A bicycle odometer (which measures distance traveled) is attd,x2 ee x0-akvached near the wheel hveekad -,x2x0ub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Dqm)/zpq.ua6 ; vymt d2oes 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 qv;2a./u dt mqz py)m6linear acceleration change?

Could a nonrigid body be describedmdp)wb/ac1-2fb8i g 3 ari*-93dx kivcvh 8yl by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torqued4evd l/-orl)lvdqqv +t8l3 3 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 difficultt 7e7iu0k 1jl g1g/ekx to do a sit-up with your hands behind your head than when your arms are stretched out in f 7gxg0k1ltkiu7/e1ej ront of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the pe);xk vr1rw oe0dal cranks. In which gear isrx)w eo0k1;rv 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 able to run fast have slender lower legs with flesh35 nwdyc)ve( b and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why tny5b3 dwv)c(ehis distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narro wzgf60 sf6e0rr2a.pi w beam?

If the net force on a system is zero, is the net torque td nk2 jo.h+6p k)um:q 7leb4h:tu q6kalso zero? If the net torque on un+m:.4kqk h u66bthkp j)d ql7to e:2a system is zero, is the net force zero?

Two inclines have the same height but makfj h.lxxn4 8o4u4 hnz)e different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball atn44olun x8 hhzj4.x)f the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) r.g -yfav3r4 w icu6eavqs5 :1down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches th:6g airu-y.1 4q v5rf acsevw3e 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 cylinde j lex0n9b* 1lvzm8t34/upegar have the same radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total ki ep 4x*z3980jnl v/1blmetuganetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most moving orqv;wca os*2: ml3kh l5 rotating objects eventually slow l*cw qhso:3la ; 52mkvdown and stop. Explain.

If there were a great migration of people toward the 3i6r8d8b akz 2p;qmop Earth’s equator, how would this affect the length of the day28ap r3k6miqdobp ;8z?

Can the diver of Fig. 8–29 do a somersault without having any initial rotatov* /in- jcy8aion whe-ycj8von a*i/n she leaves the board?

The moment of inertia of a rotating solid disk about an axis through itsy5 231bqhhy6wuiiy dn0 qf2.h center of mass is .32if hi5qu0ywhh6 yn2bd yq1 $\frac{1}{2}WR^2$ (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-kwn32 k+:468g gzci-tk fq- giiwz7d dfg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decreas+ndfzi:-zc7q k w36-dg iikg g2ftw4 8e, or stay the same? Explain.

Two spheres look identzyb)m a)5.x+z p7qjsq7 k7ku j jl9j*pical and have the same mass. However, one is hollow and the other is solid. Describe an experiment to determine which is whick)sj)px9lkb5 p q.z+az 7q7u j7yj*jmh.

The angular velocity of a wheel rotating on a horizontal m*nhmn bec8 u*lu))m48l2 aozaxle points west. In what dirucam nm*nlu*)2o b)88m hzel4ection is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large + 0va+9 -mruxek(mobvfreely rotating turntable. What happens if you walk toward the c(bmxv +ar-+ emk0v9uo enter?

A shortstop may leap into the air to catch a ball and throw it quickly. As he t ,etumd8.gn9a9 rb+ c fb7)hclhrows the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite di9hb ct9u,.er+bnd7 gmfl8) acrection (Fig. 8–36). Explain.

On the basis of the law of c+lcnoq/n-no wa zr 10)onservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helic10 wqnrnaco n/o)+-lzopter stable.

