A bicycle odometer (which me*i7ziv7zm . fsasures distance traveled) is attached near the wheel hub and is designed for 27-inv7iz z*ism7.fch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velu0zq4yvj-ea*p7qp ;s ocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radiau0*q vapys4j - e7;zpql 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 vb..q 7i;h /j)w*b:ektu rw)12t odznmpio)psingle value of the angular velocity $\omega$ Explain.

Can a small force ever exh*0vn ;z d;efc tsto*/ert a greater torque than a larger force? Explain.

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

Why is it more difficult to do a sit-up with your hands behind your head than )a ipz k2hsz05alnaga c48a.. 9ouso :when your arms are stretched out in front of you? A diagram may help you to answer thisas5 sp )uanahagoak:c092 . zl48iz.o.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the peda0br 1l;b 73cp2cdly jol cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprockej31y rl lb7p2c; cdo0bt? 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 h6z8cprch4;+4xz* cy 6 gycg xoave slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–xc6xcg4cg zc6*oy4 8y;h +pz r34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, n,:zrn)hsw3c w arrow beam?

If the net force on a system is zero, is the net toriog ag,4 x/x/ wh9z 2ri4w-0 0ualqqeyque also zero? If the net torque on a system io4qwqg ihx/a9/4x2, y-rae0igz w0u ls zero, is the net force zero?

Two inclines have the same height but make different angles witjgf)m:pc, akox2 h.dxj2 i+6ph the horizontal. The same steel ball is rolled down each incline. On which +6a2 xki,dmo h.jpfjp2c g: )xincline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) dowk1u v -3b4q2 8fo ythgvwh8,4 l8jdyman an incline. One sphere has twice the radius and twice the mass of the other. Whi2 gvqf 1db8 tyhwoy8l8mj,-u4 kav3h4 ch 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-b75+ u njjxf 2anj+bq 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 gb-nxj2 u +jj5q+afbn7reater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum :1tc3 kluop ,bare conserved. Yet most moving or rotating objects eventually slow down andltk1o:cu, 3bp stop. Explain.

If there were a great migration of people toward the Earth’s e;8uja 3 a7irta6u9nao quator, how would au ;i9o8at6 rja3a7unthis affect the length of the day?

Can the diver of Fig. 8–29 do a somersae) s*f8z8 y5e:-o(ujlfgv wzv iup0a;ult without having any initial rotation wh ; 0f-oz(8fu5e siwg* luvyv: ea)8pzjen she leaves the board?

The moment of inertia of a rotating solid disk about an a+)frevyfe u 3 z(p8by4a5c0u gxis through its center of mass is4 0ez8+ g )yufyrfb3eap 5v(cu $\frac{1}{2}WR^2$ (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 q rh e*zx,t+ axugan)/dc3*n(a rotating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrencn ru/q*g*dhtz )+(x 3a xa,ease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hollow and o gn)zt/ n-r+85 sytvmthe other is solid. Describe an expev8 y+ tgsrt-n/mn oz)5riment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axl/gc) 9 2t lufj87htbqgepv4(kw n-, s 2qgr7rye points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points elusf)kgq 224,rh7 g rpebnw-v 89(t/ tqcgyj7 ast, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standwmwtm;k zmlsg8-ms09 ux( 49ying on the edge of a large freely rotating turntable. What happens if you9mus0t ml(m kz8s w;4gy xm-w9 walk toward the center?

A shortstop may leap into the ai58hr xzkrc4o/ r to catch a ball and 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 rotate iorzrx 4 c/hk58n the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum 4tlcwzdx0zba/np,e*a5 fm.k l, ,g8i , discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the sec* g8zaemlnwa0zb5.tpk i4/x,, fld,cond propeller can operate to keep the helicopter stable.

