A bicycle odometer (which measures dist 2q.o(ps/y fi.ozx/aiance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What haps/2(io./. ozp aqfxyi pens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constantrk:3c jp33i 9jj1rxgo+o9 tkj angular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angu p9 goc3i3k+9torkj3j rj:j x1lar 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 ang7j rtnd-wz* 6vular velocity $\omega$ Explain.

Can a small force ever exert a great2j(; */5d: kja6 afsricyfqvxer 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 wk dy22c*9xvwm yvjbo8wa3or 1)s20bat(- pthead than when your arms are stretched out in front of 8ybad-w2 t m1*jv 22(wort03) ck9xaybsov wpyou? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel0x onxc4y hk5 *58atmr and three at the pedal cranks. In which gear is it harder to pedal, a smalymc5x* o4 0thk8rxna5l 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 t2w2 gvt;oi1913 lqx sv*mkynro run fast have slender lower legs with flesh and muscle concentrated high, close to the body 1q2s wix gnvvy21* klo;r39 tm(Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry ac5ga-j;2 :v / k(2lkzfldsgp q long, narrow beam?

If the net force on a system is zero, is the net torque also zero? Iftwv w c84rsccj3t/rrh;63am + the net torquewwstah3 64 c3+rmjtcr r /c;8v on a system is zero, is the net force zero?

Two inclines have the same height but make different angles with the o,oz,jc c p)qpjaatx8u *: .i*horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greaqi)u ,z*j.jtxo* a8cp :,capoter? Explain.

Two solid spheres simultaneously startpza /mo6w 5 nszi**-1b pm5uvr rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the sp15vrzz*- m/b 5pom nuwa6i*other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. They start fro ktgfgx l6;d7mp-s0,z m rest at the top of an incline. Which reaches the bottom first? Which has thek 7ltg;zfg6pmxds0- , 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 mosx 0,(agvjw+nving or rotating objects eventually slow gjx(a0v +w,sn down and stop. Explain.

If there were a great migrysn(51gnt +ci ation of people toward the Earth’s equator, how would this affect the length 5csint(gy+ 1n of the day?

Can the diver of Fig. 8–2)amn;qc8 szxr) 8i ojx*/ js7i9 do a somersault without having any initial rotation when she leaves the bo) ;r8z*8)jmcxi7xq/ assi nojard?

The moment of inertia of a rotating solid dis v9m28vtp9u k yndt )o: 0*a(q:osgdkmk about an axis through its center of mat8(:yks20d a v m) unto 9om:pgkq*9dvss 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 ro; wo vt(bv--rptating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity incre-pb-(v row ;tvase, decrease, or stay the same? Explain.

Two spheres look identical and have the sd+ bhw *0uaru p) 3z;,wl7jljdame mass. However, one is hollow and the other is solid. Describe an experiment to determine w*zrwu0jdb h,+3dp; j a u)lw7lhich is which.

The angular velocity of a wheel rotating on a horizontal axle po-/ ;k pdcrqj 15ap8trkints west. In what direction kp5r1/kqd t j;-p8crais 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 freely rotating turntable./crh- e v*o wkbsg)7+*ngpwm . What happens if you walk toward th/rh)*cweg s pv7bw m+*k. go-ne center?

A shortstop may leap into the air to catch a ball and throw it quickly. As he t(; l8u//zlo gwlo yup)hrows the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in thely8gz u/p lo/;(l) wuo opposite direction (Fig. 8–36). Explain.

On the basis of the law of coya(*vcz93kj6 0ap: q oxca)m inservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). ajy0ixvk :c9zqa6ca*3po m( ) Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (a) 30 y1jz6swnu7:s5r/ lm bz4dw)q g nn7y2$^{\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 ampz,hm;f ep id04)hot(7sd+z)zmj 3zuazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and th0mfz+ p ze)z h3io( htd4mp)7;,sduzj e Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. Thea p-j*r fd3gk2 beam diverges at an angl *agjf23pd- rke $\theta$ (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 64y0z uc 7g/uxh8 c()fpue)nfc500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow d74 e c)x8ufuhnz0) ucy(g/pcfown?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another chf7msgb:,)j 5e/ wknp eild. If the ball makes 15.0 revolutions, what is itsmg j7)wn:f,b/5 eepks diameter?    m

  A bicycle with tires 68h3uh 99dpdmn*ya0b x c-o av5, cm in diameter travels 8.0 km. How many revolutions do thd,aom0c5 -uyhhda*pb v x n993e wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameterrj.x. k39 kh;igp: lje rotates at 2500 rpm. Calculate its angular verj.l j p9ghk :;e.x3kilocity 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 revopla hb0-ztj,ao5p0s/i7f m w.lution in 4.0 s (Fig. 8–38). (a) What is the a5hz./a oftils bwpm j7, p00-linear speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity o b;dn. :bhq6b2 1b :baz lc0+otle:(ke9tihiif the Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of 3han*q ju 5jehu.p9,ua 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 skxzau 4ggc-k. ptt, c ,2:io;7.0 cm from the axis of rotation is to exp ,,t-kku ps2gzg.ox ica ct4;:erience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from smlo*9ju4 nr1 130 rpm to 280 rpm in 4.0 s. Determin1ml9 4js ronu*e
(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 radiusllj 6kmof:00*ytm rrprgk6 94 $R_1$ 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, astrona,l 3o(aayv o(yuts aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly.,aly(3oo (v ya 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 c,jt bnz)6,k v15,000 rpm in 220 s. Through how many revolutions did itn,c b6 v,)kjtz turn in this time?    $rev$

