A bicycle odometer (which measures distanc 5 czixtdq(h1)e traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if youqx) 1 t(cidh5z use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constan0:w 0 mp b4di,4kwfxeup(f,z bt 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 acceleration? For which cases would the magnitude of either component of linear acceleration change mwbxf: p(4kp0zdfwbu 0ei ,4,?

Could a nonrigid body be described by a single value of the angular velk/ute sxi+c4 gbv9* y7,sdvu . dg-*miocity $\omega$ Explain.

Can a small force ever exert a greater torque than a large5 uj0 h)rg-h o :qyk6p8of qg(f;apza-r 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 mkry0l(q /39f bgvj) fdo a sit-up with your hands behind your head than when your arms are stretched out in front of you? A diagram may help yo0l q )yk/mf39bj vf(rgu to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel a ppw+cj il95e+ wf69eznd three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? 9wc9+5 pwz flpjee+i6In 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 wicfmb9n3v0 t3w .d6lxcg3 (avdth flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advanxcwmfa3n g6vt(d03l .3 dbc9 vtageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow beam?vabn6rx s8/rq62ssq 2

If the net force on a system is zeo7wf(22 f lg g;rhno-uzer;q2qw 4 pg1ro, is the net torque also zero? If the net torque on a system is zero, is thr(o2w;-p g h7f;ow2rzq4 qg2g1u eflne net force zero?

Two inclines have the same height but make different angmqq5d(cr*q 1cles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of m5cqc q1qd (*rthe ball at the bottom be greater? Explain.

Two solid spheres simultanewj jo4)t mcu x 2fujp65li1u,-ously start rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of t64 it5j12fuu)lxwucp o-, jmjhe 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 same mass. They start from 31tpgex 4yit7 rest at the top of an incli3gti x t4e7y1pne. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most moving or*)c sthwn(p5h rotating objects eventually s*(5hpt s nwh)clow down and stop. Explain.

If there were a great migration of vi du8;rcclu7zb- pwt. 51 +kmpeople toward the Earth’s equator, how would this affect the length of t;8tuzlv pu5ird17c. bw-+ cm khe day?

Can the diver of Fig. 8–29 do a some*00xhdu;wcved 25 je yrsault without having any initial rotation when h xue0je2d0w*5 y; vcdshe leaves the board?

The moment of inertia of a rotating solid disk afc+zfh9/ h qy f,d7;ncbout an axis through its center of mas 9;c/hfyd7nh+ffq zc,s 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 z+ 7o2.c ;lbqjdkvdh61sfp3beach outstretched hand. Ics3p2.q7f d j;bblh +1dkzvo6 f you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, on15o -atbygb2i e is hollow and the other is solid. Desb aygb2i1-5o tcribe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points west. p56ts0ugz: lsIn what direction is the linea g:t6zsl 05upsr 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 speed increasing or decreasing?

Suppose you are standing on the edge of a larg+4l hnjb-cej :c5pbs-e freely rotating turntable. What happens ihnbe:sl+p-4 cc-jj b5 f you walk toward the center?

A shortstop may leap into the air to catch a ball a:5cc) mmsh* kvnd throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotatm *cv:hm)5kc se in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservatiox+eetsitp .z2vm,3,ce *5 fc un of angular momentum, discuss why a helicopter must have more than one rotorp3et x,cvz eu2c* m,f+et. s5i (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles clfe;a ,2nhq.in radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidence. Calculate l:;xb2sjin d1lx(cj- -q.l wp, using the informa2n-scl-wxb.; ( ji:pjxd1llqtion inside the Front Cover, 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 tb 80b z8k2 cy5,sz b1qcmeixj.he Moon, 380,000 km from Earth. The beam diverges at an angle bqbx8jm,2ccky8iz01 ez. b5s $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender 6( d j7uygr8uirotate at a rate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular accelera(ug yjr6u 7i8dtion as the blades slow down?    $rad/s^2$

  A child rolls a ball on tp-ti26ve/u yow m)d+a level floor 3.5 m to another child. If the ball makes 15.0 revolutions, what is itsmo/e26 tp-y+ w tidu)v diameter?    m

