A bicycle odometer (which measures distqnstj 89a*w6 qance traveled) is attached near the wheel hub and is designed for 27-inch whean6j8*stwq 9q els. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant ;)olgntg,f-f 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 acceleration? For ftogg f-l);, nwhich cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be des,aaf4 or;5w46 q pmqawcribed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever6,vcr ge(o 6fx/dl 9ad 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: l.q0m8hgm ml behind your head than when your arms are stretched out in front of you? A diagram may help you lm:h 0gm.lmq8to answer this.

A 21-speed bicycle has seven sprock loqp3;ri 6wq)oyy 4r(ets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it ho6y3qo yp;w )rr (lqi4arder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on dv0wi)(qccb eim /9d-being able to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On cvd i(b)w/0d icqem9- the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) je opr-ghhbq7.1ygw(8f0uhg. dn0 (u carry a long, narrow beam?

If the net force on a system is zero, is the net torque also ze q73c3qr67gbb ,anp itro? If the net torque on a system is zera3 7 bn6g t3,qripb7qco, is the net force zero?

Two inclines have the sm xfyyz z3(i.yrc5,im8d2p (same height but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the bamf3zipm5i( z(x2 ds8yy,.cryll at the bottom be greater? Explain.

Two solid spheres simultaneo5:ew i.sg5lp;:9iutvsx a 8x, s mw6hously start rolling (from rest) 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 the89;isxg wvps5a., 6oluw:t:ex5h m isre? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. They start jak*av:nv4gr8oy b fp+2;81rr8 hlye from rest at the top of an incline. Which reaches the bottom first? Which has the gr81a*j8 2yh8ln kr;e:gpf4rv bvyo+a reater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momeq wnxo5 x:h1 p2in6(jxntum are conserved. Yet most moving or rotaw6 hn:q1p2 (5nxo xjixting objects eventually slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equah *jne ens*1 --;x8ektjorg f.tor, how would this af et8;hsnenx*r*1 jke - .fgo-jfect the length of the day?

Can the diver of Fig. 8–2zueq5dg30jsl 80ku9m9 do a somersault without having any initial rotation when she leaves8dkg q3s 0 5zujlu90me the board?

The moment of inertia of a rotatingng ce-.rfiwg/ /abg9s6q: /my solid disk about an axis through its center of mass .- sr bi/wyefqn6 mg/: 9cgga/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 mzyhl- 0a0b 8kr;8qmy3a odx6xass in each outstretched hand. If you suddenly drop the masses, will your angular velocio6yrxdm0l ;z yqxbah-38 a8 0kty increase, decrease, or stay the same? Explain.

Two spheres look identicab,v to2ugbf9;1d8x m( d /ckqrl and have the same mass. However, one is hollow and the other isr,kb;f9dcxv/ ( bo 1dut28mg q solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotat9 aovq g;y8k:ming on a horizontal axle points 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 speed increasing or de;yvm k:a9q8g ocreasing?

Suppose you are standing on the edge of a large freely rotating turntable. Wha s+pm/bkm5l* ia+sik6c.; ppat happens if you walk towk/. a;5+*ms b+amcplsipi kp6 ard the center?

A shortstop may leap into thp,xbmbv ezl1a4bg:2ro-9 sx ,e air to catch a ball and throw it quickly. As he throws the ball, the upper part of his bodyrvo4-2ep glx,xbmbb z,9s1a: 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 conservation of angulzfxfnll*bz p78 *i.r 6ar momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the secfi7rz l*xp6lf. z*nb 8ond propeller can operate to keep the helicopter stable.

  Express the following angles in radianr+iw)zgt np b:(g-r9rs: (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. Calculate, using thqzc-/ck:3p zse information inside the Front Cover, the angular diameters (in radians) q3c zp/k z:cs-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 from Earth. y 4,-lk.1g2izah 5eyh+8roi qtg4xmuThe beam diverges at an angliu4kl ogxa hz2.,yr y 458 1+qig-tmehe $\theta$ (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. Wx(g 7 ))zwzn ymm7th9x +yq3)rd5aszahen the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the bl h(7)w75am9yzdtzxn)g3y q)as rzm+ xades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 f387rs6 wsfalm to another child. If the ball makes 15.0 revolutions, what is its diamsf3af6 w78sr leter?    m

  A bicycle with tires 68 cm icqytx(il3hk mr,0am : z (6mo.n diameter travels 8.0 km. How many revolutions do the wi0q( m x(.zh6 yolca,krmm3t:heels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rot) dj) )pryuw0tates at 2500 rpm. Calculate its angul )r))0uptjywdar velocity 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 revolutionch1 b(h c3 y8lnd:cf0y in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from the centery8ldf(b3y hn 10 hcc:c?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocitb7qj,zy0ogym)7zfb9g, :z bey of the Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a poi 5xb lr1 o 7xe)gyyeh32;tti2tnt
(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 fro g(z +j.6;uny b8vjmikm the axis of rotation is to expermy.+u b;nvgz8 k(j i6jience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about i s dq(y(ek6r5hz i.ro3ts center from 130 rpm to 280 rpm in 4.0 s. Deterdro(ryk( e5 i qz3s.6hmine
(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 mm6; l* qf0zvn$R_1$ 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, i o .-rg4un6/5fobk0jdod r 1xastronauts aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated fronj-ob 6 1dxd5ifo4.r /kugor0m 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 acceleratescv 7+spj s1:cd uniformly from rest to 15,000 rpm in 220 s. Through how many revolutioncv :7jd1c+ss ps did it turn in this time?    $rev$

