A bicycle odometer (which measures distance y.f 3btv(x*ha8t3 d ofn*+ gpntraveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle wit3 apo b(xny+t*. 8*nf3hv tdgfh 24-inch wheels?

Suppose a disk rotates at constant angular velocity. D z jxq5c6(:f5-xfclzih(f1 id oes 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 acceleration chan((x 1l-q6z5i5chz fff:dijc xge?

Could a nonrigid body be described by a single value of the angulh h dnq.la-0b/ar velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Explau/ar rmokabcx */+-2wy,-8uyo3,i uv ; t ctorin.

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 heu tk w.8h a97hibya,y3ad than when your arms ar9,7.8awatuhh yybk i 3e stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel acm +8 b+m-sjx0ctb6uqnd three at the pedal cranks. In which gear is it harder to pedaljcm-b +q0um8+cst6xb, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slend10xa8; he(c+ jrbtvyr er lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, expl;(8 h +rtra0jyb vcx1eain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, nar:06vlut9 / m-gxs e2/tpvmwekrow beam?

If the net force on a systemebhm ))jddw.*3vka8j is zero, is the net torque also zero? If the net torque on a system is zero, is jd3.jhk)*b)8wev admthe net force zero?

Two inclines have the sa8i g:. jr z7v ep0oerusm629nxme 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 ball at6 po0eijz u27n8.9rxvs:gr me the bottom be greater? Explain.

Two solid spheres simultaneously start 0ypvllp ,9jr *-3v0zjx inhq3rolling (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 l-* jvpqj,x00 9 r3vlzinhyp 3speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the sa jj y0hj5/.lva dhfe cjz*7mmp+),f4gme mass. They start from rest at the top of an imel5f+ a mj*cv,h.g 4zj7h) /jp ydfj0ncline. 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 KE?

We claim that momentum athndt ls du, onj)/w (,8k q.87v6iirwnd angular momentum are conserved. Yet most moving or rotating objects eventually slow down anh6o)qdr , tl k8i8 wsuvwd(t./ 7,ninjd stop. Explain.

If there were a great migration of people toward the Earth’s equator,rqw09 yu )5mtb how would this affect the length of the da)br590 myqut wy?

Can the diver of Fig. 8–29 do a somersault without having any initial rotation w 9h(dps gnku msoz.5,k9 byq:9hen she leaves o5,(mzud n 9hssk9.q9 :pgbky the board?

The moment of inertia of a rotating solid disk about an axis through rfk1n :f*w4lzits center of mass is 4rlw*k:ff 1nz $\frac{1}{2}WR^2$ (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 mass0a)m* nzz8oj x in each outstretched hand. If you suddenly drop the 8*nxzoj )z0ma masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identicaek +6fcy4h) a( vtaub(l and have the same mass. However, one is hollow and the other isyb)cafvk +(t6e 4 ha(u solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle pdogej p 3-o95f,r +eq v4ttt:whsnm 90oints 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 angut-pnh9dg: 5j re v9t f0eom+,t4soqw3 lar speed increasing or decreasing?

Suppose you are standing on the edge h/.-kjawe p5bof a large freely rotating turntable. What happens if you walk towawbe-ap k./hj5 rd the center?

