A bicycle odometer (which measures distance traveled) is attached near the whe6e2h,g( vspqo el hub and is designed for sp(h og6qe,2v27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular veloci xp )eas- rmlgm j8-ow+d(/ie,ty. Does a point on the rim have radial and/or tangential acceleration? If the lei/p8,+ ()a w--mdsmoregxj 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 change?

Could a nonrigid body be described by a single value of the angular velocpg :r 3 wfij,p0lb3tr2ity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Ex,t h xfqj5jb1to: 7 0zut2v(efplain.

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 yourm) nsu*eh 7uc3 hands behind your head than when your arms are stretched out in front of you? A diagram may help you to answer n)*7uc3s uhme this.

A 21-speed bicycle has seven sprockets at the rear wheel and three ae6k/bmcv 80y r.u*ybpt 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 ha *y. 6bv0epkubymc /8rrder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legs with fles4;kx(r /- b ow0pwtnbkh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of xbnopw rkk- /wb;t (04mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, 0g /-usb4dt s)hsw c+snarrow beam?

If the net force on a system is bikv)yr)i,he3 -w y9gs+ y:bhzero, is the net torque also zero? If the net torque on a systevh+i 9:)wy 3yb)hbise, gy-krm is zero, is the net force zero?

Two inclines have the same height but w(zx(9de +xqymake different angles with the horizontal. The same steel ball is rolled down each incxd+ (( zwyeqx9line. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously starpj 7s1 m.l.ofq4a(nfx:i1s ojt rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom om xs.n1osj fj(a:f l4qp.1i 7of 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 tgpi )x954-mea,ob1pnf1tb43ygc iv 0 sj/zm uhe same radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which hasb ibsnft p05ajy1 ggi ) mz4 4pco/xm9v3,-eu1 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 moving tvrk +29o rf3 aubh au9wop75,g(aq9q or rotating objects eventually slow down and stop. Ex59(ku gbow+pu9rvq9 ,t7aq raho3a 2f plain.

If there were a great migration of people toward the Earth’s equatu1m4r, rt l,/spao dzy s4b(j 9/gaal7or, how would this affect the length of the dayjd, a1u,4/ml/rrys49z sl aab7(po g t?

Can the diver of Fig. 8–29 do a somersault wicf(1k/u(gpmv*d4 ;qt qv5tcht 6ta*lthout having any initial rotation when she leaves h( tlf4 mv/(p;ucdtc q15t*aq tkgv*6the board?

The moment of inertia of a rotating solid disk about an axis through : b0lht(,imea8xekng/1ajh/ 1l 9pny its center of mass t1a9(m e 8/ 1k/hahixnp:yjblnlge , 0is $\frac{1}{2}WR^2$ (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 mass in each outstret e9dfanw o54gtc.;ic,ched hand. If you suddenly drop weto; nc5.9gc ,4ifda the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. Howevee3mb9-uvs; d/z 1g rsvr, one is hollow and the other is solid. Describe an experiment /3d ;s- gvbsue1vm rz9to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle po-yfnrp3xxz7c4j wp7v**gu + aints west. In what direction is the linear velocity of a point on the towxvjug 7*n4pzf+rx ya3cp7*-p of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turntu sx 4n3h4bs97cf gsj8able. What happens if you walk4sf8u79 gbsscnx34j h toward the center?

A shortstop may leap into the air to catch a ball andinn;( mi f+,q p8 .rw8kytm7gh throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickl,fmkngi8ntmw qr i h(+8yp7 .;y 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 ofwn m fz0o2m /*gj/r3aj angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can opera/ /2jza0 wmrgn*ojf 3mte to keep the helicopter stable.

