpb5:m/ub.,zs/;p cq j tnodc 1ub:9 e is zero, is the net torque also zero? If the net torque on adjn5/u.ct,eb ccb ; pbm:9p1z :usqo/ system is zero, is the net force zero?

Two inclines have the same height but make different angles 8rcdsjm(r*/z l (xzw2 with the horizontal. The same steel ball is rolled down each incline. On which incl/(( wc8j z dmzls2xrr*ine will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from ree a;mh*.ubau)h7q*vv gxc f.* st) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which ha. *vuu )7*avcqhehm fb.x* a;gs the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radiul+o8lhft l,/2ud jjr3 s 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 gr8 jr2tjf h+udloll,/3eater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are coo jw.oj xw0-)onserved. Yet most moving or rotating objects eventually slow down and stop. Exp wo 0oowx).j-jlain.

If there were a great migratio0vy5p7 wc ,ejrmo(zz-n of people toward the Earth’s equator, how would this affectrmcz yew(o0 p -7,jz5v the length of the day?

Can the diver of Fig. 8–29 do a somersault without having any initaeeb*bo kcc+ dk5tnw 661 66sjial rotation when she leaveso165 acnwsbk6+k*bt6 cdej6e the board?

The moment of inertia of a rotating solid disk about 4zwy.yj)0 /yy 2zazkk4v*a op an axis through its center of mass iy)y2yz ap0vk/wy.44*ozzk j as $\frac{1}{2}WR^2$ (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-k w5v *zx8c*) uj*2sdwyy2qfuy g mass in each outstretched hand. If you suddenly drop the masses, will your angc * yj8wuw2y*v5zu sqy2)x f*dular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one i sgz 2n u4aw-j nlc*.i;a9gg8h qzw9/fs hollow and the other is soli zgn*9cgaal/ u hs- zw9q84gfn;.i jw2d. Describe an experiment to determine which is which.

The angular velocity of a wheel rotatit(u g-ra,n5z yng on a horizontal axle points west. In what dirn5,zyru ( ag-tection 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 speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotatij*)tzwkkxe.6w 8sw9d8cuwv6 3 pv v7 mng turntable. What happens if you walk toward s v93 wztpvm86w.8 xvwke*dkuc)wj76the center?

A shortstop may leap into thb+9jq)q*n ,ohfdldv 3c9x )a llu. q1be air to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rota*qqa +l3f.v1xolh)9bdljcq ndb u,9 )tes. 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 consc j6 s0bh-p9wuervation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stabl-u69jc0h bps we.

  Express the following a2 a4j maq- ot,,gnzrty6ibpyf :oh2;8ngles 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 because of an amazing coincidence. Calculatentub n69j .fn8o4fbq- ji k51m .ipb;zowb7r3, using the information inside the Front Cover, thq7j.n izmnupob 8f-3fb tk4r.5 o6;wbj 91 ibne angular 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 fr md0vczqlvt)8b(+(v y3m )ms ni 6wx7iom Earth. The beam diverges at an a)7 wb xvm8vn+6dtqc)z m3( ymsi ilv(0ngle $\theta$ (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 mo2tekf7kq2fx-y3zr g5 ie/hj*tor is turned off dx k7rt2j-k zey/ f g5fehq2*3iuring operation, 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 another chn:b:zc22 dew0-t(o5y sp dvtjild. If the ball makes 15.0 revoldz -: wvo:5tpes(y 0bd2jtcn2 utions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0i d0c6zdllz+ce n y5(ldq6w/5 km. How many revolutions do the whqdd6(z6+ nlliz cl5cw50e/d yeels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculatx01fnin 3 ywd 20 sqor3+dai dw(3qfp;e its angular veloi n3(w;dsr0 +wdypo1a q f02 33difqxncity 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 -4wgb(l/cc oyone complete revolution in 4.0 s (Fig. 8–38). (a) What is the linearl gy (-ocwc4/b 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 orbit around the Sunjd49 kg(z7, iw filsh-+eue6a    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of ,2z )cvnjr5jva point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0 cm from the4* n)*uww q5rtdg a)mf axis of rotation is to experience an acceleration of 100,00 qw)dn5m*w)*rug aft40 $g’s$?    $rpm$

  A 70-cm-diameter wheel(fljmwp+/ m xs)g:he) accelerates uniformly about its center from 130 rpm to 280 rpm in 4.0 s. Determi:m /l hspgw+()mxf je)ne
(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 radius - rjk-ka*ockfr1uu 5; $R_1$ 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 Apolloal 3+ j9hzczv4 spacecraft put themselves into 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 wit9l+j4ahvz cz3h 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 frk h*) kmj7j9pyom rest to 15,000 rpm in 220 s. Through how many revolutions did it turn in kh my9 k7j*p)jthis time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calculaaimsoc-* b+t,te
(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 highspeerr- w +f) cxe;msq tw2ace+(,j1iyr)od jets in a whirling “human centrifuge,” which takes 1.0 min to tu) fjet;ywrm1-xo src+i2+qa()cew, rrn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates unzx o9 9*m scudjlgj/7+iformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have travzsd*/o 7+gmu99jjl xceled in this time?    m

