A bicycle odometer (n a+d vbjh(eg57th/u-3g.e y lwhich measures distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happ7vg-g.jth+/e3ae5b( ldnh uy ens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on the rl8b6,32 txqciqjf ;vhim have radial and/or tangential accelerat6l28jhv ,i; xqqc3tfbion? 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 body be described by a single value of the angular velocof0 9p o9d0elwqo jh;o k,6a 0brgs2c1ity $\omega$ Explain.

Can a small force ever exert a greater torque than v-2iru-eu3 (m,kv yvfa 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 fp1lxq9hkz; -ib*:d s fy+4jsd your head than when your arms are stretched out in front of you? A+ q hkif: b*szlf-9;s1yjxp4d diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the pedal )fyrmttd gxby-d z, )6b+p+ 4icrxbt,yz+yf6 -d)md i4p +br )gtanks. 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 small front sprocket or a large front sprocket? Why?

Mammals that depend on being a aqcgspwn(7. d7q;zh1y9,c rcble to run fast have slender lower legs with flesh and muscle concentrated high, close to gsw9; (c7r,a1ccqzp. qh d7 ynthe body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers g1ff.j2,(tm aci71y dutw t2t(Fig. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is t cy172 .jxjcyihe net torque also zero? If the net torque on a system is zero, is the net force ze2jiyc j.xyc71ro?

Two inclines have the same height but make different ang3h fk1;s t kj28vebvtpc*d;7 eles with the horizontal. The same steel ball is rolledv pj*t k;fes7e 3b8 ;ktcv12hd down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest0nz1fp.5j e diouqn2 o7l5h8je54j yuekn s1:) 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 hneh jj0peu1 knzyio q.5lj25de18fsn47 uo5: as 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 f76vuv4 (uf+elzcnc k 7rom rest at the top of an incline. Which reaches thevuck(fuc +lz7n74ev 6 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 2)f)n01ht,kr g g)xb+ekbmzz rotating objects eventually slow fb)gr0t)bhgem+ ,)k 2zzkxn1 down and stop. Explain.

If there were a great migration of people toward the Earth’fjm+b,twsz1 *p) jwz 4s equator, how would this affect theswmz*b1p, )jz ft4wj+ length of the day?

Can the diver of Fig. 8–2 tfq8:75tdns9rasytd .dg9 5i85 fabh9 do a somersault without having any initial rotation when she leaves the boardna8s9sft79b:qrya5t d8if.htdd5g 5 ?

The moment of inertia of a rotating solid disk about an axis tfh0 0r (tig:kvhrough its center (rhf0:ikvt g0of mass 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-k33qqu/zaeec( yn657z ae5a di6n 8tl fg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or 5n(dz5ueyia6/ 8q n6ezt ea3a qc lf37stay the same? Explain.

Two spheres look identical and have the same mass. Hu0w9vn3 x qz4gowever, one is hollow and the other is solid. Describe an exg0v94nwqz3u xperiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle pd im gxjaso2+7he5rocy6f (, 4oints west. In what direction is the linear velocity of a point on the top of the wheel? If t grx5 iaocj+d e4ohy62 fm(,s7he 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 standin4gq,u6y y4ryyl1c)j q tjk6j,g on the edge of a large freely rotating turntable. What happengc4y6 1yyl, 6tqq)yj, jj k4urs if you walk toward the center?

A shortstop may leap into the air to catch a ball and throw it quickly. As hb 8 )n5hdw p7gc2hmi9we throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hipg7c w2 9)h 8mhniwdb5ps and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angu-3 et5ri:+wp:fxw ake lar momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second prop rep+ we t3::ai-kw5xfeller can operate to keep the helicopter stable.