  Express the following angles in radians: (a8 ) uwe4nw5va/fsavns.y ,v*n ) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth b i07mzvr g cco+9;n2zmecause of an amazing coincidence. Calculate, using the information inside the From9irco vng0m+cz;7 z2nt 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 the Moon, 380,000 4awpw4(cvn/h* d 6dwzkm from Earth. The beam diverges at an anglevcz* 6 pdahw4(4/ nwdw $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rrog8,l3 cd au:otate at a rate of 6500 rpm. When the motor is turned off duri locgd3,ura8:ng 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 child. If5t4g i )k+w0yz: hixkl the ball makes 15.0 revolutions, what is its diameter? lk :y+ wx)zti0k5 ig4h   m

  A bicycle with tires 68 cm in diameternr/ b /ww mtp z:l+8nn1 p7cv-n60ivsq travels 8.0 km. How many revolutions do the wheels make? 1lcn6tn +npb/n7v p8iqz :smvrw /0-w   $rev$

  (a) A grinding wheel 0.35 m in diameter rotadbda km4, 4zn,w/igr nem4p2;tes at 2500 rpm. Calculate its angular vwdr de/ 4, nnaim2pg44b,z m;kelocity 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 on dg bck0;9tk+ne complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated +9kkc;dtn 0bg 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbit ) oyrvx29 e.zz/m-ygl around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed8eefgzwy4 0x /hiue t*, j/jtt,(f+qq 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 fkl 8 )spqu7mw7rom the axis of rotation is to experie 87ks)p7lqmu wnce an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel acbm x8jw53a -1:xmrfub celerates uniformly about its center from 130 rpm to 280 rpmu :fa3-5bmrb18xwmjx 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 radiu xq6 ( mq/2:ehv,ln3wr1lazrt s $R_1$ 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 sp::w-a07wqah,0 w fcxkzfbq :j acecraft 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 k-qzx0fwcf: j7,:b:0 waah qw a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,ds1hdy3kjk8)/q: i ru000 rpm in 220 s. Through how many revolutions did it turn insq: rh ud)/jk8 i31dyk this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calcf7); dreznt+a fz33o nulate
(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,,-m )e0f r5knxd*k ehtvuw:j for the stresses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 comp,*h:fxew) n0vkm -de,urt5jk lete 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 unifom 4e (,1t)hvfuv vzi2nrmly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wh)h z vtv4 mu(inef,21veel have traveled in this time?    m