  Express the following angles in raddd5m21qlbew2/ 9 + x rl5bvyimians: (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, usi5-no7 s-bq /7noq7n*zjsope wng the information inside the Front Cover,n e 7zsn- j w7-boqo5o7s*p/nq the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from j9 )sn()xoczoa/q/m 0ty7h de Earth. The beam diverges at an a()0 )9/so7ehyjdmt/qzxc noa ngle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rot9n2vfynp*r)4mqfu8h3i s z (rate 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 angularpin(4z um 82) srv3rhy q*ffn9 acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another v q8id(7jrw6v0vq)x*( a b4j keeewdoj e8/g -child. If the ball makes 15.0 revolutions, what is its diamete jd)oiw7d(8-w4v*x ( erj gvak8ebj6/ e0eqq vr?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many g-w7do, z1fm+. hnp)p0rn dqluom hb )xk j17.revolutions do the wfjn.bh m 1x,+o0d 7.p-o1hn) wkrgl 7)mu zqpdheels make?    $rev$

  (a) A grinding wheel 0.35 m 54bfqu t6feb utt5(a)in diameter rotates at 2500 rpm. Calculate its angular velocity in f(te45t atfquubb6 5 )$rad/s$ $\omega$ =    $rad/sec$
(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 complqdq 1a8m0 f7eosd 0o- wyx*z-nete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2-noewsd m f*1x-yza 0q08qod 7 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 olwebye2uw 01qz t0:n)a *i0igrbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a popmjc (dbhio)ut*o+92q4+juc y 7eh .p int
(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 centrifug9gn7ayeqkm u op v1v3/mc71k 2la 7m:pe rotate if a particle 7.0 cm from the axis of rotation is to experience an acceleration of 1001n:p3oepkql72g k7va v m umac1ym7/9,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its centerfp(8raes1t 7wcc0 1b n from 130 rpm to 280 rpm in 4.0 s. Determin( 1cn87bfr c01tsepa we
(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 vx5-tuoi ff3h4d s0;us $R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard the Apollo spacecraft put thewnl : shnp vy*-xd;2h/mselves into a slow rotation to distribute the Sun’s energy-; vyx/ hh2sn :ln*wdp evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm in opl8g ))lu. nns,gr3 d220 s. Through how malnrgu8 o,l))gsd. np3ny revolutions did it turn in this time?    $rev$

  An automobile engine slows downgi ac1:u9li)z 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 focsr ars+zqvs92ag3h /7p- 4 tlr the stresses of flying highspeed jets in a whirling “human centrifuge,” which7h r2rp93 tl4q -sv+acg z/sas 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 diameter4du 6)hbfg wmw r y-c6(r5r8rv accelerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in thisf cd)rv6hg 6-(urr ww yb5r48m time?    m

  A cooling fan is turned off when it is running at 850rev/min Ipd*1ux2p:tkx wi 0ypp u.87aqt turns 1500 revolutions i : pyua1q7 *x.twup82 xdpk0pbefore it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions asb+2 ,mazqwvjt3v- p8 y the car reduces its speed uniformly from j 8m bqw3-pzytv +,av295km/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 unifqtb8k 3h 2zi5x7uebz7ormly from 95km/h to 45km/h The tires have a diaxh2i 7t7z5 3 qbbzek8umeter 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 px ;t z0-anv fx bqrz f:z4--aazv3-v2xedal when climbing a hill. The pedalt zn f- x4fva-:z xazx0;2q3bavzrv--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 is the je(g b x7dt2hh iwwh-2r)f 521pe; ufymagnitude of the t)2d1ex heyrw-pj fih 2t ;75hugw2b(forque 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 torquen03orul9( t pvwj2l/ 0barkk0tnn(- w about the axle of the wheel shown in Fig. 8–39. Assume that a friction torque of 0.4uktnno0 (l/jr lb-ww0p(ra0nkv9 32t $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attache 6zo6h t.,ifeya(9shh6 sm/ni d 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 tor i6s9nfh6imyt.ash6oz/(e ,hque on this system.

