A bicycle odometer (which measures distance traveled) is attached g wok *lj0ff0r 65+gjnnear the wheel hub and l* n6j0f0fw5o+jgkr g is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on the rimjszb( rg 7, ib53n(zkb have radial and/bb5 zrk7 gj( 3(bni,szor 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 change?

Could a nonrigid boddyap. gh4c6 h4y be described by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exsl;b, - dhlrjo*5ffe8 ert a greater torque than a larger force? Explain.

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

Why is it more difficult to do a sit-up with your hands behin o5ydge: *tnoftuba u639z7r1 d your head than when your arms are str a3o5n tgby * :9ftduoeur61z7etched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockb(vzeo z;xb+qv b*upje0w2-w)p6fcs2m .e9 k ets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a sf-2b b9jeok (+uwe)p e vwz.*x6z0pbv2mcq s;mall front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lo0 cin z);2dlj).dfe9l(pk st -fdlv4wwer legs with flesh and muscle coc)9ld. dtjld2 ffls(nwv4i- k zp;0 e)ncentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a longd6-badf2g7yt+ opx)cuye s vnc) 0v65, narrow beam?

If the net force on a system is zero, is the net torq.vwax xd/7lu7b ut3 d.ue also zero? If the net torque on a system is zero,. 7dva t7ux /uwbxdl3. is the net force zero?

Two inclines have the same height buqsf -ec f.j8:d0wqk0f t make different angles with the horizontal. Th8w.0f -q: fkqscedj0f e same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start 0yo* -m3krbibc8udqm4p +v-g j fb s63rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the * fmmjr s q46 y0gcoi+updkb33-b8b-v bottom?

A sphere and a cylinder have 8p add b +t d-)w7j*xozsk7q)othe 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 has the greater total kinetic energy at the bottom? jdzq7ko+ t8* a )ddsw-obx7)pWhich has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yeor (,b+6xp u: xofkp)ht most moving or rotating uopb6x:kh+p ,(o)r fxobjects eventually slow down and stop. Explain.

If there were a great mi6 e51b hs)hlaiso0 sswvb* ,j(8ix q9kgration of people toward the Earth’s equator, how would *se) ks(o x1lvs b0iq,waij69 hb85hsthis affect the length of the day?

Can the diver of Fig. 8–29 do a somersault without having any initial rotam*dc0-cnm+nky4c mro.d. +lotion wheno-o+4ld mm. *.crmcn+ dyk 0nc she leaves the board?

The moment of inertia of a rotating wd4swd:u37x o7 k/ bvesolid disk about an axis through its center of massue/k7:bxd37o wsd vw4 is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg mass in each outst:1328mhwvk;v zm llxc retched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? 3: zvmk2 wcv 8h1mlxl;Explain.

Two spheres look identicalxl b1erl-wx*1k-3+w n3pvv:qbjg:j a and have the same mass. However, one is hollow and the otw:jnxw k+b3* gl vp3 :aje1q--l1brxvher is solid. Describe an experiment to determine which is which.

The angular velocity of a xc(d*n7cr;n-d1krny l:p ue , wheel rotating on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular rxdel*7pdnk nc-1r:c;( n,yuspeed increasing or decreasing?

Suppose you are standing on +8jmr hbusc.b 5c0 a.nthe edge of a large freely rotating turntable. What happens if you walk toward the centerb.b5m80ccraj .h un+s?

A shortstop may leap into the air to catch a l:kz zf o;;n8elu -f:gball and throw it quickly. As he throws the ball, the upper part of his be:k; lzfo -zl8gf nu:;ody 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 ci vm5ub 1-ghgx ;dwj,)onservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss - jbg 1g5;x uwdv,mi)hone or more ways the second propeller can operate to keep the helicopter stable.