  Express the following angles in radians: (a)xoas6v8j yvst8*g* (n uf7k+ n 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.,u l* + hbmoy+n+e00civ-yrlk Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun k0co-0lbm vuih yrey,ln+*++ 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 behwp2e90) c3 2 qferojom9r t-lam diverges at an ano p)eh3com -f99q0lt2 j rew2rgle $\theta$ (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 mqhmnnxtz786 , .,4,bm . o,yjzlc xyzpotor is turned off during operation, the blades slow to rest in 3.0 s. Whatc ,oq.4xym.,zmp68 lnz t n7jybxh ,,z is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a2r(7xc pjzxb/ level floor 3.5 m to another child. If the ball makes 15.0 revolutions, what is it/r7pxjb xcz2(s diameter?    m

  A bicycle with tirespf6s9vpy * i5v 68 cm in diameter travels 8.0 km. How many revolutions do the whes5 *pip9 yv6fvels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm.gf/i,* o x/i yz(4o1uqk 0plno Calculate its angular ve/ yfqxooz ki, ul4i/(0 onp*g1locity 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 revolution in 4.0 s (Fig. 8–x ;+k9r :)n r;mte ikjzmsp:/ u5oyg/k38). (a) What is the linear speed of a child seated 1.2 m from the centeren+t;us /g :iro )j:y9/;p5km zrmkk x?    $m/s$
(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 .tpz l.)d9dw s) cl fwrr a/9wp63xlm3Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed o 7getr+9w q0q) vogy5jf 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 cent1yxk ) f7(7tos/,pmrv+nsxhv rifuge rotate if a particle 7.0 cm from the axis of rotation is t+)hvx7sp(v7 o sny1mtxkfr ,/o experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpm ys: az pt)4le1to 280 rpm in 4.0 s. Determi1z)l sep:a4ty ne
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radiusu:7xl+vy 2akh4 rg/m j $R_1$ 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,fm88n /bol6n8ag *zuuft6 ,ka4krd)q astronauts aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft canumadft8k,8rl*b)6g k46q8o/nz fu a n 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 kdvr8 uthfm0l* upa)-:t+ c,w220 s. Through how ct+u,-afv m:l)wpkrd 8*ut0 hmany revolutions did it turn in this time?    $rev$

  An automobile engine slows down from n y4y)zdy3. xxzii6o 6 p0kf;u4500 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 high ip8-e-abir+zei l8 3d f3og8vspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reachio38iv be8a-3ep rilg-z i8+dfng 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 to 360 rpm in 6.(i,ve/h;9cqu0z sky hy kn 0f)5 s. How far will a point on the edge of the wheek0(/k uyniz0yc)9vsh qf;,hel have traveled in this time?    m

  A cooling fan is turned off whdr /d6x xf;(/kw4by naen it is running at 850rev/min It turns 1500 revolutions b6dadfb//(kxrx;4 y nw 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 q7l rm zq3b4c,the car reduces its speed uniformly from 95km/h to 45km/h The tires have a lb4m7 q3qr,c zdiameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions -n:v pg,q-q-c3zq+r8sl ddvd as the car reduces its speed uniformly from 95km/h to 45km/h The tire dqrc,+qgs-8:-vn ddlq3zp- vs 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 puts all her weight on eaclp 9:.8i(fnvnu v3v uxl 9p:drh pedal when climbing a hill. The pedals rotate in a circle of radius 17 cm. i9p.8vn:llr : fv93 n puv(dxu
(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 8kyd4-kdh8pr bz .zz8 c)uvp) wide. What is the magnitude p-)z kpvh8rd. )yc 8ub8z zd4kof the torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of th r9r9 d -lg7hs/)xazcie wheel shown in Fig. 8–39. Assume that a frictigrc/7izd 9r- a)sl9hxon 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 of a massless rod which t1bb c8,hwi.bpivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on thiwbi8h.c b,b1t s system.

