A bicycle odometer (which mea..s +m1 *x juf+aoblccsures distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 1 mjs.*xo.lfbu+ cc+a 24-inch wheels?

Suppose a disk rotates at constant angular velocit+gq r wvae;)j1y. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear accele1+j weqvg)a; rration change?

Could a nonrigid body be described by a single value of the an )7o7q:e-dmt+eohuo bgular velocity $\omega$ Explain.

Can a small force ever exert:y.fj7zg6 i (xz2rf n2bqb4gq 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 no i smj.e6i0-0+wheodo 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 anw.s o j-e 0h06oiemn+iswer this.

A 21-speed bicycle has sevj h-/wdw,lnkjna69bi2c u7o; en sprockets 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 harder to pedal, a small front spronnl bwh6i2; kwjad ,oj/9 -uc7cket or a large front sprocket? Why?

Mammals that depend on being able to run fast have st7epcnmw xjv r :7 0xe -rb13x-p9a.oglender 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 .w-tg9bea37x 7pcvnr xpm- 1:oj re0xis advantageous.

Why do tightrope walkers (Fig. 8–35) carry tsglh9fm3c) *a long, narrow beam?

If the net force on a system is zero, is the net torque also ze ga4x6+;af9c7/n mw(ott*w(k jx ilutro? If the net torqlaxw to7tt* m/(ku;a4c9wn6+j(fix g ue on a system is zero, is the net force zero?

Two inclines have the same height but make differentab-;a,gb0n ud 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 grnu -;g0ada ,bbeater? Explain.

Two solid spheres simultankne lquzot i8,a2/gwo k s9e:(6ovw9)eously 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 there? Which has the greater total kinetic energy at the bottotno ( l gesovww/9 qke89uizoa ,6k:2)m?

A sphere and a cylinder have the same radius and t nh l (6(d75k2ltoxsyjqq:8wyiy2 + athe same mass. They start from rest at the top of an incline. Which reachqly( 2s y8q2o:h(t5n i6l7tw yxkj+ ades the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most m-pv6.:sqxxkp9(y0l1;lt e zgy 4 dfmb oving or rotating objects eventually slow down and stop. Explaiqs l.pl46gt m 9kpbx-zyed( v 0;y1:fxn.

If there were a great migratu0c6 odk+e wlg:3fdh -ion of people toward the Earth’s equator, how would this affedkd+ 0 uch3 gf-:lewo6ct the length of the day?

Can the diver of Fig. 8–29 do a somersault withlq5h l5xjs/ 0zout having any initial rotation when she leaves the bzl0hx ql55/js oard?

The moment of inertia of a rotating solid dirwt/z79kx 6td53sd qmsk about an axis through its center of mass ist6 /m53 ddtx q9wskz7r $\frac{1}{2}WR^2$ (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-kgkksa6jnj4 wt*ekmfkn;p aa)6yg6 561 mass in each outstretched hand. If you suddenly drop maastnwakgkk65pe) ;6jkf6njy64* 1 the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the sax9(o 84avzm i7lxe5dx. cwkl) me mass. However, one is hollow and the other is solid. Describe an experiment to determine whic8xl7 54xv9kxal idz)c.(woem h is which.

The angular velocity of a wheel rotat0qh0i/dh yvj j4 q: ywwec:(y(ing 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 tangenq(iyv /h0w0j:qhe:w dcy(4 y jtial linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the e45dy .tvbd tj fb22gv.dge of a large freely rotating turntable. What havt t.bbv245.dydg f2j ppens if you walk toward the center?

A shortstop may leap into the air to catch a ball a287ak( q zegaind throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hi 7z(8g2ik eaqaps and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angult nj ):n(zspt.ar 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 heljt)p zs(. nnt:icopter stable.