  Express the followin6r;ubmv: lwe*g 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 coincidence. Calcul x1sw.yki dn.5ate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and ts wki.1d y5xn.he Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 38oe2a/*5bflu1 (qyaog 1* qttj0,000 km from Earth. The beam diverges aa/uq1yote*(f *jo2ag bt5l q1t an 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 6500l wf:ims:awyy: d+sh)+8)p z.2b uqg b rpm. When the motor is turned off during o:bs:up qds ylia.y m f8wwzb 2))+h+g:peration, 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 floorq 5ceu6mpadl5-r.;m l 3.5 m to another child. If the ball makes 15.0 revolutions, what is its diameter?-aumpe.65dr ;l c q5ml    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutio4eb 1,zyqx ih )06hl q(ecum*gns do the wheels1uh6h (e,c*mxz gyi04l ebqq) make?    $rev$

  (a) A grinding wheel 0.35 m in diamet ,ymf(vc/f2 ;knvjmi )7h vlh0er rotates at 2500 rpm. Calculate its angular ve,h mfl)yh/v7i(n c;2 kjvvm0flocity 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 (j 3fq39ljz dk7Fig. 8–38). (a) What is the linear speed of a child seated 1.23d9jfj3kq 7z l 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 orbi(jxj4kp,d6vvc w(q - ;e1 koeqt around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed ofiwvw75z. psv/ 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 rotatw(0h4 qav oqyqu3;dh()5fxp f e if a particle 7.0 cm from the axis of rotation is to experience an accelerati)hqw (( 3y vof0qdqhaupf 54;xon of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about x5 go+oh; . hna2wha(gits center from 130 rpm to 28 aoo;x.h hg w+g2na(5h0 rpm 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 azmrf -c9x6(;m(hnb b$R_1$ 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 aboardl lf90cw+3f3(fcy (v*zuqc m g the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Dezvu0 cmf+lf y(wcfqc3g93* l( termine
(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 2ohc7 ktw 17r4akq a+; uh*.+ruacd*ym20 s. Through how makh77 ru o4cyk;1td cwa q.+m*h*auar+ny revolutions did it turn in this time?    $rev$

  An automobile engine s 21lowhby 4n:p,*qkd5p cjln )lows down from 4500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed jets itnpg w4 uo27+in a whirling “human cewu2 gp+7 ino4tntrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diaqzhlv8pw+7jvaa 46l -c:t oc7meter accelerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on acjaol-6lt8 vzq+: pv7wh c47the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 15m5f9ce80 vekc00 revolutions before it comes to a stmc8fe kvc5 0e9op.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed uniivee zy8)hp(e7l6gb .formly from 95km/h to 7 bee l)8vz(6hygei.p 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 re-2suv( va(e( qmhgly5b 6br 1pvolutions as the car reduces its speed uniformly from 95km/h tlbv 6eybrsp gmu-h (21q(v(a5o 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

  A 55-kg person riding a bike puts all her weight on each pedal when clina9.w43yd+kpx +xguoig n zz6u0h )5w mbing a hill. The pedals rotate in a circl3nk)y+auuw 5n 49z g0 xpgx.ih+z o6wde 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 is t+ fh p)lx+. 5hqi 9p2uv4vdts6yzkb1nhe magni 4xd2pu6+ls.5 t+nzhq v bpv9yfh )k1itude of the torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the ax-bv (c2 5nj0vgu)pevylv 7 b3ole of the wheel shown in Fig. 8–39. Assume thatv3-pb50lu) vvngcb7ojv (y2e 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 ah db7c0;gft i.w.k6 tottached 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 magnh;bif dw60.g .cktot7itude and direction of the net torque on this system.