  Express the followingloj w72d *s0c4 ,kjzum angles 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 coinciden ul+q2-5 fi dqc1/bx9cykdbv; ce. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of kdb-b15iyc/x + uq9clq2d f v;the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is direc yx/l2c 9v4holted at the Moon, 380,000 km from Earth. The beam diverges at /2l lhxvyc 9o4an 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 f1h8)x-ao p q;n;d ylrmotor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow hr 8 a)nq-ypo1fdx ;;ldown?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to av;)zi/widj;y nother child. If the ball makes 1vj; z;)yidi w/5.0 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 8 axp,0hbnd hk n-t mrs4.j/9dkm. How many revolutions do the wheels make?hhmsjd0btn, . -9r/dkpnx a84    $rev$

  (a) A grinding wheel 0si3y:w c+8 0:uuxmsndo ) t2u bzjxo5).35 m in diameter rotates at 2500 rpm. Calculate its angular velocityo x:zn2b8y uc):us i duw3t0x+sj5om) 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. 8–38). (akmmyw k/z6 rtt ,,1lk+) What is the linear speed of artl+k ,kzk6wy1tm, m/ child 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) in its orb4,x)*4hv0ei g ddc uyqit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a pom2pl blj )f; g3c.+qzaint
(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 cmr 4yb xpq tdwte+oi+bz12d)+( from the axis of rotation is to experience a be qotizp+)2+4 d(+ ty1bwdrxn acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel acwb(v +ozq)zyo)3 w.jscelerates uniformly about its center from 130 rpm to 280 rpw(z3 w.+so v)jb qzo)ym in 4.0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radius z+wat8 ybqxfb 92:7lp$R_1$ 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 1cj3u pjz*k8pMoon, astronauts aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s 8 *cj1pk zjpu3energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm in 220 s. Througq;;yn8xa v01a n2tnech how many revolutions did it nty1cqa ev;a;xn n80 2turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calcunna09 ib*s z+t .:ks:qvris l*late
(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 4o 4)j. d*rva/mygf/ rsbtq9mflying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final s /f4mdg*/ya ).roqb49sjvtr mpeed.
(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 dia:.v/:g8nokyq; tzzhjmeter accelerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have .: jzzqnkg:yh/8o t ;vtraveled in this time?    m

  A cooling fan is turned off when it is runningc0wa6uc;()zezu+hv, m+paw s at 850rev/min It turns 1500 revolutio+mvcw 0uwh z+;a,zc6ue )s( pans before 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 itb2oyhugpzb zn mg8l/(9. 8j z3s speed uniformly from 95km/h to 45km/ ogh/2zz gl(j b9zbp8 .m83uynh The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the cp ;zdhe3w )w r5+7cv02wri4csn- zcw car reduces its speed uniformly from 95km/h to 45km/h The tirc5c +c;cpe2rrws7n wzz-h d 40)3 wwvies 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 bivrx*x ,gb ,fp wed-ddo22c.7h:d(ig +izu v)oke puts all her weight on each pedal when climbing a hi zx. 2o ge,d p +(7gvfc2bd-hivu,:oix)w*d drll. The pedals 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 force4ww qt3l u6bwptvks3 q;+y-b) of 55 N on the end of a door 74 cm wide. What is the magnitude of w l 3-ttw kqpbsuywq;+v) 6b34the 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 about4dx l8qy.a0 5ipm2at.lg t*e z the axle of the wheel shown in Fig. 8–39. Assume thl5dxzatg .08 tpqm l *.y2aie4at 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 of a massless rod g*tb kv-m+ pns1z0jx(6 yur0q which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitudey10xp6 * +(jk rq0nszmgb tu-v and direction of the net torque on this system.