  A bicycle with tires 68 cm in diamet qc 5vxu3bfx2-er travels 8.0 km. How many revolutions do the wheels make? bu c vxq5-x2f3   $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 25-wi+ti (u:wv: wfif+ v00 rpm. Calculate its angular velocity itff+i -vi i (:wv:wwu+n $rad/s$ $\omega$ =    $rad/sec$
(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 compl7kmgdg2:2sp6-enws26cp.2 3k x etn qmd, hunete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed2 xe.hnuc3 ktegds26,md-gnqw6mn p:72s2k p 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 the Eafpl73*0 fttco-c: sv erth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speqs7*)1o.vwa paurvrl sja2.59x p *zged of 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 centrifuge rotate if a particle 7.0 cm vxqq)mx-y m1/rik(u 6 from the axis of rotation is to experience y(k6 mxu )1 /imrx-qqvan acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accecojzr be.1g/10xfazz5an , h-lerates uniformly about its center from 130 rpm to 280 rpm in 4ar0z,jhn5c bx1fo .z a1/ezg-.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 pobzs6 3svmj. 8gx2 $R_1$ 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 spi1a s ,bvm(54gyzsu3op6msj 0acecraft 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 r0v(apb3g1 sy4 i6oz 5mmusj s,evolution 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 unzkb4,jm7pr 0giformly from rest to 15,000 rpm in 220 s. Through how many revolutions did it turn in4 bk7,gm0r pjz this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. C,g( to8 ol jbq6(uzqd2g;ktu/alculate
(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 flyingc pb1..qmf),prs.hu p highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaphqm.f )p1,rsp.c b.uching 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 6.5v rj-(xax, id1 s. How far will a point on the edge of the wheel h- rx 1,vj(dxaiave traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 1zhv 4d8o jpge ii4.6u5500 revolutions before it comes to gdjuh 6 58p.e z4ivoi4a 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 b a*40kkd;pft car reduces its speed uniformly from 95kd* k pfb;0ta4km/h to 45km/h 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 car reduces its speed unis2r*dps1 i3+ui2 mqdq formly from 95km/h to 45km/h The tires have a diameter of 0.8uiqr1 im ds*2q+s23dp 0 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 bojpg8:nq3;r 5ok tc(m ike puts all her weight on each pedal when climbing a hill. The pedal3g; omjkq p:nr5to8 c(s rotate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cm wide. What ix e3w:cu+,s f.mwjx3b s the magnitude of the torque if thxxs+fej .,3 w3w: cumbe 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 the wheel shown in:cs3v ctp/f-,keke z) Fig. 8–39. Assume that a friction torqu v3 ck/-zfe)p:tkec,s e of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are atta ud xi0 m-xp5jv.31 iatqk5j3zched to the ends of a massless rod which pivots 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 ondx-km3p5vtj3 q.01 zaij 5iux this system.

  The bolts on the cylinder head of an engine require tight g32dp w2n-h(.jq grp u)hqq6ak/j9r aening to a torque of 38 jh622qqgg/)(k.haa9q -nurp w3jdpr$m \cdot N$ 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-kyd- /6d)awm x,*lzzctg sphere of radius 0.648 m when the axis oa)wtcxzm,l *-zy6 d /df rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicy0r- +3 olowi; lpoe2feo5 1paicle 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 ignored (why?).+-e1wo o 0e2 piof5;oprl li3a    $kg \cdot m^2$