A shortstop may leap into the air to catch a bh +qag) a34xkqp q)0kpall and throw it quickly. As he throws t)xpk p+0)q3a4qq h gakhe ball, the upper part of his body 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 angular momentum, discusp;sk +.hcpd:5ugvl 8g k. b*yts why a helicopter must have more than one rotor (or propeller). Discus;cgkh+g. yv:s.8 t* plk 5pbdus one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles igg(hyfa6,3 ,r+ab an sn 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 Earddx4(:xpny)qy te+, 6 5gum ryth because of an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of 4 qn) 5 dgrt,d:epmx(x6uy+ yythe 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. The beam divs8 hg v/m)w 4gkv:-hvverges at an angle s8vgv-4/wgv:vm)k hh $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotaa(*4xxx 8 eqljte at a rate of 6500 rpm. When the motor is turned off jxl(a4 xex q8*during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If-k ilgk f * d7kndvdk,q1j.o 5qd1c4-m the ball makes 15.0 revolujq-kd gd 4kl,kv.*ndo5cf1 iqd1mk7-tions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolqy 05peu+h,jiniet0. ei f) /wutions do the wheels m /)tj,pe iinuy0+.5wqfe0 iehake?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at amy)wd4ne; pd6) ib)g2500 rpm. Calculate its angular velocity in 4;m6)gdb) a)ydep win $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.0 s (Fig. 8–3 zkli6p9nq/t)j-6op *yc w hsn *r;pk*8). (a) What is the linear speed of a chi *h* nql;k9pyrco- /jwi6*6)zpknt ps ld seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth a 6iv+cc0 xdm*1 km4wpgs5mg; (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 po, mcy56u (anbjlv+hfof v(j4*int
(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 particlel2 h-p/4ibal;q7k f;h(apjp b 7.0 cm from the axis of rotation is to experience an acceleration of 100,0 ;-(lbl;qabp/f hp7ip j4h2ak00 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 13ls z tkr5;fdr0j8lb,-y,hlut6:p on-0 rpm to 280 rpm in 4.0 s. Determine r8zjf5t;6y 0tllprhdb,-o-u ,sknl:
(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 s .kn,+mw/zcx+hf2 vwa)0wbed -0gku$R_1$ 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 spacecp jc7t g(xnf41 /tl8xfraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated p1gx8 lc/nfft7t4 j(x 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. Tjjrry yuagh7y we s;0,x+54p*hrough how many revjp +syguje0 wy5r*7 h,xyr ;a4olutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 .aqxb sg :k( f t.h +cw.l;/kk2a(nxqtrpm 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 thev4c)+ kw5e im qk3 k(y4kmp8zn stresses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.mk4ywe )+kvk84(mpnk 3ciz 5q0 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 from 240 rpm tup ,lj( tuknkuug3:7 9o 360 rpm in 6.5 s. How far wi(u uju:g pt7u9,lkn 3kll a point 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 It turns 1knfd 8os/isy xm5,m3m- ql. 3ig +djm9500 revolutions sj .s9gkd3m xid/, 3o8yl+mi n5f-mmqbefore 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 reduces its speed uniformly fr:qm m v *les2*mcwzy- aafv*,(hc; fd3om 95km/h to 45km/h The tires have a diameter of mhef qa2,3w; fzls*ma:m(dvy -c*v* c 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed uniformly 78 gde olvk8a;from 95km/h to 45km/h The tires have a dlo8;87 ea gvkdiameter 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 y vg7g1mier,7 sn)hx/x*(1ol kx. cc pbike puts all her weight on each pedal when climbing a hill. The pedals rotate in a circle of rad*y/ xr1(.p, kcng77 xc1g sioe mv)xlhius 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 is thd,vrp 4da 4x./jeb:r86 ge qfwe magnitude of the torque if the fo jafxd6 /drgwpe 4r:ve,84b. qrce is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig. 8–39. Assuxm0( qi)mp-d kc;q +rgme that a friction tox;dp mm( ki+qc)q-gr0rque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of marxfxfo1+r 8gmju+ 0b2 ss m, are attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Iniojx++u 2rg0 x8b 1frfmtially 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 cylindsmh. s r 5+c4cxhn 94a*vvsbl,er head of an engine require tightening to a torque of 38rc9svsb xlc5s a+n,m4 h*4 v.h $m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radius 0.648 m whepw62r9 : v theh.myl(kn the axis oe6 r(ym:h.l ktp 92wvhf rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. c9b, zkj of(+yThe rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignored (w,f(bc9 +yzkoj hy?).    $kg \cdot m^2$