  The bolts on the cylinder head of an e1a (sfntz ( kvv3j0fm-ngine require tightening to a torque of 38 v(k 01v sna f3(jmtzf-$m \cdot N$ 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 nu.qj;h4 fh)va 10.8-kg sphere of radius 0.648 m when the axis of rotation is ;q u)f.4v jhnhthrough its center.    $kg \cdot m^2$

  Calculate the moment of inerti8k4 c3frx6el t hyzw9ox n7h3-a of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be igno 9fr3n 7 oxwh4h86 elxct-3zkyred (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of ac3r r*ctzapk0x s87l 7 thin, light rod is rotated in a horizontal circle of 78c l0xs73ack*tzp rrradius 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 34c 4 pnbfsbulg62bh6 a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force b bcp fu3lb6h42 4s6ngbetween 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 tho-+l;koz,ho/ eiwj3 s e array of point objects shown in Fig. 8–43 abou; loikoosew h +z-3/j,t
(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 twobryjmu+l49q3/o ve8 pc *1 brx 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 satel5r/g gstiyo,e2 qyb9v-c 6p4q vqp b.)lite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If thi b qgeq . cgv,)2 r56byypot/s49qp-ve 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 cyd uds31xe6/kq w bpm;;vgh;1zlinder with a radius of 8.50 cm and a mass of 0.5803uw m/1v1ks p ; bg;xd6hqdez; kg. Calculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerating it from rerrt)h9eo6 (ezy8 o ,vcst 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-driveg/p9k ,e)nn v 3.-uhten +ktiin 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 torquekii epenn3).+k,/-nttg huv9 . $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shcxw vp4b t3i jmzh97 z f1je5(u3zo6*dut off and is eventually brought uniformly to rj5j4pfu 3z9tb(6*mvwx1o 7izhd z 3ceest 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 v cnh) bqq530 vo.,/cj(ujxawat 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 thrrad ,*qh *5fmxrsl2f 3own solely by the action of the forearm, which rotates abo*d5 h2a mlx qrs,3rf*fut the elbow 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 Fig. 8 r ;)ijs4/ut(od9capn–4o ui(;/s9)j rt4anpdc6.
(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,c.kp:w:ezmphbe)n 9 /lg+dq two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest within fotw:v2q,gk+qtln) dx0t/*7ul jvi p o(ur full turns (revolutions) and rel,+*:qgvpdi2(kotwjqlt/ 7u t)n vl 0xeases 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 ofm mo2im0nji( fnv)8h : 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 torquvh-)whi yr8b+dd/ k.x*t dxf* e 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 9.04/z:cfw7 l, l( drtosq cm rolls without slipping down a lane at 3.stwozdcqlf (/l, r47:3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respa 52tz;hdezkzb njmr 53:*3asect to the Sun as the sum of two hsa nz5 bd; kzj3r:tze2 35*materms,
(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 lnf;exfib:(3k h6 gj0mass of 1640 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Asse63(n: ijfglfkb 0 xh;ume it is a solid cylinder.    J