  Express the following angles in/oqpe; 2:ybg79exqh u 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 coincidvt k50bi-p9awb9 n ;o 9s.lx)msb ul*rence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the 9w5b; )k pv9r axusstb.-o0lil9nbm * Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 kmuvg 1n6 e-28bmfa nk7t from Earth. The beam diverges at an an2-1 gtemuabfnv k8n 76gle $\theta$ (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 cmr gn1 ; *co5;adt+hlrate 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 bd*hcgo1c+ t a;n5r lm;lades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to f 10to9yzg ;wdanother child. If the ball makes 15.9g w;1of0ydz t0 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 kmfb9s*1ueu w oa1cy28to tj.u9. How many revolutions do the wheels msb.182 fy wtu9*9oje1 ac uuotake?    $rev$

  (a) A grinding wheel 0.35 m in diam f-kxz( k6uu6v*l.k8 hrg9yuueter rotates at 2500 rpm. Calculate its angular veloc9kkzkx* uuuyl h8g-6f vu.r(6ity 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 rbwwwy7 emk)f+5a. yt)5nuw e, evolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m y bw+, wuy met7w.enf) 5)k5wafrom 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 j3or)pdcz.:c w9z, we Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speo8a y., a d051b9j8uromlnk teed 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 particle 7.0 cm fromac+b, vfwo (1 -dzw9plvkoz 83 the axis of rotation is to experf8v k1 l3c ovbw+d9(aw,zzo-p ience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from bl1 49g6 p9ejr c(6yiw)wvflbgb7 +zl130 rpm to 280 rpm in 4.0 s.6jvcf)(94by pb llgir+7w1gbel 96zw Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radiul xx cz8a5lr(,l b3p2by 9hu4u(6hoj ).uljtds $R_1$ 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 spnsc qe70 h/hv.acecraft put themselves into a ss. chnqh /e0v7low 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 cj*7h(v;aglf many revolutions did it turg vhl;j(fc*a7 n in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 12 is5dz09 -(,zhtygd ig00 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 thet,mh kx xi*).lgt,preh13sipfq,x 23 stresses of flying highspeed jets in a whirling “human f3sip ,3 2*gx.h1kprt,em,txl qhxi) centrifuge,” which takes 1.0 min to turn 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 uniformly froeudeyc )7.y;f// zpkg m 240 rpm to 360 rpm in 6.5 s. How far wil7cfe .euy;/) yzkpdg /l a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is ru*56bna dl*+d y.e9 rb g)bxfponning at 850rev/min It turns 1500 revolutions bbd 96db+yrfnob.e5l*p*)a xg efore 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 car reducesv smdiyk0 j(.a4jju4* 17 znqq its speed uniformly from 95ksvj1q40q *i mdzj.jun a4y( k7m/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

