A bicycle odometer (whiyz5xk7xk8 w9wqx*k w(f. en3qch measures distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What hapwqfnk98k(. qww x5 x3 *yxzk7epens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angularki71ye yqt/q y4-y+5frq xi+d x9q-r c velocity. Does a point on the rim have radial and/or tangential acceleration? If the diskq qy-- +i9krq/7 1yxyr 4ief+ycx5dq t’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of tr:fcz7( y ozos2sd gxhfj8 3)0he angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Explic uf w-,rbxll),4o8vh dv 0.gain.

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

Why is it more difficult to do a sit-up with your hands behind your hep9v6dd 34cebs3jotqwf0 nj 17ad than when your arms are stretched out in front of you? A 67jfd04o3p9 1d vtscenqwjb3diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the pf4o8dhpoo9l2x1 /, kq jwat t/edal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is f,/2 lhoo 8daj9ttk /1wqp4xo it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower 0;,zti kl5 gtnlegs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of r0g ik;ttn5zl, otational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fibvfmoo9u5-e8 (8k3m,oj(mhvtbt u. xg. 8–35) carry a long, narrow beam?

If the net force on a system is zer 5wsmsei.mq7gm(cmd0h( l7 8 fo, is the net torque also zero? If the net torque on a system is zero, is7sw7md0.l(gf m mm 8e5(hiqsc the net force zero?

Two inclines have the same heightu; uv7y wh9x:r but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball auhr:9 v7ux;y wt the bottom be greater? Explain.

Two solid spheres simultanesi,ja(e23mb(;adu9z r 1tfy1as 4 xg tously start rolling (from rest) down an incline. One sphere has twice the radius and twimyab4(jarast u z , 9efsg1;x 3(t2id1ce the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and theqarc64ud2t s 09.ny xz 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 greater total kinetic energy at the bottom? Which has the greater rotational K.zr4su t6xqynac09 2dE?

We claim that momentum and akr;jj+tsye4-e. xo f(nf iu:5r- say;ngular momentum are conserved. Yet most moving or rotating objects eventually slow down and stop.(rterx.+okijunj se-sy:;- ya;54 ff Explain.

If there were a great migration of peopyitrhi y- d sd.(rk4j f/o61h+le toward the Earth’s equator, how would tsf rih1jy id+r( 6/tk.h-d4yohis affect the length of the day?

Can the diver of Fig. 8–u0ba.(pys0q k 29 do a somersault without having any initial rotation when she leaves the boa00bpu.ys k( qard?

The moment of inertia of a rot qht b38+/fpetating solid disk about an axis through its center of mqb3 tf/p eh+t8ass 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 holdingfq d h2k28c b-f(i ea7fy,2tgv8uk x+r a 2-kg mass in each outstretched hand. If you suddenly drop the masses,2(x2k h8ykr8 fbqtv,f2-ceg+diua f7 will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and havpjkiq-gptp .1s)j1eg(r ( x4/t jp;dme the same mass. However, one is hollow and th)dj pxtrgpjg/1e(pk.(mpsq ;1-4 j tie other is solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotatinz7j q ,i5yz.wkug2x5h g on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the x, qzykh wgu.z57 ij25wheel? 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 a.1cj (rsafck *6pooyx;i6g 7the edge of a large freely rotating turntable. What happk6ay6 ; 7xogsfproj(cac.1 i*ens if you walk toward the center?

A shortstop may leap into the air to catch a ball and throw it quickly. As he 9q*m jsm:hpxt;9 *mjhthrows the ball, the upper part of his body rotatestpm9;* 9*:jhqjmxsh m . 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 conserv gh1 b 5lpi6m,gjghh1+ation of angular momentum, discuss why a helicopter must have more than one rotor (+,g 1mh1l ghpgj h6ib5or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following anglek hglq4 6eh+:0gy 3odss 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. Calculate, using ob;2tq vz 3br-the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.v2t rb 3zq;b-o
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,0vuq)ibi zf 2n66iu7p ,00 km from Earth. The beam diverges an iu6,)2f6pq7iu i bvzt an angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. When the motor isq7v;jrz:-8h ubz b0hnwp2 ui7 turned off during operation, the bl2hz;:i 7z8-bu bpjwn0 rqv u7hades 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 child. If the27 dax(ktun .egzeoka, j4 6e/ ball makes 15e62eta4 ,ou kz7kxdgn (/j .ae.0 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.h5 wa7vid7t 9m0 km. How many revolutions do t5tdh v9mwai7 7he wheels make?    $rev$