  A cooling fan is turned off w2b 7y3 62*pimrx e/ ma-fvnegm6t-m uwhen it is running at 850rev/min It turns 1500 revolutions befo*efxre/ -i - ymv37t mw22mn6mab6pug re it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the cj,vy*yx( )ouj ar reduces its speed uniformly from 95km/h to 45kmy*uj)yv ,(x oj/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces i-6 -5nknu3:ty wgfagoe o,3tats speed uniformly from 95km/h to 45km/h The tires have a diameter oe- o k6gt no, gn5a33wuty:fa-f 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each pedal when climbing a n3cds4q7rabrsyo /. +hill. The pedals rotate in a circle of radius 17cqdso 4rs+.an/ 3ybr7 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 -odk8; *lna7pdkrai:guh;7i on the end of a door 74 cm wide. What is the magnitude of the 7khl8a- rd*da7o;k i p;u:in gtorque 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 shown in Fig. 8–nsag an1qb. 0fr na;8539. Assume that a friction torque of 0.4 8fg5a.nqs0ba n;r1n a$m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of.;gvm hwan( k: a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction o .gvk nh(wma;:f the net torque on this system.

  The bolts on the cylinder head of an engine require tightening to a torque -8 loe-*hr,oc e1gibrof 38 egro8l cbeih1o-* r-, $m \cdot N$ 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 in3, xa/fl3(n+f1(q+o ifsx g in0 ralabertia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation n0,3+aa l1x(sganio q(+f 3 xfli bf/ris through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diamf*,w,.p eer ajeter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignored (why?) j*erw ,epf.a,.    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in a horn 3.u31a7 xbrdmqr 3ifizontal circle of radius 1.2 m. fanu3q 31rm3.rb ix7dCalculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating at constant angular s))k(z2v gi ty6jgpw)z peed (Fig. 8–42)t)izkwyp 6 (v2z)jgg). 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 shown ih . pfdc8::dgkn Fig. 8–43 aboutd :h:c8.gkp df
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen atoms who2,oi ym4receyn-4 r; qse 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 correc9o(b-2m yfi+f qngvk( t rate, engineers fire four tangentiav+ioqy (bgmf9kn-(2 f l rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of x, zp+dw,;zawr tih.58.50 cm and a mass of 0.580 kw .;,hzzrdpxai5+, wtg. 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 f(;v+rgc sr j2lrom 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 and isus*ej. 87pqv.x:qoq0zq n ko (hs:c3e 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 q.o0cu:k.p q :8esvq hj3q*s(zxe o7nthe 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 s)tf ;fqxsu t1.hut off and is eventually brought uniformly to refq1f)tt . sux;st 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-k 5 v4wd5i2b6lr(v zue8n)n drug 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 solelyebmui2 l2x0 0f 9+r0hyuysa 9t by the action of the forearm, which rotates about the elbow joiu2902 +x0rts f9m iyay ueblh0nt 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 thin r5c:i 7wssdz8 upjs 5glb-sn1s 4. xj 0ux8.vwpod, as shown in Fig. 8–458 x ..7-n1:xpj8w cszwiv dlb5jsu04ssgsup 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 two masupink tk2p,em )/a4d*ses, $m_1$ 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 fraz *li.;ym s*gom rest within four full turns (revola .zg ym;*l*isutions) 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 moment or+ fjjo50a qxrnn o-,-tju92of 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 develops a torque ofd:dhzi-/kvoktaus, : epws 2q/5-3a n 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 slw r/omtx: u6zrci5,n)ipping down a lane at 3.3 ,/cur6rtn: zw5)im xo$m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun a: zpb7-ic.zgueu s-k t;+x4abs the sum of tw u c+b pg:ekaz-xbt.4 zu7-;sio 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 k v f9ou 1b,1:o n8-kocgso5adradius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a gn,o 1v1 : 8fd9oa c5koubo-kssolid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg start98 zny :d5a veag6ui-cs from rest and rolls without slipping down a 30az6:e n gyi-ca8d5u9v .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 vertic s81gi pe8vj qy8x9os2ally 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 ju21 is sye88ogqp j8x9vst before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.dw :t+6fk 2,tpcn dna(210-kg ball rotating on the end of a thin string in a wkn 6+2tdfcnd p:(at,circle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical g:eohnkgil; 9hj 93b8b rinding wheel of radius 18 cm :n;kbo j e9gb 9lh3hi8when 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, hands7f ybjt(i;fmx5,h:a g at 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 rotation decreases7g;,bi famytj5x:h (f 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 Fiqt2-k9 t e6tv9cb g-qhi,o3 jyg. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular s9jbvtqoqc92, 3-ke6ytgiht -peed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial rate of 1.0 ke-y.nwvi 4 +ba4of7 7v9e+bpyj o y/crev every 2.0 s e i4p+jb-a 7nfw+ov9k eyv.o/ b yc47yto a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis t8viz3v di:eo +hrough its center at a frequency of 1.5rev/s The wheelv ize8:+vd oi3 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 wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinning at 3y-gkj/ )i, +n5zs6qvb,kc di*ubdxb 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 ofahpk5b92sx(po7lv)9m m9 bsor 3p ga9 the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I r(:mc3,gvic ais dropped onto an identical disk rotating at angular speeca , g3vric(:md $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns alvp /uxw0qe3.dc7q1 s t 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 rnnxya 5ysz5 zuv:b1l5ojrj*8s ;xm*r(52 qpstands at the center of a rotating merry-go-round platform o 55 v1(z*zsl:p5u 5ma;rbyjo8 xs2r qrynjn*xf radius 3.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-ofmej1im0h(a 64r+r bgo-round is rotating freely with an angular velocity of 0.8(r 0a+j 6mirf ebmh4o1 $rad/s$ 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 dwarf, losing about ha,cbe; sc zu+n.(h 1ktalf 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 t snbhu z1k.c+(ac,e; to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in excess of t)rk3 rn,2szw9fjv n(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 r1tvg, l7gf )xml(n ldd2jy*6 $ 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, thatq5zbhs3 dy,vil .x b/6 can rotate freely without friction. The moment of inertia of the person plus ths.q/ 5y3bi6zlxdvh, be 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, syr ,-wtup,d*;fv b i)xwfn,-round turntable that is mounted on frictionless bearings and haftydxivbpr* u;s,f ,n,-w,)w s 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 e9wufq*dp ewax8u*)fy 1k 2.v knd of the rope lying on the top edge of the spool. A person gru w2. qduk*p*yew1)v kaxf8f9abs 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 center of mass move?