  A cooling fan is turneduq s( nxlvit++(r va7) off when it is running at 850rev/min It turns 1500 revolutions before it comes to a stop(7)r+sxv(qaliv+n t u .
(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 szu9e-gai:x; a 0w2d gapeed uniformly from 95kma;g2 9gzaeu :xd0-w ai/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 ca anc3 6hgxb m()f+b;ylr reduces its speed uniformly from 95km/h to 45km/h The tires h(x);g6hyf mb lc3n ba+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 putsk zo++6f*f.nsbwsm ) z all her weight on each pedal when climbing a hill. The pedals rotate in a circle of radiusn*fz) w+ +z 6bom.ksfs 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 doygh7b+dzmf 8*,h 66tz mdn ym;or 74 cm wide. What is the magnitude of the torque if the force is exertedfm86gmzm bht6,+7 dhdz y;*n y
(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 Fig. 8–39. m /+yhvhi s(u m4.*dzhAssume that a frictm vdum(hi*./syz4h+ h ion torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the e2qvqkb ugn.(9y(+q6kgeebu*zjag ldx9/0 :w nds of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Cax9q/ 6zegued (:ug jwkgbqqkln 9.0(vy2b +*alculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tip+ynl.ym/ icrjtvef1 / w 0.6aghtening to a torque of 38 yavft/c0i l6j 1+.m /reynp.w $m \cdot N$ 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 sphere of radius 0.648 m when thxmk;b7dg6 1una v,*uee axi *kx7e,;abum duv gn61s of rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inerti;vq yq* kw)ac+a of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub c*; qkq+yv)awc an be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, ligra;m)jqqy lal ;,o...orz8ww ht rod is rotated in a horizontal circle of radius 1.2 m. Calcula.,wo lw);aq a o;.jlqymr r8.zte
(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 rotating at constant angular spk)4v6mk n.rqfmaj: ;2a h/oljeed (Fig. 8–42). The friction force between her hands aah6)j4kq2o/ nj.: fravmlm ;knd 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 poihvvyk/2m 9 x0vnt objects shown in Fig. 8–43 abvxk 2vh m0vy/9out
(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 maa-/r:v6b1e fn tysh6j+m(q agss 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, unifor5p0t1wd zik)l5 k2x ha 6u4hivvk,lp/ m cylindrical satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3)dk4ut1xhk ha/6kl,iwlvz pv2p55 0i 600 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 radius of 8.50 cm anldbr99 (+grqmja(ic8d a mass of 0.580 kld8( (rg+q jamci9 9brg. 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,33;p-dqa .od jb 3oiqi accelerating 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 merry-go-rolcu atj;xkz z0-,vcr dqu3-8 3und 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 vrudjzc;c--u3x kat8lz,03 q 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 odfh;xhq wg)0*f; vavp:c(t0dff and is eventually brought uniformly to rest by a frictionalv *xfhqd:g;a( ) 0t;d0pwvhfc 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 acceleraff 7*(4 qd mypzx-a;fdtes 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-k(z9le7 udej l:g ball is thrown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 1lzl(ej : 9du7e0 $m/s$ 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 beq3bmuc*z.3z g7/8yug yr 4ege considered a long thin rod, as shown in Fig. 8–46.qee/3y cz 4 u8zuggrg. 7myb*3
(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 mbba;/wc4i-i;)y zx ukasses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest within fou3m 8apgj7tk iq7bj vg8 8,uhi.r full turns (revolutj8tm,uk7gq .vhi8a ibgp3j78ions) 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 moment of inertisr l (r7 i;pal 4j)1ne9l,;hys mxso.ja 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 an)r19av d1t n)s/kq v3*o khdk torque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls without slipp4xoigm91h cm9/ etn-pl njelyy.,0hy+s+2 so ing down a lane at s1h 9egymh/.+ llmc iynnto24 p+-90ose yj,x3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy+ri+qf7yh/no 1x .f8qnr7iwa of the Earth with respect to the Sun as the sum of twfq.hr18f7in a+7qno+ wirxy/ o terms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a m cfg7et1 ha g++rex4k)ass of 1640 kg and a radius of 7.50 m. How much net work is required to accelerate eh7 agrf ge+x)t1k+ 4cit 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 f )nfv ph:x*w*vvapmwyo4 -d/zk4m ) em1 +yi0lrom rest and rolls without slipping down a 30wl m id)f)*avz+h 1mm0kxv 4wpy/*vy e:4-npo.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

  Two masses, $m_1$ = 18 kg and $m_2$ = 26.5 kg are connected by a rope that hangs over a pulley (as in Fig. 8–47). The pulley is a uniform cylinder of radius 0.260 m and mass 7.50 kg. Initially, is on the ground and $m_2$ rests 3.00 m above the ground. If the system is now released, use conservation of energy to determine the speed of $m_2$ just before it strikes the ground. Assume the pulley is frictionless.    $m/s$