  An automobile engine fl,h:*r.apip w udd8( slows down from 4500 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 thz8u /aftui t /n5wbzg fq+r/76e stresses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn /a ui58b7tnu+qz/rg 6/tf zfw 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 e)n+ gpeq1l0+x6 fo9w zdvik 8uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a poi1v9 g zl+iefe60 px8qo+d)kw nnt on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min Izje3s7i9+n7wfkwrm:9vv9 i9ly b n 6qt turns 1500 revolutions before kwe f+99j7q s7viz969nvb myn:lwir3 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 revojqw k67 mzmo/) r-1xdhlutions as the car reduces its speed uniformly from 95km/h to 45km/o/61m-)d 7zjxq k hwrmh 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 reduc,n ( uabs5/dmies its speed uniformly from 95km/h to 45km/h The tires have a diamd(un5aim sb/,eter 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 9yaw (2a8yl ,ub i1bxbbike puts all her weight on each pedal when climbing a hill. The pedals rotate in a circle of radius 17 91x , uy8lba(i wya2bbcm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 5kn 1b,4dp qbb15 N on the end of a door 74 cm wide. What is the magnipb4bnd ,b1 1kqtude of 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 wheffqj2s2 r-9zaa k8k;f, t (bjnel shown in Fig. 8–39. Assume that a friction 9;k-tnf ksafj,j 8ba(2q2 f rztorque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are ielnk/nhmah5 *1 6ue0attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direclu /05nm henahek i*16tion of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening tbav-15k, x ;m/qdf0w 6lm2tz:kbhlf y o a torque of 38 ,1t fzkvh 6xb50;lmykf/ba :ql-wdm 2$m \cdot N$ 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 ra )t,j(pgfn 3utdius 0.648 m when the axis of rotation is tj tt)p(3, ngfuhrough its center.    $kg \cdot m^2$

