A bicycle odometer (which measuresbyp9krn(1 b5 v distance traveled) is attached near the wheel hub and is designed for pyb9(rknvb 15 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at coysgzob1g. ;f*bpbq0 /4d7 goanstant angular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential fp7go/4g ba szgb0o.bd;1 *qyacceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a sinao8 zuvoi1knu24g 0t6gle value of the angular velocity $\omega$ Explain.

Can a small force ever eo(c myom d:sx i252-hk,ubit+xert 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 behind yourwmh xad;1 (r*y head than when your arms are stretched out in front of you? A diagram may help you to answew a;rxd(1 *yhmr this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the ped9kh:it3o -,k km.syj tz9s k6e6uyyx )al cranks. In which gear is it harder to pedal, a small rear sprocket o3yxem.:ss6uizykk-t9k 96ht ,kjy) 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 slender lower legs with fleshym7/ : ljyi r3d*ryli +) zl-2)uarxyn and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageou - 2li3:yyyriu7lzxr )/ jr+m)a*y dnls.

Why do tightrope walkers (Fig. 8–35) carry a long,oko 05u iu-zb*9df )xa narrow beam?

If the net force on a system is zero, is the net torque also zero? If the net tqvm;w3s, la-m orque on a system is zero, is the net forcewql -m,3mv;as zero?

Two inclines have the same height but make diffomsx7o, k ;iy9 pxr;/p b;cot)erent angles with the horizontal. The same steel ball is rolled down each iko x/ry) po;;m p;t c7s9o,ixbncline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an inclb+; kc+9vh4 sw)kgnudp9 hw :xine. One sphere has twice wuph +g+4hbc9 s;kxn:k) vwd9the 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 bottom?

A sphere and a cylinder have the same radius and the sadoenh667 )krnme 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 n6h)6nd7 oerkthe greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet mos7)cfji:uuo k.t moving or rotating objects eventui.)7 ujfukco :ally slow down and stop. Explain.

If there were a great migration of pin9 ae,(gc;yove bu6mdg,tw. ,69y, c*chg s qeople toward the Earth’s equator, how would this affect the length of tg, wtye,nec9yqv co. ( u9sh6* g; aidgm,b6,che day?

Can the diver of Fig. 8–29 do a somersault without having any initial rozl;ez c +lxb:y eo3 : bq2w4fbz4aa/+btation when she leaves the bo2 ;z3b:fl+a4cezwq/l +4eozb:b xybaard?

The moment of inertia of a rotat(qeph ,h.n.qp ing solid disk about an axis through its center of mass q nhqh.p pe(,.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 sitti72v5y 7+ bdxtqx6ilrong on a rotating stool holding a 2-kg mass in each outstretchx 6xy7+52qoivlb 7drted hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and h51xh 84b+bcr kra2,b ja p;acoave the same mass. However, one is hollow and the ,caa hbxjao12;c+prb r85bk4 other is solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle p nfn qii/07oag 0qhk;2:uc;yqoints west. In what direction is the linear velocity of a point on the top of the wheel?cgnh/:;7 ;i0 ya 0qqkni2 foqu 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 8-k nu7r.ux )4dgqt.cxj,uf) y5w kmoof a large freely rotating turntable. What happens if you walk toward the.)qwm.t8jox )f5kcu-4 k u,7gn yrxud center?

A shortstop may leap into the air w(sij1q:t1 eqto catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you1qtewji :s( q1 will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the la nax)07ie jcn x s8lk ax/n8bz+.ns/t2w of conservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeas8.itsbce / 8na/ l2n0 z) x+kxjxnn7ller can operate to keep the helicopter stable.