  The bolts on the cylinder head om8 . u bu:zn3fcr2sdl.f an engine require tightening to a torque of 38 f3c8.u.zmns u bdl:r2$m \cdot N$ 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 inertmhfmq9f uz1-l5h y6lj2:,aez ia of a 10.8-kg sphere of radius 0.648 m when the axis of jqaylzf m9hzeu51m l6h: -f2,rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of ywc8j6w4jvvk vkq8 * c(ef+(-ssiw5 sinertia of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. Thq84c-56fk+y(vcsevj w8wvjk (s*w si e mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the ewfd o0nm4z s/z oa+,2a*ywl0yl4 qq .vnd of a thin, light rod is rotated in a horizontal circle of radius 1.2 m. Cwns mw, 2a.004oz4lz oflyq */qav y+dalculate
(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 wpzue .kkt*3scii ( u6/heel rotating at constant angular speed (Fig. 8–42). The friction force between her hcsz tp .k6uk/u3i(* ieands 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)mc,lx9se t ria : yq8ip6i0(l point objects shown in Fig. 8–43 aboqa(i)sl8 9m:x cit, rpile60y ut
(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 .bv9btosj;h0:9zg no of two oxygen atoms 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 satelly 6w+v6 -+ubyaii7 byyd hij*6ite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the saah6yiw7i*v6 bdi u bjyy-y++ 6tellite 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 n + :ndh*ydxw3a radius of 8.50 cm and a mass of 0.580 kg. Calculate d*:w3 d+yhnxn
(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 fromr)gtu5 s5hlj* 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 tangew. * wcn+6fktbntially on a small 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 (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torqu n t.k*cwbw6f+e. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is evenrn2 3 0xj(hdsvtually brought uniformly to rest by srh3vx 0( dj2na 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 accele n9tonfda :,f6c42djtyq6 l 3irates 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 z 7 z lumi10qs9fo3rd3zh;nc6solely by the action of the forearm, which rotates about the elboo zz izf1;3mn9c3husq6 dl7r0w joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as shown in Fiu;qgl.)ibes 2 iyo5n8 g. 8–46;os5bnqe l2 .y8i)iug .
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consists of twoc:d2 ys)pv:rjp9x 7. bp6hhvt 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 aw3 + x9ydf:,1:n nkx+*bh jkd m;euwlwithin four full turns (revolutions) and re*+y fnw:9:l;n wd eumx,kb1x djhka3+leases 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 inertia on: d k bm))zn4uxun*a.b,a c3rf $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine d1):sx vzcb; xx c)1g kob6bm3mevelops 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 slipping down a 3o1 l+ohbyz6f9dqp2 /ud lc,glane at 3.3 2hu odd/yblp+ f3,1z o96clqg$m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as t bx*k, d:vhcc1 sejgas y,g. x,-7,ktn :uykt7he sum of tw cxs7 t ,eagsxk, 1ng:*-cyvbutkk.hj:7,yd,o 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,ryhq bs:g7n f84f k-p 1640 kg and a radius of 7.50 m. How much net work is required to acce4 rsf:kqh b-8nfyp 7g,lerate 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 masxxprm8 1 *;g ;yznau9ps 1.80 kg starts from rest and rolls without slip*gzxp8 r9;1;pn yu maxping 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 verfk-s(( kjxy:;o5 adqb tically on its tip. It starts to fall and its lower end does not s:a; okj -dkby(qsx f(5lip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end of a thin stl,g 8;bb, g8 dyxjs4bfring in a circle of radius 1.10 m8 gbg ;xbfs yd8bj4l,, at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grun(t. c u; tyrdu4b r32:9sgqrinding wheel of radius 18 cm when rotating at 1500 rpm? su;utnr23( rbuyt.cg9 4d:qr    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands a40tv1dky6 ib ft his side, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the spe dkt0vfi1by 46ed 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. w*v8q3v/ n4rvo0b f fz8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to 84b fozq3 nfvv/w0v*r the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increalel d69 w)kkatw27. psa:1pytse her spin rotation rate from an initial rate of 1.0 rev every 2.0 s to a:ye wkl)salwkp9tpd1.a 672t 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 aa h lhl7 i 4ccju;hb84ejny g9q:p6j4+vgn a67xis 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 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 7l pcgehj 46 g:na8janq+ 6vlu9c4ib4;h7jyh of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure s)mxm3ezq z5ik4 w7 fd+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 Earth jt+*xv /e4o jrge x/n)