  Calculate the moment nvbpvtf1sk1 4oyv; ); of inertia of a bicycle wheel 66.7 cm in diameter. The rim and tire havpkb s;y fv1)t41nvvo; e a combined mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the 5el s(;4h ;zcnyon6keggi5: t)bwc1u end of a thin, light rod is rotated in a horizontal circle of rakc( se w6 g:t bo1c hnn45g;eylu);iz5dius 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 herji 47in5i; rotating at constant angular speed (Fig. 8–42). The friction frjin i 54;i7heorce 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 s0crcw(4 lff f)hown in Fig. 8–43f)4 l0cf rfc(w 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 atoms whose t;baa5/ y5wz3x4wgxz fzhv a4 -otal 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, uniformw0c qc5c wtyg36- r:6uvdvu3,p.kyiu 7 tfn3m cylindrical satellite spinning at the correct rate, engineers fire f.0ycy3vtt upc 3mvwnf6k: i, g-3 6u rdqc57uwour tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required 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 iq 7k z+012kwz yi:rwah 7 ng(ehmmu53a mass of 0.580gzk57ae z2i k mh731 wy+qmuhn :w0r(i 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 i9wipy;..qe l 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 merr2r(3q.q ek8qi bmmxg3y-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the accelerabqxe.q3 mgk8r(23qim tion, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotatinjs/p+p r f*mr 4s8-8pz*rn0n3b sjfhcg at 10,300 rpm is shut off and is eventually brought uniformly to res4b+8mnpz*hs prnjj 3*/8s-0s rpcrf ft 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 ball ate, /oxqf t+-lc 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by 0iw2l(qh ) z/l;bvjdgthe action of the forearm, which rotates about the elbow d(/0l2lhjvw)i; q zbg 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 bla 8b gkczh/ ud01y*cwk0de can be considered a long thin rod, as shown in Fig. 8–4ckwz *hd /00ckgb81y u6.
(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 o 5 fb:oytqwk+)wzn2e 6f 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 acceleratbgs vhvv +fgzu+vd3 6s1++ vdexz l866es the hammer from rest within four full turns (revolutions)xub66sv1 sld++gv dzv+zvf + 6 ge3h8v 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 ahi yz0*+nc/(hk9o; ga b zbd ipiw)1gyo9v7p8 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 develops a csr2c6d2zi tl0y .fdw1+k sj, 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 9c-bbgkz9swh(p xn4 w 9dzq1 10-k5fx s.0 cm rolls without slipping down a lane xk- n wbh5dw4gk cx1-zf s0z99p1(sqbat 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth e fu)f jo/ 0ceukco10;it n-1vwith respect to the Sun as the sum of two terms01kenfto/ oi0cj;)c e f-u1vu,
(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 kh :dpk,,v emrl +)o9kvg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a kv eh l+kro,vpd)m 9:,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 alm2z v y5xtxsd (y/g:t35fy 4end rolls without slipping x5tely xy 2yt:/z4 (5vf dsg3mdown 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 balanced vertically on itso plp ,xh91gu0 tip. It starts to fall and its lower end does not slip. Whatolp u,01hgx9p will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the en.wy ,ijxb jua acp20*5vwf-7 kd of a thin string in a circle of radius 1.10 m at an angular speed o.5ki0xw j7pa-2y,vu a jfwc*bf 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg u*4s f8s4 durkvniform cylindrical grinding wheel of radius 18 cm when rotating at 150skvr*s4f4 8 ud0 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 his5fxx:2yld k/b lpkuno uism)0z .75 xj 9ayn42 side, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms zl n/jn yibk 4m x5y9 )lo.xs520ad7pux:ku2f to a horizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce her momentx03 slf ccfc/9g/ -*x(1flt,tdueb mw of inertia by a factor of about 3.5 when changing from the straigc 3cl*9-df (1lsew t m x/0b/gffcu,txht position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spi0ox i;-)ql dlg3ig 4rqn rotation rate from an initial rate of 1.0 rev every 2.0 s to a final d; lgqrl4g)xiq0 3io- 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 through its center at a freq85wvhi d o/pb)tv, hpfa.-oy -uency of 1.5rev/s The wheel can be considered a unifw fbp )h/t5vv-hp 8,yi. da-ooorm 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 spinningi8sr9-sf,pcalv r7etz 1 x52t at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular mome. t(;pmh 8fk 4zof(nfantum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is d.oa wtcb7c:8 propped onto an identical disk rotating at angular speed 7:cb.c8o wtap$\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4 wo6 h-;wauh l2sj2e1y$rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotata qse9032k; )d0rmqzfp :;faj mnf*qeing merry-go-round platform of radius 3.0 0*ef9qm zsfdj)qem3;nraf:2 qap; 0km 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 velo )0ci fz o5s1u. )jfbidkg/z3rcity of 0.8czs)ojri.)igufbz 1dk 3 /f05 $rad/s$ 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 hmvbsq/zn w50vd t, dw 4an-ua4;,x5 bfalf 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 ratea sn,0t4nfvww,d bqav;45u/ xb5 -dmz be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in excess o 9u9j c3bsuo9- 4 .ubze1-2+rws gbkuifs frg*f 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 dplhjst:d( dxy0qc m9;f h6 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, initiallty.0* s oeiq:e:+vgwvy at rest, that can rotate freely without friction. The moment of inertia of the person plus the *v+yg:wt oie0.: esqv 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 edj0fx6jf ef, l,ge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless beari,f,lf6 je x0jfngs and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with thed m3,r 0 hg-gxydt(/cp 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 permrd0dy-/3tg ( h cg,xpson without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces the Earth. Detk g;2tx 5xtxt)vk1a3vo8r3 rbermine the ratio of the Moon’s spin angular momentu8 b xv5t;32rraxtogx )1vt kk3m (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 rate of 1 m/5,onj ceqv1 :b$s^2$ 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 0.2hlh-1 f dzbz7tmbzvw8g/m2; :0 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the tom8bz7 dgtw:1zl; h/f vh b-m2zrque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made ofgcue em7i o00f;/ m0lx two solid cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (cog00l0mefix e/7cu m; oncentric) 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 the angular speed of the bc/* kx 4psk6krear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as great, were r9vf8 ei -md:rut-qa,p otating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 1u9 : pv, -itfdqear-m81 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 it to use energy/ehwd b ::cw8w stored in a heavy rotating flywheel. Suppose such awdw8: /wb eh:c 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 illustratugoi(n :v8.v/krj) pees 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 horizontal sur y u8mahne).:j x6:a,jun6yrv bd* wd*face 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 g w+jqn6im9mcns,* 7w rm*gq9length L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizon s jrcn*,*gmn 9imw+76 mqgqw9tally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the floor, s ul9 qgpyoi-3r 1 -1zaojqw55and we want to exert a horizontal force F at its axle so that it will climb a stepo-i1w5ryquj-q a9g 5z s 3lp1o against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4enk d/rw-3 qnx0e)2f l.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist 3qewde2-/x0l n nrkf )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 lbi/) 1.3sa-gvia nxhy ength 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate ofi /-yh.a xv n31i)asgb 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 soli o yeig9bp95k+d cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tighb+ey9gi9p 5ok tly against her body.   

  You are designing a clutch assembly which )k*o1szvtnf k/+mk:a3ja3kz consists of two cylindrical plates, of+k *v3 fza/3tzkjk1on a):ks m mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped rough track dq wv 0l 1/afposfm56+of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it ilw1 /mf5 vq+sf0 6apdos to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not tm)bgs6 5l7kkassume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reducqoft(7ux-,gy p m,qg .es its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) Whyp,7m f(q qux-g.tog ,at 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