  The bolts on the cylinder head61pnq 9pwmh*gkdo 1/w of an engine require tightening to a torque of 38*1d 6/w nh9pqp kmgow1 $m \cdot N$ 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 spher7oi e(-:mc /gy i(xstht/k.a te of radius 0.648 m when the axik-c./xe:s( o7ig a/hy (itttms of rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diamxk 0,w0lfc-speter. The rim and tire have a combined m, kw0cf0 xpl-sass 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 a thin, light rod is rotated in a horiz ;rk,b6c), puzddte ,wontal circle of radius 1.2 m. Calcul d6 u,t,ed; rkwb),pczate
(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 bw5 8p3-tv eqjkowl on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force vj5pq-ewt3k 8between 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-syp pl4uy3 gpn,.dx; . 8–43ndp4sg-l uxy; y3p .p, 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 coquwyxp 5v:g75nh.6ff+ d r z4k. 5gjtgnsists of two oxygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the +myv+p cw7 cb8correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius8+ vbycpm+7wc of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.5i4ngl/p./ g8:jwxrc:o yk hy80 cm and a mass of 0.: kyh8 .nl:8ciyoggj / wp/4rx580 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, accelerpxu1dh*ist)ge4cbx0 1 1)t ,sc ypr 8lating 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 small hand-driven mer*cn/po en9ez-ry-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 7609nncp-e/ zo e* 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 torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off a89pyo()i qh :sc-9+dxsesckb nd is eventually brought uniformly to :c- s dsb+s8xcpki(e9 y9 qo)hrest 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 accelvt(g fxg93 c:lerates 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 solely by tjz7 boh 6o2 xn v:0em90ih j:/nfoqw:uhe action of the forearm, which rotates about the elbow joint under the actionqoo nzwju 7f 0:vi0:2:bx/h ej6o h9mn 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 ca ux 6.c zfdadx-0r oaas9a3l0y4yu: fjve463 cn be considered a long thin rod, as shown in Fig. 8–43f rd3s uj-eaxca6d0 vzca: 449a u6fo.yy0l x6.
(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 masse ppbribpz vrg ;5mu/mb*vd7hdn,)e(h i-e4*8 s, $m_1$ 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 acceleray4m5ak n lg: uhvs8hmcobc6hgy ,,9 60tes the hammer from rest within four full turns (revolutions) and releases it at a speed of 288y6h:9 50b,nmhc vlya c6h sgogmu ,k4 $m/s$ 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 w:rh7 dc++rcw moment 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 ivvdzgh +xe m 3z:0d2fm*uzk.dd:s-/ of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass u -t.kmrs).mz 7.3 kg and radius 9.0 cm rolls without slipping down a lan tzmkr-.smu). e at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Suni;r1 kmu-l1a z as the sum of two termsl; a-i1z1urmk,
(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 164o0jlwv qv,*+ h0 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? Aq,o+vj v hw*l0ssume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rol+5(b +ns +rm1xkcx ojdls without slipping dow ok b1x(+m5+ dnsjr+cxn 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,hlvf 6,smhuxxd6s6. 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 before it hits the ground? [Hint: Use conservatio66l, h.6 svx dhxf,sumn of energy.]    $m/s$