  The tires of a car make 65 revolutio72s8 ,i mc,j* d7cz jqp+dbk.xpq;h afns as the car reduces its speed uniformly from 95km/h p m7.kz8d, jbah2cciqx +7*,;sfpqj dto 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 puji/ gde -k4xa1w)rp1bts all her weight on each pedal when climbing a hill. The pedals rotate in a ciraid1w g k4) 1xp/-bejrcle 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) 9pbfddy9 z0ml +spm* door 74 cm wide. What is the magnitude of the torquy9lfms9 dm0z p d+p)b*e 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 wheeldlsh - u/ ;ief,dnjfqbf1wr/xgn/g b2a51/o4 shown in Fig. 8–39. Assume that a friction torque of 0fjr 4ns,hei x-15dbd/g aq/;l2 g1bnfu/fo/w .4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to th16g yrgmn3n v6e ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then rg nmgr66vn 31yeleased. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tighten 5:pn;vpg9rnv/hi yg .ing to a torque of 38ig5n9yg p /pv r:.;hvn $m \cdot N$ 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 7knkp(v)z g kryz6,m,radius 0.648 m when the axis of rota 6g7kv)mrpnz( , ,zykktion is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. T)+i zc t:fn(ukhe rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ig:ucktfizn )(+ nored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the en1b3,8tccdk01ggnhp a d of a thin, light rod is rotated in a horizontal circle of radius 1.2 m. Cag k,3bc h 01ntdg81pcalculate
(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 angula t,t.guq jd75nqzkp)9ztvj)5r speed (Fig. 8–42). The frictun)5qv .jztqdg9tp,7z tkj )5ion 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 oo pq)z kw*kr5;bjects shown in Fig. 8–43 awrzk) 5;k oqp*bout
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen at:y,q3zwe y t3uoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the correct r*3+t.obxdm yuate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket ifm *uy+o.tdb3x 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 cmtec z1,bjjo(b.(n(zhc dn hn /kp /s:y(ayj3, and a mass of 0.580 kg. Calculat(ze: 3h(d,j1/y bakp,h( z(yjn.cnt/s oncb je
(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;nnro w3 m/:4;+ttvmdat-m c/xyzc s, it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merry-go-round and is ableaz.oi0a-2l:kw p zzy. 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 radiu2l.a.z: ow yka0pi-zz s 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 10,300 rpm is shut off and iaf 3e:v vc*lq7s eventually brought uniformly to rest by a frictional torque of 1.avevq clf 37:*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 aeyynce57/4 ozcvn 8 6yt 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-khoa n:;o(su7ig ball is thrown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8 os(i7uh a:o;n–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 bl5s ljrsu80t1 c:ztd4gade can be considered a long thin rod, as shown in Fig. 8–46rtz4lj018 dtu 5sgsc:.
(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 a/ 1 -7 ygj *r;8surmyr:zgwnwb (sxm3wb)6di 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 from rest within four full turnrzd+r:nu8w d trsv ,cpvh9;,-5ad q8hs (revolut n5d;av 8dh-:u+ rwp9t, cqv,d8r srhzions) 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 of iner2hjl u 7*mg: y8uvmpi*tia 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 devel.w 5g0y0.cngehh6 kb9 czaqh.ops a torque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls without slippinctsy 2yujho hk22z))4((o/ pnard+ong down a lao)sojy(h222n un dk/apohy4z)rt +c ( ne at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energyvm (ds 7kr:xy4 of the Earth with respect to the Sun as the sum of two terms,dr:(m7xy4 vks
(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 rai)7 svpwl8m k/d0lzot6fae- 0 dius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1fpmio e 0kaw ) s8d60z-t/7vll.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and*h t/jste74d e mass 1.80 kg starts from rest and rolls without slitds 4 te7j*/hepping down a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

  Two masses, $m_1$ = 18 kg and $m_2$ = 26.5 kg are connected by a rope that hangs over a pulley (as in Fig. 8–47). The pulley is a uniform cylinder of radius 0.260 m and mass 7.50 kg. Initially, is on the ground and $m_2$ rests 3.00 m above the ground. If the system is now released, use conservation of energy to determine the speed of $m_2$ just before it strikes the ground. Assume the pulley is frictionless.    $m/s$