(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 droppedr1; c 7dotolk u ah+)0q*+o7xg9fvfpd onto an identical disk rotating atop7 fg *or+79 1d)dqu0hl;fak+cot xv angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2 jh-5knxn- ts md62l(tvph8 v6.4 $rev/s$ 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 rota 1pdm ( d rzso(dg87nfct6p6o;ting merry-go-round platform of radf nzc(p6gt; r186ddood7 (pms ius 3.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an ang,5vi +x,f h t 7fzcc/9j8miyqyular velocity of 0.8,y 7/fhi,vmyqcc 85j9+x it zf $rad/s$ 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 abojr)km;u/lvz 8 ut half its mass in the process, and winding up with a radius 1.0% of its existing radius. )m/ lj kr;zuv8Assuming 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 windke r+r; (mbk1) u)pvats in excess 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 (o5a,4 omyxxwo s8zm*xw ))p /s0mb hb$ 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 ctqzbuh.4 fkzw: - jq/1an rotate freely without friction. The moment of inertia of the persouwt4hq:/-q 1 zkzfjb.n 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 w 5yatsr jmb gb *7;dv7ejn.;(mnpa68the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment pb7adtj se7rvam ( mjbw8n*yn.g65;;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 lying on t5ot)v5,o+wc)kdi x mg he top edge of the spool. A person grabs the5cix,t)m +5wog vokd) end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces the l-u u:a(l7i3zd b8,nri.t,l qu0privEarth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbir(.,qrid-b upl znu3i7al t ivu0l8,:tal angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from re*i(d .ygaa(gb st 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 0.20m3 z3(y7m(oytag/bel */kf 3oxszm *u m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by /7 3s/o*m xteu*3y k myg(olaz3(bzfm the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0,n356 0 ihi;g:ielxphg x3ane.050 kg and diameteehg ;65ihl3a i3 ixnx0n,p:ger 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 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 ahm sghjqn;r.:3 ah2.zngular speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times 85aw(ly7 *faz nx ,adbas great, were rotating at a speed of 1.0 revolution every 12 days. If it were to uy l bw,anazf(8 *a57xdndergo 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 momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use energy stored ihyv)d-)q2;q dzpi5o tn a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a io)ht dp2-dy;v)z5qquniform 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 antr7oy) wl 9tp(*jc11em3t iw( jufhe- $H_2O$ 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 horizont lqskz*s9i(svi9(q z/ 10iclb al surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot fre 8)p cg+nza+qyl h:6mipl/-c vely (i.e., we ignore friction) about a hzc+8m:i-)6 l+ /pcvnqga ply hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the floor, -w4 bm8 /iw6yg+hrsl; freip0tbz9j*and we want to exert a horizontal force F at its axle so that ihy ri+m4jbt wz-b0sg rl8/p9*wfei;6t will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with do jqg6mdz4f7 nqt62+zkl/5c speed v=4.2m/s on a flat road is making a turn with a radius Thej7 ln dq +fd64mgtz6qcz k5/o2 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 rock into a sling of length 1.5 m and c+*ji.5 nrlc82pyw k0xqva6ybegins 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 acxjlrwc0yyn+*iq 5v k 82.c6 pahieve 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, makr +tn jr(vo:e7sa nkk2.3* pfcing reasonable estimat+ rpae3( j7n.o:v2cf krsk*tnes for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cysop9qd at n)+:ic; 0aelindrical plates, ofid:t esq9o+a;a cpn0 ) 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. 8–58g-h( / )cz;tk8xpi cct. What is the minimum value of the txc)pgz ;(/-it ck8hc vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not ass-8eanaqznlb7+ 4w2+ov 1s mirume $r\ll R$ h =    (R-r)

  The tires of a car makbq07n. 4nqet23monjo szew8.e 85 revolutions as the car reduces its speed uniformly from 90 w 0e bzoeo3nj7qnt.8.4ns 2qmkm/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