  (a) A grinding wheel 0.35 m 1.sh5 smt y2hbin diameter rotates at 2500 rpm. Calculate its angular vyt .512m shshbelocity 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 revolutionc( (6i kof+qrf in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1. f(f+6 coqki(r2 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 itk8qi77 f;vl7cs/zkp x s orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a poi 6wwn+q0btg wt* pwi- kz).*ugnt
(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 ue ,5fkk15unmvjy j .6from the axis of rotation is to experience anj16.emkk,5 5fuyj uvn acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center fromw,udyah xj(oxo-: /.g9z ri*d 130 rpm to 280 rpm in 4.0 s. r(.9wo:yoduxi ,/d a-h *gxjzDetermine
(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 radiub6ek*;d s/)x-nf+ * yiqnyvoru,8c jq5c - ztzs $R_1$ 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 Apollocj2c d m:l7ri8-bp1ub spacecraft put themselves into a slow rotation to disbblj 2dp-71m: ciur c8tribute 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 acceleraon5oq2 imu y38tes uniformly from rest to 15,000 rpm in 220 s. Through howoo mu q3y8n5i2 many revolutions did it turn in this time?    $rev$

  An automobile engine slows doukw 3).d 6h t4f8biqiswn 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 the stresses of flying highspehy2s x95pp :qyed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 209 y25sqyxp:hp 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 diatk64m5 z2,f4xo7 h t/dmy1dtojmf(v z meter accelerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on thfjt zdxf (zmdyt645vmo m 7 t4,k2h/1oe edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it u;xvplkd c ; o))ke/+;wfn0nka3q,qja xi . 9mis running at 850rev/min It turns 1500 revolutions be;/n w+ ax9 ) cak0fiqvl;;3jeokpduxq,) kmn.fore 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 reducesbzrh a 8jo)0kt;fc/e9k/cko0 its speed uniformly from 95km/h to 45km/h The tires have a diameter oekrk/bfzc cth0/ jo08)ka9o; 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

  The tires of a car make 65 revolutions as the car r fvl2dp,q +8ezta 9nl1educes its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.d lant+pq291z8, lefv80 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 ).7qkxe dc t.vall her weight on each pedal when climbing a hill. The pedalstxqd)k .ve.7 c rotate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cm wide. What isywsxr7h t+.4s 9prndcn6 6f)h the magnin796hx 6cpys+drn ws4h) r.tf tude 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 aboun(qil2m8 /t hw5j6ga7 xk7 3(akhrxr yt the axle of the wheel shown in Fig. 8–39. Assume thgmi5 ak /t h(nhrq7k3 (j82y7 6wxalrxat a friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends 7m h 553lz3gma+tag mbv gk(c3of a massless rod which pivots as shown in Fig.l 7k(gh3m3mgz ca35 tvbg+m5a 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tighteninmn 3b)i6spptx 4 *y5kag to a torque of 38sx)akn 5t4 b3pyi6* mp $m \cdot N$ 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 radiu df):huv65(6ev4lx2i nkqac ds 0.648 m when the axis )hv qax 52n(v:fclke4id 6du6 of rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. The rim jnw ;a; v4i,yaand tire have;n,ywa;i4jv a 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 end of a ao :)nrkc1y s0s+rb)q thin, light rod is rotated in a horizontal circle of rad)oarc b):nr yk0qs+s1 ius 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 ok(9y8tsqy, ma+; vojp ja+. bm h8vp;a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction fyam+ 9+t;ky(po, ;.qvp aos8j bjmv8horce 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 theaf; wk,a*s.zdc:*p qf-ak ciyu/ww97 array of point objects shown in Fig. 8–43 absu kcd fw9 i,f7p;q **/a -waya:w.kzcout
(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 toleq)chm 9 g/ac9 zmdi723rgs*6i 1s yctal 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 cylindrica8fcs ,u pt8zy-l satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a rads ptcu ,yfz-88ius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 cm and a mass omb4bg c)0ml .7u1,vkx qspvb, f 0.580 vk07v x lp,)gcbbbum.1 s q4,mkg. 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 fiv ,zcuzp:+sxa6 4o2lrom 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 smalpq mgd3wcx: *f/la eo 6nym1:(l hand-driven merry-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 (eaeq/cx: f: gapnmlm6 *3o w1d(ych 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 is8 8ijh ly,2*ff xrmt;u eventually brought uniformly to rest by a frictional tf; , 88y*htjrfl2uixm orque 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 accele m1hn 3tbepv/3rates a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown somix b6; yiinae o7.crq:7 nd53lely by the action of the forearm, which rotates about the elbow joint under the action of t ;i a7dc:5 o eiri.36bmqnnyx7he 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 considir)v0:duqlng*m.s9 pf :jx (8f8it lm g bp.-jered a long thin rod, as shown in Fig. 8–46l v(bi .pf:jsgtgn8q*:di . -lrmu0p9m)fj8 x.
(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 consi 37ww(w0u ifmksts 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 fr 2yv ( 28ou5acnvhebo,om rest within four full turns (revolutions)ehya o528,vv2b( oncu 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 inerth6jmcpg-p x *2 f,/nrpia 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 of 20ewfn;l q4xm ,80 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass m ef6/8bydokbb51vab8l 5 4vc7.3 kg and radius 9.0 cm rolls without slipping down a lane at bc bb 8/a5e8k 46lbm vdv5fo1y3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with ,40lpq7tf hpk respect to the Sun as the sum of two thfq 0p74kpt l,erms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius of 7.50 m. How much net wor0+re:isrs)xjq5e x :xk is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.s:x+sr)ei0qjx x :e 5r00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rf9j *0rs1su -ou ub yb.;eha/best and rolls without slipping b.fhe s/b0uab *uyrjo-u1;s9down 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 vahrofug,qrx l, h7 8;,ertically 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 polx, uo;7r,8lg aqhh, fre just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 1h 5;xzd.yyds0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an anguladyyd1;hz5sx. r speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindricalt 8iag+4xv ryk*3fs;c grinding wheel of radius vcr3 f*i kxyt+; 4gas818 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side,jeqr2 ku54k2 / ht.hqd4xf0d u 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 dt0/d4dfhr. eq 25 jkuk4qhu x2ecreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce her moment uw : hcy186svzof inertia by a factor of about 3.5 when changing from the straight position to the tuck6u: hy8s v1wcz 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 spin rotation rate from an iniu7rk .-4 zldzjbjj;.7hzn:u* ermz )o tial rate of 1.0 rev every 2.0 s to a final rate4z)ok7 n7jzm.-.jzj*r bdh: uzure;l 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 rfis(6nzp6y y4d/ c h7sr y7f-aotating around a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3cy767yi (6a/4h- fyspz nfdsr .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 sj,p8m(kubg; j kater 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 Eavc, (p3 cqpwdj4a-g g-rth
(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 cylindricle qr+-w/w+lg-;hy gral disk of moment of inertia I is dropped onto an identical disk rotating atl ;w-rwge++/rg y-qhl angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns atey a.sk+g; kp3 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at t b7 t(upk2yhu(z(ha7 :rpn(57 *yshtp mlv2 imhe center of a rotating merry-go-round platform of radius 3.0 m and moment of in 7 tn2hm7r72h azp*u:hplyp5s( tmvbu((yik( ertia 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-b:f**f;7hv i*fnf nbrm70,z huug*h oround is rotating freely with an angular velocity of 0,;ifo7hvhr :f *m*buf0g nu f*n7*zb h.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 eventually collapses into a white dwarf, losing about half ib(64;k k-//l- fs s he//us8kr6mleyzqavz w uts 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-a/ (eyks4kzws ks e/v lubr;lm/-8zqf /u66ho 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 e; tamsuds3;i :ncho- mw++ck( xcess 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 qv nl0nl:88g t$ 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 can rotate freely ;h7uizzeasr3z /f. ;cwithout fric s;rz;e.fiau3/ 7hzzction. The moment of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diametergmej0unxi b7 ,4 v+e..ki 5fptf a7oun;4ws8m merry-go-round turntable that is mounted on fris. eab;pu05feug7okt 4 +fjv m.nn 8ixi ,w47mctionless 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 the ground with the end of the rope lyip-r r8ih/ezz 9ng 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 behih /ie-9z8 prrznd 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 si+gn- jdz,l,hgde always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (aboutdzjgl g,-h ,n+ 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 5a-q-ru/evyapm *b )crest 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 oldd f+4vj vhev7bsx1)vhva13 o5 y i-cs0.)zjf a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque deliversa)b1d1evhjcv +v. d5s7jyx4iv zf -3l)0hvoed by the motor.    $m \cdot N$