  The bolts on the cylinder head okb1e7s4z fhs4b* ay, lf an engine require tightening to a torque of 38 7,l* abzs44sb1 yhkfe$m \cdot N$ 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-k hqtt (3t0uxk6g sphere of radius 0.648 m when the axis of rotation is through its centeqx3 t tut(hk60r.    $kg \cdot m^2$

  Calculate the moment of inertia of a3jf 3yg)oc l/u bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 l3)3juof gyc/ 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 aqot40em8ds9d(3 5k2oujai l h horizontal circukjq2slm4d3io a5o 8hd9 t(e0le of radius 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(czm *rz0uu 4faq/yr -+fqbg)’s wheel rotating at constant angular speed (Fig. 8–42). The friq4*()z z+ rr/ga f-cf bqum0uyction force between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objects shown in .tkvx9c m0gqd1a yl0 /Fig. 8–43 a.a lv0g c1m k/yt9dx0qbout
(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 wh5+1lrgjz 3h qs3f5nbz ose 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 sen l .a-g*6xf * y3h/uwznns5oro4ka /pinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass*l.a5ueo3aszgk6nn wxy/r/ 4oh nf-* of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 gpez6 lzk 2y4co- 3xv nb)f) 9n3esou;cm and a mass of 0.580 kg. Calculatzlgsu oyz4e f9 k-;c26 )ov)n3pxebn3e
(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, accelet*se/8 2eadjda e 3-nqrating 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-round and is ab2fhjdnsiu3 6g3s.z12yh g -pagw5kn;le to accelerate it fp3 uj f2yhsswiknd.z;g6hn g5a-1g32 rom 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 torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is eve)u6z+o8ft 4-m fq vza8j*sojo( orm s2ntually brought uniformly to rest by a omr62 8 8-)uoa4+(qsjt*jos mfofv z zfrictional 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 accet su-q vxhp.bxbcm0w8 ln sf;z7k9 i4n*m3 n38lerates 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 ptys)t;.ntf 9j6y1 thhz6 n+ g ltom6,by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, h;t .t yyfjmo,9t pz66 tnth)l g+s61nFig. 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 consi1f c fn;sz88te9 emkz)dered a long thin rod, as shown in Fig. 8–46s9 zec1)t;km zef88n f.
(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 yq bj*87d jyu0q/:y0v vzq9m sof two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest within four fh)(w63foc4u,o*1nvfhea es dull turns (revolutionf noauf hwe*, ()c6e 14dv3shos) 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 inertia of ;oh5gpx o h h7roipd0/,* hz8m$3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine 8vjtya2ow )-kyz ;c0fz *(wnu develops a 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.( 3hcwwg m(hs and radius 9.0 cm rolls without slipping down a laneg(cshh( . 3mww at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth wvnouax1 -/0 dv k..8xjfjkm ,rith respect to the Sun as the sum of two terms,u a- kmof1 x.njv8v,jd0r.x/ k
(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 kg and a radius of 7.50 m.5d0/vf vp 1n/z mx*qrr How much net work is required to accelerate it from rest to a rotv5qr0 /m1vf *pxz/r ndation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and roll r7u+tq ss*x mgnn0)-xs without s*- )nnsx xt70g+rums qlipping down a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is balanced vertically o5ou n -ep)gfrj /*zy1hn its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of theoe urz*y5fhn)pjg-/ 1 pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg v56g;m s vwj,qo5)dvak +;ezxy91p:t) zzsjn ball rotating on the end of a thin string in a circle of radius 1.10 m at an ang6mojavw+v vt19)szy 5e;)z k5 ;jp,qx:zn ds gular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kyja m ;wd9z*vq*u7wb2-j kvc wm834abg uniform cylindrical grinding wheel of radius 18 cm when rotating aujmw*7wam b4c jv q yz- 3*b;dwka829vt 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 hibivy( l :00m5a;ne fjus side, on a platform that is rotating at a rate of ;v0:jeabf 0yl5iun( m1.3rev/s If he raises his arms to a horizontal 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 inertia by +/q2x me6ucfd a factor of about 3.5 when changing from the straight position to the tuck position. If xm /d+fqu62c eshe 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 rotation rate from an initial mg)sn8* ccjm,j+8 gs rd;ome4*rf g0prate of 1.0 rev e8j)jr fo*g8p,dcrmns*mmc+04; sg gevery 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 7hwe,c +g79l 5df- w mql3cdzefrequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay stil +emcwwz93dqgd-flc7 h,7e 5 cks to it?    $rev/s$