  Express the following angles in rz,bo:522se qx rw)xjcadians: (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 vx9slgsx-4; i0u+cuq because of an amazing coincidence. Calculate, using the information inside the Front Co v+ 4cs0i-sgq9xu; lxuver, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km fro4 rzk9zaq c(jrwldm u*;7); fo1pf8j xm Earth. The beam diverges at an a;q8 pljz*1fjw9k uz;d f)a xrm7(c r4ongle $\theta$ (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 650snr 90bm94tiw,vivh /4+2f7xi vhel)q.u kh f0 rpm. When the motor is turned off durshl2q4iievwh h9 txr4 u0f nb.v/9mfi,7)+vking 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 c2ij5u2(o3xi a p 9rkzlhild. If the ball makes 15.0 revolution2 9ir x(z pk3ujloi52as, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 k3 6ejxpf,6smu m. How many revolutions do the whe3us p6jexf,m 6els make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates azz,x1mqk r 6zis3h /i,t 2500 rpm. Calculate its angular velocit 1zq,k6zi ,rm is3zh/xy 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 -p;g jpqcg 3-d+ *wqz2n.obuqcomplete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speedg pqq-g *j 2 p;.oq+und3zwcb- of a 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 .ags6 e,in:hw the Earth (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 pointh- ih0eok9dh dyu17) b
(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 r mbfzsrk* 2i,/otate if a particle 7.0 cm from the axis of rot /,br*z msk2fiation is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its centq(rtpnc 11. xlzq; ew1er from 130 rpm to 280 rpm1 xnzp1crl1.e;( tqqw 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 radiurz8k ss-(bqcs,yplm4*5s y5zs $R_1$ 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 spacecraft put7 g sy:zrnfx )x:l:ckg74anjm7vig 0- themselves into a slow rotation to distribute gy:f7 7:xsm z)ain4xcvkj-g7 g0:rln the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates unifocndeeuf 7y6v pb +y vf3-85o/krmly from rest to 15,000 rpm in 220 s. Through how many revolutions did it turn ik df/+o86yn y-v b75ecpuevf3n this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Cauz ;( s+crhp-xvt 53vqlculate
(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 highsp2 k gzrbc;x0:eeed jets in a whirling “human centrifuge,” which takes 1.0 min to cz 2x:ger0;kbturn 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 2m:/jiwi w 6vxb;8.g :soti1 e fclmj:,40 rpm to 360 rpm in 6.5 s. How far will a point on the edge ofcj:e: l w8.bo v:/;mi6tg,ijim s1 xwf the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It pe2ku-/,mek 3j*gaoqturns 1500 revolutions before it comes to a stopp j-o/ eqag,k3k*u 2me.
(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 aoqr,-+ e3 umvns the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.80 m.,+3e muov-n rq
(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 )8iae/: enm bm0 )oka0g )jebtreduces its speed uniformly from 95km/h to 45km/h The tires hgm ei)bkoaa eej0/t)8 b: nm0)ave 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 wtvzbh0:nn2g58+ g dq aeight on each pedal when climbing a hill. The pedals rotate in a circle of radius 17 c:2b+58nnatg0dzgh qv m.
(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 the mag8j 9nagr0/b- obibe 5va/ ehe:nitude of the torque if the force is exe0-5://9ebbhgr vbia 8ej nao erted
(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 Fir-x)mmqoo/m 3g. 8–39. Assume txr)o3mo /-mmq hat 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 t rinp;5thp 6b8he ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizont t;8ipr5nph6b al position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head o 9f na7vgg9cul(.kdpey(2.w hf an engine require tightening to a torque of 38v72gkg ea9l.(dn(9 ywu ph.cf $m \cdot N$ 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 spherd vha)bs w4h)muaf6rnrs7*bcp 1 7b)+e of radius 0.648 m when the axis of rotation is 4ba7b+s a s1pwfv ) nm*crhrhb 67)u)dthrough its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. The rim v00ig,t bz j:3m4w8 nc/ lihek (qk/rzand tire8kt0 v:kmnq/l4 ir(eigb/,c3 w0zj zh have a combined mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of qv4jx9 z fx(b7a thin, light rod is rotated in a horizontal circle of radius 1.2 m. Calculate 7f 9jxxqbzv4(
(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 bowciq f/ :buvj27c:jb :c3v.owwl on a potter’s wheel rotating at constant angular speed (Fig.q/ic.2:jj7bcuf: bc 3ov vw:w 8–42). The friction force between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objects shown in Fig. 8–43g7 vwx u,7kio( u(7k vgiw 7ox,about
(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 totalf(dl2-6cjhylh j1 zb*nmg. d, mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinnh 0mif( cbxxt, -,zd0he*f mo0ing at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has h0cbe-xx,0(*om0f zd t,hf mia 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 radiuc(d(17zsisw6 j d,uais of 8.50 cm and a mass of 0.580 kssiw(dd, i7( a z1u6jcg. 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 fromz)wmlp38 ej) pa aq9z79dg 7om 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-drilbt ogud(g7.5k5 zh/cven 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 tz(c gthug7b .ol5k/5d he edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is eventually brought x7,ld:h9 e i;7l jlcu63 sqmfluniformly tds97e i l76 hlu:lq;,fl3xmj co 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 cit g;6teg/wm+ 3b1v*wwcsx *accelerates 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 balj gl 8ftz2a*k -(o0uvzl is thrown solely by the action of the forearm, which rotates about the elbow joint under the act 2k0fo8( z-gvjaulz* tion 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 am(7bf 5pyazb4quo40o long thin rod, as shown in Fig. 8–46 q( zbm0a4ob47ufop 5y.
(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 two masses, t3fkzviz ax*os:5o., *pymr4 $m_1$ 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 thep/0sjrf t (b:euvs9; l hammer from rest within four full turns (revoluti(srtbfl ;:jue0s 9vp /ons) and releases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a momentr sj3rk1-*ifr of 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 tw61v o z81n,btdxjzsy 8;vg4 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass wo1 rs7 kt3tg17.3 kg and radius 9.0 cm rolls without slipping down a lane at 3.7g3ksrtt w 11o3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth wite z7 hp/cm;y6 3ptxj)0en+ erwh respect to the Sun as the sum of two zj p0pw)ch+n7m3y;xe /e6t erterms,