  A small 650-gram ball on the end of a t4v5sdaf.p2 wq;n m 6vrhin, light rod is rotated in a horizontal circle of radqfv5;wvr a.6ms n2p4 dius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotaycrm ,c21m8v x io5 ujet./2mq.sf hy4ting at constant angular speed (Fig. 8–42). The friction force between h.. 8i y thq,jc5yosxc22mm/v 1u4emfrer 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 Fig9t bop(s g8n5b0ngqs3. 8–43 no(n3b 0g8sq5g9ptbs 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 total mass iaw r6i2ghi dh13e1( zbs $5.3 \times10^{ -26}$ 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 cylindagp. :ic ol/:zx o8h7h:,jz yirical satellite spinning at the correct rate, engineers fire f8 : .ci:op: hxzohz/ia,y lj7gour tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is /o 920d,zywe 1yy)c d/ln gtbwa uniform cylinder with a radius of 8.50 cm and a mass of 0.580 kg. Cal/o , cnbwgy)1d9ty 0wy2dezl/culate
(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, accelera;5m 9cgiy1 v0 vj2x xsx9b6nwating it from 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 st48xm .)vm,iifo /tqlmall 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 torlx vit,t/iq mo.)4mf8 que. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300) 8(s-ca vqa yca6si6i rpm is shut off and is eventually brought uniformly to rest by a frictional torc6ciaa- 8qa(ys)s vi6que 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 acceleratesv6ofm7 gaiv(3 kmd;-6 oojc;n 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 solely9w i)ta/x4wuy o 0nb9z by the action of the forearm, which rotates about the elbow joint under the ac 9/bn 0iyu4tzxa) w9wotion 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 s 9;ahh /y ; kprs4diumozu570shown in Fig. 8–46. ypdsu;/7u4r5hzh;ks0 9mi o a
(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 consistb6s)9 b7:juhpznodq 5s of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer frombwt48f* pry h0 )ogc-ism34am rest within four full turns (revolutions) and releasahyp)oif0ws-m44rb* 3mt c8 ges 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 c ;1vuy(xdoa4q1yr xj5b-zc- 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 280 *k ikwe5 q4mo9$m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radiul:dtpb20h0me y6 dijf;6wk 0ihs 0o8x s 9.0 cm rolls without slipping down a lano; l8mhii0yf:6ed260 xd skjt w 0b0phe at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth wit 3v)g/ tqb*r+jf+b caqh respect to the Sun as the sum of two terms b+/)a +3jvqrqf tbcg*,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 :v4-ysfi p b7)9kd y lg,im(wyvcbx;,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? Assume it is a solid cyl,kb l7 fygx dwm(i);9y ib:vpc -,sv4yinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls withor(/-hcnvxd,v ut slipping down a 30d(-hnvr x,/c v.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 e g msbax0sm0b( 8ca9,l /-dtwvertically on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just bef8b,ga9 b0x /mw l-sa(tme0d score 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 ;n zmbvf* n)f zl9: /wue*iuvjxnu *830.zkeua thin string in a circle of radius 1: n9wnkv evizfe3.0xzu*mjfl*un* );/ zu8 bu.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 cylindric :du2brk.w-ax al grinding wheel of radius 18 cm when rotating at 1500 rpm?xw-2. dua:krb    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, handqo4tb5f 1zd)me9k++uxx9 za +g*bdhxs at 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+t9)+xdgo 9fkx e54u* 1zmxd baz+qhb. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one sx dkv,z)eb ms:f4 v-4r v.u8pqhown 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, b uzmke48vr:q4dp.- v) xfs ,vwhat is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin djk: e;5(zlknrqce rp1407 wtrotation rate from an initial rate of 1.0 rev everyje zp1rk0c n;qw t 7k(e:4ldr5 2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis through its center at a freku j3x.iw8.nxnrtc wnj 9-,e 4quency 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 jncxuwwk49.8nxn3 -ir.j,et a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinningb0v+0 k)umin) +owurwg4os hi5 .i-aa 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 thuldn( (:h6fjo7fj /ex dhq;o;e Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I i*d1 gjz(bx6gis dropped onto an identical disk 1 z(j*xidg6 gbrotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns ar wr)awz9nryt q,:nb c :*ij 45m09bxdt 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center6/h)+0uunmlod2c7ne u if1r r of a rotating merry-go-round platform of radius 3.0 m ar0cu+fm 1e) in 2u6uh7dr /olnnd 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 ang8 t9*j d(0xblylhq-icb8b- kxular velocity of 0.89 jxlidl0* 8-xh(bc q8- ktbyb $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, losing about+joc(0 -a5rc; iswqwbs *ua 8m 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(i5*-acuws r + ;cbmso0a8wjq 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 winds in excess of,gryupe*l+*zsz 0 j u1 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 g*,iuru8a y*ih/z*l z$ 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, tf+ yi fx(faqzr (o())c ky10xbhat can rotate freely without friction. The moment of in)rxcq 1x 0)i(y +f(aybfk(z ofertia 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 diameter merry-go-rok h enapwd6z/:rdl8 pj.47qm )und turntable that is mounted on frich.)papln/7k r:d qmjdzwe648 tionless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the (lx5ull51 dig:jp5fk ground 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 without slipping. What length of rope unwinds from the spool? How fap lx1 kud5li5j5: (fglr does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces the Earth. t 56en9hcn1 mj,vwso (sl ur b2d*vnqxzq05: -Determine the ratio of the Moon’s spin angul0,*ecznv-q9 s5b(1j :o2 qthrn5vmlusx6 nwdar momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from 6-fh 6mj*z*lhu/q bvs 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 i+1 cy0vy ;:nvn o7rasvn the shape of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the moy ;onr v10svy+v7na :ctor.    $m \cdot N$

  (a) A yo-yo is made of two solidr j) k1qq4bpph2a2 ptb:0x3reg yo;7m cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy 2a:1xgqt4jr7kprp o b0y p 2;qmhe 3)bto 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, h:opr7ig4 ii 2pzi +;rfow is the angular 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, were3et4am ) 4hyah09bt ay rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutrohtbyyh4t e 3 )aa0m49an 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 t-m m o+sct3gtkvflu; ef:5. 8do use energy stored in3t+ dm:su 8otmf;l e.k- 5gvfc 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 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 an v ts,vv23uy(w$H_2O$ 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 at 79t)d1vo1 gev+t 1x gagvtbx2 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 len ddru.3(: dmou)b xxtm1p:m2rgth 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,x r: uxbp 2dd3r:)um1.dt(m mo 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, aw4x f5 +oa5kjznd we want to exert a horik oz 5xjfa5w4+zontal force F at its axle 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/8s qwjb t8 pl ta29pdxd(,; g7pa*o1mksg(e n2s on a flat road is making a turn with a radius The forces acting on the cyclist and cycle abtxk9dws8(ane87pjmo,1 ppsg ;t gqd*2(l2 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-k2uw u7 d p9bglvcac;(*g rock into a sling of length 1.5 m and begins whirling the rock in a nearly horizontc(d*apu7;9w2v g lcb ual circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s bolu67 z3c y r4ho/axiz91ka6vedy as a solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calcuyrxkz7/ v69zhea3ilo 6au1c4 late 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 cylindrical plates,mw6 ffkn)x9c 0 of macknf xf09m)6 wss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 tra 7 /fh/1k7my-ms q n7q0 q(stzxqgn.ouck of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the hig7q0 y fmnn 7/zgxso 1qk.7um/st(-hqqhest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but d 00)sa 2+fj h.hywfn3de3uocyii. f,d o not assume $r\ll R$ h =    (R-r)

  The tires of a car m3 l6sgp pa4r*pake 85 revolutions as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of eacppp3r64g*a lsh 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