  A sphere of radius 20.tzj*8wn9a i5p 0 cm and mass 1.80 kg starts from rest and rolls without slipping do nwt 5*zji8a9pwn 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 tip. It starts to fall and. ae/fz,u5* fno fjqv- its lofe*fu/j,- qn.5ovf azwer end does not slip. 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 agfrwyr27 8p(l+8 7cu ore5s ustring in a circle of radius r(g r 587plo+fu awre7y28 usc1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grinding yzql chkg:588v/1 uea wheel of radius 18 cm when ro8ha k8ylz1c/q:eu5 gvtating 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, on a platform that is rotating at a rate dsicgoh fh(. +4pn6g+g / wwhdp ralh,-p20 e4of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, th6-w /fp l +phd20ho +(cs4ge,hrw4ginhag pd.e 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 oneq03wx wg4+ xe9ii,0 j36l.tegqx irqf shown in Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing ,0i0iwtjqw 4 q9e6 lxqf3.+eg gxrix3from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase1ytwi q82/jq gto1-y +8gdmti her spin rotation rate from an initial rate of 1.0 rev every 2.0 s to a final rate of 3ggqwq8t totiyi/m1 -8 2 1+djy $rev/s$ 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 rota 1,q)pwu+e;lkjpaxg1f 0 rh21fc 7okl ting 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 3.1-kg 2 x1hp 7ju10r qkk;cgl,wfo)fpa+1le 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 gn*s 0e;air;lzd;-x vfigure 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 modgdd .. nhk;v j3u6s.ementum 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 n,mz4jnj.fa qie3n;3 ja-v )fv) mdo6disk of moment of inertia I is dropped onto an identical disk rjfm63anavj,z ;q)od i -mnnej vf).43otating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4 *lf y3jiu vv/advt;nn28,4 s4zz bx;c $rev/s$ 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 k7p7g5q5 m1kgauyma *r g stands at the center of a rotating merry-go-round platform of radius 3.0 m and moment of inert u517mmypaqg7 r 5ga*kia 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-roun6m kkpawto-kq*l;l :kymo ,vp56 ,kv, d is rotating freely with an angular velocity of 0.*l5 k oq k;k,vwpyvmt6,-kpo,k:m l 6a8 $rad/s$ 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 whr;py ( q:wicxa+ui.-o ite 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 new rotation rate be?(round to the near wii ux(ra.;:+qyo-cpest 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 invol)hfk(rqptcg3o ok o*73px oq /14dea8ve winds 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 gw lhekrdk1:x c) l g.1ubi/19$ 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, initiao a ofg70 a-.8mskx1hdlly at rest, that can rotate freely without friction. The moment of inertia of the person plus the platform g8.xao7adsohf- 0 k1mis $I_P$ 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 edxa44x0wd0u2en dv* jn ge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment on0exj0ndd*x 2 4 wvua4f 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 wjf: /hrkq0j t9ith the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? Ho9qt jr:/h0k jfw far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faj,v2s5g nc6wipq x j1ub)o6a 8ces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital anguvuq) 2p,6o b iwg65 x8sc1jajnlar momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from r8jfvfi n 6rz+ bp*5km;est 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. n,4+wax(xq di20 m acquires a rotational rate of from restxx4i+,wd a(nq over 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 twop nd a9kd a8zr8a . f-h*79jqpl:ku9io solid 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 ene:.jafkpk 78d 9-a* laz9di89rp un hqorgy 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 aniohw9 o3)jfznxbd9+0 gular 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 as great, were rot)gxe voy:a1a vt6c82h ating at a speed a8gtx:2ych )vo v ae16of 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 momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a zn q0rn/8ppn*h/ zry7low-pollution automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywhez/nny0 *qrp7znrh8p / el “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 vuzi,4ys w;5)y0:jl rup3zwzan $H_2O$ 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 gy a,ah+dfcr khj- 59y1ki 0/wat 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 lengt)f7k2nt ;*lregf 9da ch L can pivot freely (i.e., we ignore friction) about a hingefe k;)* flt2na d79rcg 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 thdo:94 zv.j xqse floor, and we want to exert a horizontal force F at its .oszx j4dv: 9qaxle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s /x tgtr/14r5 ax xbxfd *v2n,zon a flat road is making a turn with a radius The forces acting on the cyclist and cycle are the norm*, /rgx atzt5fx/rdn1 xb24vx al 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 begins whirli+uak 21ib w vnh/yor95ng the rock in a nearly horizontal circle above his he9b/ i5y2w v1uohrk+n aad, 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 does the torque come from?    $m \cdot N$

  Model a figure skater’s body bdt eqjr-w7-8 48y5u 4pzsu 0j3besyras a solid cylinder and her arms as thin rods, making reasonable estimates for t j80ep-sue b3yu-ys 5tqjd r8bz47w r4he 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 cylisjl 3iifa;:+4 99w3kdomi bvunncw: ,ndrical plates, of mass fila,3b +m34; ijkdvn9co9 wwin:s u:$M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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. 8–5d21 jlzz9w n/p8. What is the minimum value of the vertical height h that the marble must dr z9/p1jlnzdw 2op 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 assume (x ;fd fqy2l s;pp+1zo$r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed hp ;ad)huge.6 pugoj :2rlp 00uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What wegg) 0 dluh:6;pu 2.hppjro0aas 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