  A 2.30-m-long pole is balanced verticalll43yg1nt agk (y on its tip. It starts to fall and its lower end does not slip. What will be the speed of4g (k1nlta3gy the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular mo- 3yr,k lf eej*2rkqhblg,69h mentum of a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at9ejq -flhy2 kr*6g, lkebr3, h an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindr/x7v09f u qauuical grinding wheel of radius 18 cm when rotating a fvqxu/u907uat 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 his side, on a platform that is rotating d6 h*j/0floyp at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, theo hj*0fd l6/yp 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 momentkze bb30flzip* /yhb0 3ysx:ggy6 g*6 of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makesg3**3gyse/ h bbzyl6 zk0xy:bfi6g 0p 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 spin rotation rate fde+lh.)/.rr az a2v fqrom an initial rate of 1.0 rev every 2d)a. zhf qv/rr.l+2ea.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 around a vertical axis throu)-l*lk j7lwmrz- 0wtz .d(ucc* rm ak1gh 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 fl7aw lm)*z0t mzw -1uc*rlc.rd (l kk-jat 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 momeny1m j b+m/hm6ntum of a figure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of the oh(nmk ku3n8 ,3mhcl(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 fxmmyd +arl 5no.5j ad4u.;u,5eygs+ of inertia I is dropped onto an identical disk rotating at angu5m +ral,;g ys5duaf4du.oj. xy nme5+lar speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.nk1p 7xp5lhh )4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotating merry- bg;m4+hmjwz ee,3 qo8go-round platform of radius 3.0 m and moment ofq,z4hom8 +mej;g 3ebw 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 freel7 j f*6 k08zilulnl(ipy with an angular velocity of 0*kpu(znl70l6f 8 ij il.8 $rad/s$ 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 eventual ;9vtfaul (g; y .s3sioeu6bu7ly collapses into a white dwarf, losing about half its mass in the process, and winding up withioag3st.6ue9(uyl; b;svuf 7 a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the 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 excesstaa. f4x)feyfr .1p ko5r sw33 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 lqdg7xwbz7;wf y kusj *.60v- $ 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 c++kp( vbl dsu3an rotate freely without friction. The moment ofpu+d+v(k3 b sl 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 ah+frxims9, 6;)bf:0 p e6n z dpqs9udet the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless 69duxb rsize)e,f+ 0dqmp6;ps9h fn: bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on thmf(m jvit+eubeet 6(: f9 h3u5e ground with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behin:9t 6mh j3fb (eufi tueemv+(5d the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces the Eartqq1bk d,apy/3azg94f h. Determine the ratio of the Moon’s spdqaya4,3 bk fq9gz/1 pin 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)bdhld ,c u5 (lkych0nl0ec3(.r7igq from rest at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of radiu*2 y+cyuiw9k5k( v epcs 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Cucekv+ k2(*i9 y5cypwalculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of massm,v cw 1hx;t wh(5/bis 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Us1v,b wh(hm t/w;i5x sce 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 rear wheel fn)7u /vbg s2g.-es,f rrdafroi-,*q($\omega_R$) 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,awzzp u9lqgsh**9(3 s but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star 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 mo *9aspz39(ulzh sgq*wmentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollutio lizh8+)ghkot 3m1;wgy xc9f /n 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 350 km without needing a flywheyhi hl /8)c;x3 w9gomzfgk1+ tel “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 illustrate. :gp/7uk, w49ho hi9-btz3sv evzi wgs 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 g:w x*ut;i:j bhorizontal 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 ls: wum :3ncej,ength L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of t3uwj::en ,ms che 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 r0 nw1i;dc et-jk 7s*rtadius R. It is standing vertically on the floor, and we want to exert a horizontjnr*ds ti7kt wc-10 e;al 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 traveling with speed v=4.2m/s on a flat road is making a 0ywob+ufjb +wh -k+1w,ah(e fturn with a radius The forces acting on the cyclist anbwyowuf+ je+hwf,a( 1b0+hk - d cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5 yz r .0tdtb*8yoxw4 m.m and begins whirling the rock in a nearly horizontal circle above his head, acom y. .dt 4tyb8wxr0*zcelerating 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 thin rods, makv*ohq-nx;m ;/o6sl qking reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly o*q;q/hlmnox ;ksv -6against her body.   

  You are designing a clutch assembly which consists of two cylindrical pla(( twlj2ie4sc tes, of mase2 (4cs lwi(tjs $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 -rp *gr)q(oj 8xyvw3t - nkw5grough 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 withouq xwkgwj-v85g - )p(r3ntory*t leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but dojcy( +tf) sod*kd8dg c+:8qjg not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uniformly *nmdzz8-04 gbwqvd c:4x 0 skxfrom 90km/h to 60km/h The tires have a diameter 8czxw 4db0 zmv0sn 4k*gxdq-: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