  (a) What is the angular momczde0 ttz 1+8h.jbo:pentum of a figure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of the f4e ) f)ugk+nbEarth
(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.)l ny1h ey*d2tvcz/v I is dropped onto an identical disnv/czevy.2d1*h t ly)k rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at zd8.k s2sfy;mok/ v 6s2.4 $rev/s$ 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 cente gde;)x.6 qspxi3dgx2r of a rotating merry-go-round platform of radix3psx.x;gdei)6dg 2 qus 3.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotatin )i4(u kdck8wak( g+xdg freely with an angular velocity of 0.8uxkik dg (+ckda4(w)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 collapses into a white dwarf, losinxt2dh qvex 8+1adfb- 8g about half its mass in the process, and winding up with a radius 1.0% of itsxav tx2dq8+-8eb 1 dfh existing radius. Assuming the 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 in excess oo;kuw7 dbjcvrd8 +9:c f 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 ;o; r znsjf3nprnf ,9)$ 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 d3q lj;mj0t6*ft8ma crest, that can rotate freely without friction. The moment of inertia of the person pl;c6dmq8m jtfa3t0 j*l us 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 9; lnmqpb +) ,hblyxi5at the edge of a 6.5-m diameter merry-go-round turntable that is mountql,b x+yh9n l)ip5; mbed on frictionless 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 ground with the end of the rope lying on thz 5ns1 9, kktp:7wnsogp9ir:be top edge of the spool. A person grabs the w1 79n:9sizk sp pb,rnk5ogt: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 far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces the Earth. gqnhh:6ufzq7 y ouv(l5c (h( n9kk+ 0kDetermine 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 particy+nk vqh7:u6k9lh5( g uc (z0oqk(hfnle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 1 m s4dfhx 6 s+u+jw78edj/$s^2$ 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 cyl,lcwi m/gwwd-jqv/ 97inder of radius 0.20 m acquires a rotationalvmjw9/i/w ,-cw dqg l7 rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 kg anumi3jnt6( s3f x6srw3kb 0/hfd diameter 0.075 m, joined by a (concentric)i0j36t3bfm63ss (hwkx ufn r/ thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the 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 of the rear wqqz, ohf9q t4:1csxb8heel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as great, were rotat-xx)e zvk j.sk*8w,m zing 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? Assxe sz.- )*kmvwzk,jx8ume 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-pollubd) mt) t,(qcqru8k/ ztion automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to dqrtum8tb(k)q/ ,cz)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 m8g7dh mn (rl/wm .pcuqw5/9j 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 horizontal surfag:f cp(udsu59e. )taz ce 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 (ij ,-4 jnobf4:-r 5vtos:bxxct.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the li:x- 4cr 4btb,o n-v5oj:x jtsfnear 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, and we w; ll-iv jg)s+sant to exert a horizontal ) j vilgss;-+lforce 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 vs q fo:m6k-0w. isfh*seu-- ij=4.2m/s on a flat road is making a turn with a radius The forces acth- 0u6.:sfq seisof-w ki *-mjing on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5x/6+obty3cror f .me ( 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 to achieve this feat, anr/r(eyfo6bto +.m3x c d where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her amob ae8p.e;p epu *4f*rms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angpmbep4o. ap 8eeuf** ;ular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch an--xwo ;al+00wc aux assembly which consists of two cylindrical plates, of mw nc;alxa- a xo+u0-0wass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 rimu5jfwvtio9kor 1 nj6yjzpv 2i -z30d 9 )b))olls along the looped rough track of Fig. 8–58. What is the minimum value9p6 t)m 0)vf5k ziobv-zj nju )o29j3d1wyiri of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not asy ob;u0(hghmo)8z 5dtsume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions jcfny ep86i1 p/1nt l,vb ssyod ca;:l5:cv76 as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 mle/8i6asdy c c t1c lf; v1 pj:p765nyo:sn,vb. (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