  What is the angular momentum of a 0.dagp c3czjs( :-0z xsn5m8x r/3.doxw210-kg ball rotating on the end of a thin string in a circle of radius 1.10 gs8m rzxac53 sj/ 0xoz3x.cdnpdw(:-m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 7 s*dtd6xr(ad2.8-kg uniform cylindrical grinding wheel of rad s67d( *dxadrtius 18 cm when rotating 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, handszxl2n2l th9 slo/oboyuc z(dw20i* 7- 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. 8–48, the speed of(y 2iwl- tl hl2/zbox0nzcd* u9o72s o rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–2e3et rg* cxd25)vlhd 87akg*jl; ce8c9) can reduce her moment of inertia by a factor of about 3.5 when changing from the straiggg v25dc3e8el*)d;a ch7*x tjr 8eklcht 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 increase her spsc9m9+hhi3 hr ot ii+2in rotation rate from an initial rate of 1.0 r9i iis29+mro3c+t h hhev every 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 c rxv:hrdru:w+*/;gr t dm3i .center at a frequency of 1.5rev/s The wheel can be considere w/cu3. grrhr:mi:+dx *rtv d;d a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinning at 3.5*b ku1bnazv24q2 f+ lhyfgq-4 $rev/s$ 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 mav/r vl5e).03nu.yx5mjx wv at bu)j7omentum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk on.b+igzu 33h uf moment of inertia I is dropped onto an identical disk r+gh. 3i ubuzn3otating 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 q ty,h*yyehym xsy5o3l( m +mr6(2f/e$rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotating merry-go-round p bvolo/4fa/ p 35cg;eslatform of radius 3.0 m andfvgpsb/3c4e;o/ 5 l ao 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 angul i lelyq28w*7+eb hiw.ar velocity of 0.8el7i2le i*w 8hw+.ybq $rad/s$ 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 iby-y5p0kxin-)y:ee o2y(bldnto 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 o x2nidely-y0e b yb5(pky): -lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds,2 g5wk;e7rq:zlbvaztq zu ++ 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 v+ qgx 5 rn1z(h.epj5d$ 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, wpuyoq /i+5 e vft18 xzj0n2,+u zrk.vinitially at rest, that can rotate freely without friction. The moment of inertia of the persne01zp+oyt ruvj,v5k/+q28 xiwfzu. on 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 y,4g)5lv m -huwn glg8 jf,9xemerry-go-round turntable that is mounted on frictionless bearings and has agv)yglwg,jl n, 9h5e 84u-fxm 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 rolls3 vuuczeu*w/ ni h4)b* on the 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, holdin)4hv /zwu*cu3nei*ub g onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces the Earthumff v+x.+2z bj78:szqsta6z . Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the .sqa xz +8m fzf6ztsu7:2jvb+ Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 1 ok p n: zbcf8p ub6 ga(r*b4.+;g/rg::exuqzm 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.t+h8 pd,e) gva20 m acquires a rotational rate of from rest over a 6.0-vpgeh +da,t8) s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, ea 9f(-i klnmz p,cc0su (t i0.gtm*z*srch 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 tfsnzpl 0,m0grm((cziuc* *.s- k9tithe linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed ofyr c*th4,fh5i)7 ptsc 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 gre7 yao w90+5nzlrwnzcq4 r) hzlq-i7e2 at, 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 izq0)w 2cr ne4-onz+rl q 5hlw a7zy97the 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-poy py)hmntc(m1i ) )8 yx 72fabtnx8b:vllution 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 (t1y8 n) :y)bpxt2 x78 y cmvnim)bafhof 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 illustratepu (m4d 0 dv2k 2jsro*d6mdx/us an $H_2O$ 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 * qt c/pul-,rkhorizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we ignore frif- k3l8qt *gyzx(xhg( ction) 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 of mass of the rod, as shown. [Hint-k(qhfg 8*yxlgz( t 3x: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically ongrv 4iwcyi9uvtgg3j7h **(k- the floor, and we want to exert a horizontal force F at its axle so that it wil ig*-wtyj i3uk7gch(v 9r 4v*gl 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 on a flat r q2*34uro mksf)o ,a-3jrl3ij) bhua boad is making a turn with a radius The forces acting on the cyclist and cycle are ther3a)s3)*ilok4 mu2u-3 jfhao jqb rb, normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into 0c mk+: z(ysl*af1bge 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 required tol+ 0fcem1:k( azyb sg* achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid .yri j)y (p4ew9c,vify pa5e/ 7rnv0jcylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched ar,e c7jai iy9. v(rywnjy4p vp05r)ef /ms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylin6 --bex havb(ndrical plates, of vha 6-xbnb(e- mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rpty+w9sfxy31js 4 ix(olls along the looped rough track of Fig. 8–58. What is the minimum value of the vertical heig pisf wyy4sjx (+xt931ht 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 d ob/mn:eq1djpt( ,-w co not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolution)qtd ijf))pjs-en : 8etvlv62 s as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tire?6 s ev 2)je8 lt-vpjq)t):ndif $\approx$    $rad/s^2$(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