  A small 650-gram ball on the end of ab9/izx :rt7twkc d 7)v thin, light rod is rotated in a horizontal c /kcrzdw xv9 tbt:7)i7ircle of radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating at constant ang95ldjo : hcuz/ular speed (Fig. 8–42). The friction force between her hands and the :/59 dojclzhu clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objects shown in Fi cgp/ f;4i6 :5knwbj 8xc1vzkbg. 8–43 aboc5 fn pjk14 :/;gbbki6c8zxwvut
(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 tota2 df3u/;4wd7z. e f k9:mg sxlwi9ko-mqy qep2l 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 cylin;ox9quw2 h/ zogwb+, t*k* njndrical satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–449, *kqj znohb x+wt/gwn2o;* u. 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 a:ro* ugd-l mfrqb3e yj2h1+vlh/k g24 nd a mass of 0.580 kg4kqhl :jbgd23r- uolhy1 v +*/2gmf er. 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 qit)v xr. 32hxfrom 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-iyl alm(u5ic6b 714o pround 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 ed1bi(4 iup 7y6 oaml5lcge. 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 jdw;1 0( mcmck10,300 rpm is shut off and is eventually brought unif0m;cw1 (dmkc jormly to rest by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a 3.6-kg ball at 7m ub72ln(vd t3fk;yo9 $m/s^2$ 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 the action of the forearm, mis -./rb6n iuwhich rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball b-rsim 6/nu.i is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a lo zgw:nu51 6xkfng thin rod, as shown in Fig. 8–46. :fu61 kwgnxz5
(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-q49ufjp 94-b y*t ltw4v amfdf 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 resl5 :7y/t9kt8kyg i0qguz)+ gbyumq ** 1hg ayst within four full turns (revolutions) and relea+z t9)*g tk 1l8*g57 qgy y/uikqag0uy h:ymbsses 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 ofba 4 yz*htc 1qvwqf-:3 inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque of -zjldqwzj /v91 ek, qd2v;s6w 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 withwiayz5* ) f xyg85xff*c l:v-vout slipping down a lane at *xf 5:a8yv*wyfi- )zcfxlvg53.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the suir r:: kxc7)giqo9fa4o riv16tupe 91m of twor:611g)c9 4q:a7 froirvouex9ptki i terms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1 48x -r7r ph4jld3q 3 qvp+lr*nybg6hy640 kg and a radius of 7.50 m. How much net work3 y jx7hr nlr*8hq6blp- q4 vdy3+p4gr is required to accelerate it from rest to a 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 r/2.qp7 , p nu.dfcgaxrest and rolls without slipping d,apu/rpq2g .cx f n7.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 vertically on its tij kvnrxy/w.;5+wl ek :bf2h3w +jj uz4p. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Us wz3; +ux4kjy. jrek:5vl hjwfnbw2+/e conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end of a th q51sxm uo izc/l/q13ain string in a circle of ramc ql5u1x1ozq/is /a3dius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kuea;n e4 28pzog uniform cylindrical grinding wheel of radius 18 cm when rotz 4o2ueep; na8ating 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 ,)uiw5nj2bnhgg cla;+k+af6 at his side, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a laj wgbc)5gka,;2h6nuf +n +i 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 zjl6)h-epz3- h- ud wtin Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is hhuhz-dp 36 z-tw)-ej ler angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her sp cu1y)u(us4z fin rotation rate from an initial rate of 1.0 rev ez(u y)cf4us 1uvery 2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axn 53tbku8w; hk dx)q2his through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.08btdx3 wnhqkh;2k) u5 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 angula is/roww4h7t7h 3 9hmqr momentum 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 bwxotwzx-t l/3 5ay 4(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 dropped ontotwim ;6k/zf/kdn eak5ic t9rboy 5 6*2 an identical disk rotating at angular specr6 6od/;/kn2 95k ia*zkbyt5 tem fiwed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns agedkrus5.lcz t8 *yn:, .sy/ft 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 the centtuie59 b q9plc-/m(eyer of a rotating merry-go-round platform of radius 3.(tqm lye995up- c/ieb0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is zz/8( ko sp 02+0guoaxt+g xazrotating freely with an angular velocity of 0.8 a80t(/ ga0gs kzpxz+ o2ux+oz$rad/s$ 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(iy;j e,5kgka into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s nyiaj5, (e;gkkew 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 excess 6-* ow+rm-r:p6e (kbmh4gvukk rsl1ibe )-z nof 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 v( 213qy0nbzz qru,h0 y9ottk $ 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 fre,ffga/hbx j 4+ely without friction. The moment of inertia of the person plusgh/, 4+afjbxf 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 g *zn3vo63gjqx :+ 2zse; xy9yk gp1ijof a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a gexxg2i9 ygsp 3jnz;ykv*36j:z+1 q o 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 ofc0x6ihbiu g825c sap/ the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding on xap0c6iubh s g82i5/cto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces the Eay1v v 2.lo7kw.9ogmzyrth. Determine the .yoz w moygvklv.129 7ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates fromqm ;+nd8 va0cxxp06 vz 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 radius 0.20 vh yq8vcm3lj9 d./ vg(zze1,lm acquires a rotational rate of from rest ove1v(/lzyj9lvvgzm h3 c.d8,q er a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical xhk ,r gzqg66(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 diamet(6q, hrzgxkg6er 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 7ijqq3 /kwup1.a bn9h2icu4a rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of o0y-8h8u0slch2h pp jgur 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 carries off no angular momentu c8jul28sp hhg 0h0p-ym.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use ezitq+ o: ffcs cfgk3(x956.f anergy stored in a heavy rotating(:fcf3t cfz+sig9o 6x 5qa.f k 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 flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates anq)pz4 /6k tvaq $H_2O$ 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 surface atlxj)jj;lr1dx 8rf4m; speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we ignore friction)*jpog3,0*c 3ampt. frduquuo:5 e3bx 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) thej: u pbecfxouarp,3u3td* qgm0 3.*o5 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 bnfx;;j ik32gxgu2fls8 5*pq, and we want to exert a horizontal force F at its axle so that it will climjbil2q 5fx 8upk *f3;x2gnsg;b 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 ism hn9 vwr384 dytw7j1q making a turn with a radius The forces acting on t8vtqr n3ydwh9 7j1wm4 he cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of lk xir0x1f4a x26fi9xqxr, . zlength 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a ratr,.xxffz1ixka9rix0q 46lx2 e 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 skateh t/zy6 f/npm xu) :arkr27vw8v l4ed;r’s body as a solid 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 tightly against heuvfw n742xrk t /y /rzpmv);8ha:de6lr body.   

  You are designing a clutch assembly which consists of two cylindrical mn :03of.jp phplates, ofjpf0 nhmpo3:. 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 of Fig. 8w poc9yw:2 w zy1wmh-0-bg*zq–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Ass-y1gwcbw-:yq0* ozhm2 wz w9pume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assumk biu-ze cb,9;8*rvog e $r\ll R$ h =    (R-r)

  The tires of a car make 85 revol*zus1 m nb7s9lp)c3*2xhxljnutions as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tire? *237b*x nsmcsz )jlp91nlhx u$\approx$    $rad/s^2$(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