(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 of0vlfz- dgo;,e 2yi j r9ey3g3s 1640 kg and a radius of 7.50 m. How much net work iyf;3g2 o 9z,l3gyidje0rsv- e s 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 rest and r :sqj:7ywk z0jr:vdt.2lgy gj+a16 dvolls without sls:j : zya k.0vg v:qr1lwyt6dg2j +7jdipping 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 verticug.7h 1qygp/7cw/q tsally on its tip. It starts to fall and its tpsy 7gug1q/w.c /q 7hlower 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 (zqy 7c 9.;h kkxgyh5ihazla 6h+, zu0ball rotating on the end of a thin string in a ci(k ,7gz5 z+hhy6hau0l h;y 9.ziaq kxcrcle of radius 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-kg uniform cylindr8ko6/c2p-5 cca9gevtn(mz( zmh; luaical grinding wheel of radius 18 cm when rotating at 1500 /cz((2zhna56om 8lcmg9keptca-uv ; 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 platfoq 9jq. ftp( y),fd6kmyrm that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of df yfy), p.mk6tqjq(9rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one sh sf/(ee2tt3c4 ibc vol2vp7 (c:mu0 zxown in 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, whazc uc l7t (icm/0:xv3s2oep(tv 2 bef4t is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initizgoo6 ;bj8hxt1g ok*zbts 9+6al rate of 1.0 rev every 2.0 s to a final rate of 3 ogh gx+1tzo b6bt8 jkz*o9s;6$rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis through its center a s k53ki4zt4/gfmf fqfa7d: 5ot 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 frequagd:4f ozt i 5fq sf/mfk3745kency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular t-mhepgg;7sqp5.j l-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 of ur x qf)jw+a3)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 z)g8i 8vv (ezpI is dropped onto an identical disk(vgvp zize8 )8 rotating 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.h+ shy8s4i *gq4 $rev/s$ 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 lr6 zpwmq*31 pkg stands at the center of a rotating merry-go-round platform of radius z qlrmw163p* p3.0 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 frpln5o a qu :)/jr9hkv. k :qih35lmwyb(p,r3reely with an angular velocity of 0.8l 5 9(h,: rk3qy:3br/iu.ph)wlop nrv 5kaqjm $rad/s$ 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 coll-(qw ad(jkt .hnju.+fcx5-fk* yi 0wcapses 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 anx ckhki*-qwtj -n ay0jw.+ 5.c du(f(fgular 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 wind7fu k5 6fm(vhts 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 (gixho6q r00o $ 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 platformdmcg;dv;* lo., initially at rest, that can rotate freely without friction. Themdl;vcd* .;go moment of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diametechbdip+5g t /gi:o (i+r merry-go-round turntable that is mounted on frictionless becbi5(:i/o hp + gdgi+tarings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the grod,n+wwqowlf is( 6x;,vb l+/i und with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person wiws /w;w xdb(,q 6lovn ,+ilf+ithout 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 fa*xox3 a-:nw66ckd b zgces the Earth. Determine the ratio of the Moon’s spin angular momentum (about itbzxn d:c x 6wkg*a3o-6s 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 19f . z+gll6 kg*4ezoqtj uhno*cg)q:from rest at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of radius 0.20 m acquirej:t qqt50-j0pnazdz+: 3tdgv s a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculat+z t0n3 p0 5vjtqj:a zg:dq-dte the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindricalk ut7 3bxni:f9 disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the ex k7niufb 39t:nd 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 resj4hu*v* l ,x;* m(tmlm)jtl+cdavec,+igk; ar wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Scjt6bg7:xfa+w 8i b+iun, 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 stab:af8c i+j6 igbt7 x+wr 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 it6c9pn0lwe+xk u 7q;wbn +9xmn 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 with+bkx c9;qnxn 9 6n+lwu7m0epwout 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 an6 f--kn,8ipujjdxja * $H_2O$ 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/w(s b+8e r9, in*s0hcwqfnd/cp7imbal 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 l-r8t57 m xy(zpg 98fpdml kiw2ength L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center ofz8kfd xiptl9 gpy(7m 5- 8r2wm 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 fw o56 ,z opna,-)elvarloor, and we want to exert a horizontal force F at its axle so that it will clim5oe,awr,n6po ) alz-vb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.mfa m8wz*7ed3 2m/s on a flat road is making a turn with a radiua8dm*fmzew 7 3s The forces acting on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rocpnv y*dx 5(s: ayi4p8lk into a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque requp*n ix 54v: 8lp(daysyired to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinjz-(p t gb4d u hv0bju(,k2b9xder 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 outstretched4tb2,-bp z(bh k9jgu(vxjud0 arms, and with arms held tightly against her body.   

  You are designing a clutch assembly 7 / q0mzsoajcq-62cum/4fva5 yzgc *j which consists of two cylindrical plates, of mass q 0c -7a jmm /*4/zjoy6sz qvgcfa52uc$M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 rougwbpqi3pb j/hl1(- uc(h track of Fig. 8–58. What is the mij (q-pi/31bluh bwpc (nimum 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? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assulyhj6 :vze)+)o7x tr;gqm1rup bk 6x /me $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uniformlk,ys, )wc/b6hiqasfd fx;:ej:9md 9 ly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of eaci w: :y,9c m;dj,efss9alkfhq6) bd/xh 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