  (a) A yo-yo is made of two sol/x2 ryo81qegd1y)gwub2 ,yziid cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1gqy y xz8bwig 1)y12e,2d /uro.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 ree+ x1rj.hk8vf ar 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 oz-5vujhzsi+ 1n7y f9y,1b0)-fysa yk8dw u szur Sun, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star 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 carrisjy y z- fy8vdzn15k0i sby7wu-fhzs,)9a+ u1es off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use energy stok(cannib /k la (btv9on69 /:ored 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 needi/b(k 9nl69 btc(/anv oinao k:ng 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 illustratc,(qsne i.s8 mes 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 surfacmlx cx/k5d), gvh* gae0t7rk ;*wpvjac-fn 58e 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 L0b:,(s1l o5 lo/z llqv/saeji 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, ,(0:ev /o azo ss1ll/lqjilb 5and (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 mbs290 2m8mmiv 9qxgrtg/,t qr qfn7* floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which 7g2 qvtm,qf rms*2qx89tnrgm 0m/9bi it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling qz ,zu6v3sk; ywith speed v=4.2m/s on a flat road is making a turn with a zs z;yk6q ,3uvradius The forces acting on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock3ms ;eyvpuvd07ad92o into a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where ;vp03e9dsd y7om ua2v does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her a-y;nbvw 1b th/rms 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 whv-b/b; nwty1ith arms held tightly against her body.   

  You are designing a clutch assemblymikiv 4t, 4do: which consists of two cylindrical plates, of t4dm:iko4v ,i 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 radiuhhg:i7b jl5l4s r rolls along the looped rough 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 lo4ihj7h lb g5:lop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not asszea/9r qa0w9hgt x 9:zume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as h-6gu1oeb/:plh* l de the car reduces its speed uniformly from 90:e6h l ob*delup-gh1/ km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s