  A 2.30-m-long pole is balanced vertically on its tip. It starts to fall and i4si+-6kg i d)uw riwj+ts lower end does no)wgwk s4iuj+-6r+i dit slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the eo(; m8 pu8odeg;erm psk/n2i6 .ar pc3bu7gi /nd of a thin string in a circle of ranb p8 por26se d/m7opr/;3gu m (ki.ic;ga8eudius 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 unifor;iem s-k .3g*c)- :wgwhymgu om cylindrical grinding wheel of radius 18 cm when uwow:cmeg*- s ;3m gghyk.-) irotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that ova qk- 22(zemis rotating at a rate of 1.3rev/s If he raises his arms to a h(-2amo2kqvze orizontal position, Fig. 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 shown in Fig. 8–29) can reduce her moment of if1jd(*9lu x2 t lxxzuqf3g)7j nertia by a factor of about 3.5 when lz* q2x)x jtljdf9u7(g1uf3xchanging from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin ro(8vwit7j . 8fgicgj1 utation rate from an initial rate of 1.0 rev every 2.0 s to a final ratej88i i.7 ( vfucj1wgtg 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 throrn2- oq-, sgad4uzn7b ugh its center at a frequency of 1.5rev/s The wheel can be considered a uniform-s, d u4rqb2n o-a7zgn 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 we9 b5gv 9j o/rr*)dypmomentum 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 mom-3;p 8glt8augpda krzi(zo2 6entum 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 of moment of inertia I is dropped on.2 9ah)n 0twur, u/kpjzubd)xto an identical disk rotating at angularut. h,u2 0 jduazn/)rkpb)9xw speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2t 3jw r07acdfo(z*b1 f.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 3zyn4hm peb+icu9z (4at the center of a rotating merry-go-round platform of radius 3.0 my9c e+(4m ubp3z4izhn and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with anm fc*f4tg/9+d i6ieead nxqlfa4 y )80 angular velocity of 0.)da4ml 80 c qe4i +tfngf ie9xfy6a*d/8 $rad/s$ 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.ih) llw *fq4,ru963p o3rcf4 tdhg zp collapses 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 angular mome ,d 64hgi ut) 33pfop* h.f9q4zlwrrclntum, 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 involglyx0svi1+ /uve winds in excess of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass a2)8e szvt1 bn$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, that can rotate freellr6ts.go d+f.-rwl kmn9-l 1 ly without sfrgwt r m+..d1k6l-l- l9lno friction. The 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 standsgy9idh;bv7vx/*3ppq.e ch . p at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on fricthd/ycq7 xp ie ;vb *gh.p9pv.3ionless 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 ona ;dgez ;6jn-ibqks3 ,,y,rrfk6m:y h 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,-ziy,m,jyefk:an k6;sr 3 h6;qbrgd walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always fa:phq/o oxtp./rovxh u:;h. .ices the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axh;r/ov/ . :hp.xp hixuo:qo.tis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates /ae :pusdxo/4 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 kfe;gq;3tb0y yx+70 pu ly /mlthe shape of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval+7uxe /;3qplfgkb;0 mtl yy0y at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical diskx*9szmtcn3lb4f2 - n31clcw js, 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 thetcnlx9cb cs3fmj l*4 zn23 1-w 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 angul5d 94o a0zz ,qn46 rlv9bcskhear 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 .ryswlurm5v 5e.q g10of our Sun, but with mass 8.0 times as great, were rotating at a speed of 1.0 revol.y0ev5 swm5rqug l1r.ution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollkzt ek9 r**;uwution automobile is for it to use energy stored in a heavy rotattew9kkrz* u *;ing 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 9ktdpu * nul,w+w5:v i$H_2O$ 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 0cn.vnmyf0lj: 6k k;smdr;)+d7b rrsa horizontal 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 fric9k 2drd zc ipo4)4voy0tion) 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 o4rpdcd 4k20zo vi )9oyf the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is staow vizku4jy 9sq:/9ys00b1 nknding vertically on the floor, and we want to exert a horizontaln 40ivq9:bz0 jw ssyyok19/ku 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 i+4at (93hvee sdkp9k speed v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist and cycle are th(v 3aeh94ts keid +9pke 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 avbn 5tw,z3 yn* :a6 hj3wzl*fg 0.50-kg rock into a sling of length 1.5 m and begins whirling the rock in a nearly horizontal cirzbv w:zw3*5l*gj3ath6ny f ,ncle 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 body as a solid cylinder and her ar4 ;:0 xmgx ixzhnef,t mef(m(3)r,ewrms as thin rods, making reasonable estimates for the dimmmtn3fer;f(ex:,eg) m(w0zxh,4 rxiensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a q*5 a/amf dq1ld6i 3jxclutch assembly which consists of two cylindrical plates, of mass q/ q* fad 6lmad3xji51$M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 1 ic-mydhtm/p5y :8as .e nvk1track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop .-n11 vmcid sy/:85ahe ymptkif 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 :ap amagc/bc:gk;+j h:qvg3 4do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutivu1 n2z jmuz q7sb5d8 nwyt;27ons as the car reduces its speed uniformly from 90km/8q12mwnnz7u uz7 5vd 2bjy;ts h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s