  The Moon orbits the Earth such that the same side abp kmk3: a7(oxlways faces the Earth. Determine the ratio of the Moox mb( 7k3ap:kon’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 ratvun sutyd:xl ;u34*8f e 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 b tly2x9ziub5pm ko1i-w0c(;0 y ui/duniform cylinder of radius 0.20 m acquires a rotational rate of from rest cbu xy0/i lyku209p-m( id;itb5wzo1over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of 34 t+ hw29pkgtovpyt *5jf u5ztwo solid cylindrical disks, each of mass 0.050 kg and diameter 0.07 5+yuttgfow z93v 2tpkj45p h*5 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 released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is th+c2m/;2bbzz ;s+ avwp/l am cge 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 mass 8.0 times akq 01/+uqtlvrs great, were 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 inqkr1v0q+ tu/l the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use eunb m3 48fbs)unergy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses 8bbf)snm 4 3uua uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates a )b gs4pzvsfy;hpj)j)lm:0a7 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 rolling o0i5x pcpxsdi.;e+90vhce vg(*od h 9un 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 91gucb+sg cw: ( rq1k su)0h2 yjodu8ecan pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular 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 robu18hgq+(0wkd2gcsse)c1ru u jy9:od, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing verticalgra )evi*lc 0,+p8 -ggvah4*zvmun s7ly on the floor, and we want to exert a horizontal force F at its axle so that it will climb a st-gc7n)lm,av *arp+zvi84 g 0*h guvesep against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a turn wi8qd6ew(8kmo3hpd2 ud(( y ipeth a radius The forces acting on the cycli(qokw2i(68e u 8 dpye(pddhm3 st 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 intby o l:5f9 frx3dkx1x;o a sling of length 1.5 m and begins whirling the9xxyl51;: d xf3 kofbr 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 the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as t *u+,+a)za5rorrhega*2ea* jy.fuya hin rods, making reasonable estimates for the dimensions. Then calculate the ra.roy eya* r*a u+zagfrh+u)5 ,eaa*2j tio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindrical pl3u )fzkrb 8itm4tde ):ates, of ma dmk:bui8t3 ))ze f4rtss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 loobo5j4 vh;ak 1wds:m m ie;j;8nped rough track of Fig. 8–58. What is the minimum value of the verticaonm;s8wkbd; 4jmh1aev: j5; il 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+ n a+5sp+ o/zubrf)sk not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 k7noy02x) nwsjxql (:revolutions as the car reduces its speed uniformly from 90km/h w:ks7x x(nnqy)l0 2joto 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