  An automobile engine slows down 2nzwj 7 (soth;s.f,k kfrom 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 in a whirlinnd-vk zt g0b:(9hd xb7g “human centrifuge,” which takvhgb b:k0 -xdd7(n9tz es 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 diameter acceleratesrki sf5:unud h w2*q4/.mu1v c y;ebd- uniformly from 240 rpm to 360 rpm in 6.5 s:2 -wisq* cre u.4 bf5yd/dmuhnu; v1k. How far will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 8g dj8vk yp+b(450rev/min It turns 1500 revolutions before it comesp 8 +4jv(kbydg 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 revolutioel:pdoct n5/j*7/hc h ns as the car reduces its speed uniformly from /cdoctn5plj7h/e* h: 95km/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 hz jjp jtno yhi0k.nv-.qv2 9+ow2a.4reduces its speed uniformly from 95km/h tonkih z2..0. jjvwjay9 2hqtp +4voon- 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 phbm rbg:b, ;(kv j)4hluts all her weight on each pedal when climbing a hill. The pedals rotate v,hm( rj4;:b l)hkbgbin 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 maft2n 3r+s8bia-o , iyzn1fwb5gifnr5 f8oz+b 3,awsyi- tn1b2 nitude 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 axle of the m5aost5l 6dig* yxe2;wheel shown in Fig. 8–39. Assume that a friction torque of o d5a5m2ge;*6slxit y0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, arezzxakebc*ci;6lz g6tjl 6)03 attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initici z6a3ec* 6jzb zx6t)kl g;0lally the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening l0e*zm5 q+cxp to a torque of 38 lp+zcq0m 5ex *$m \cdot N$ 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 2.)hgk l kxg0 6f5cxo89ynwgcof radius 0.648 m when the axis of rotation is through its centw8lkf5yngo2 x.xh gcg906c) ker.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in d9j*9sw5 uutmr iameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub c 9um9 tr*5wjusan be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in a horizonta hd,(d1l3d.idcp jx c j*3ms)ql circle of rad ls1i* h)qmd3jpdcc,d3. (jxdius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating at constant ah2t)g2e3jx bnu hui4fwh6.7 fngular speed (Fig. 8– g 24uxfu.)h6h7if j2nw3teh b42). The friction force between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objec(tci7 gazuscc* ;q8p2ts shown in Fig. 8–43 (;sqci 8a u p*c72gtczabout
(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 atomv3 pt fggrvi5. abe+y4(a+ b:ts 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 a*klv vg if 765)kglvbhy sq(q9*v mc(*t the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32bkk7fq( i6)*gv5 m * vvv(*yl h9qlgcs rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8. uyt0-.p2ka0 okl:g*c ipkuc 950 cm and a mass of 0.580 kg. Capk2 t. gc*9y-aoiu :p00k lcuklculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerating it from rest t 1ld.gla .cqd+ :rdg4to 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-gkyb t-6r t 9wx;f8ae99s b)ntwo-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round;9t-w f sbk)69eb9w aytxn8r t 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 va.sql)c ;v24kg;u i lrpm is shut off and is eventually brought uniformly to rest by a frictilik ;a.g4v)l ;v 2csuqonal 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 accelerates a 3.6-kg b*wgnq 128x g;hrg)vmx all 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 the action of the forearm.gd( 1 nc-9qo;d4swgt xla y5m, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated un(o 5.;l -dqdxg mywsgc9a1tn4iformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as shown in Fig. dz qplp f0t1mgod,i0 376wyb38–46.7 oitdd ,f3 wp0mzbl1p0q6g3y
(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 two1m7b rebm-lt fwi,,n: 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 acceleratesp;vz 90 .rtbdd the hammer from rest within four full turns (revolutions) and releases it at a spepbdt0 v d9.zr;ed 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 mjs v bhqvtbyp5 8*,-0 tngtrsvg. o.+6oment 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 a5ag-yra4e,a5 ugf e( t 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 slipping d v er6,he5g,cw h8v8ykown a lane at 8vckw ,hgr h68eevy5,3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to thkfx3 jw -y7df:r zm (tbsi(xmtw;ke wt2m9/99e Sun as the sum of two t-/9:kw m(w ef29 mtw m37r;sfk9 tjbix zt(ydxerms,
(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 3/ eafgxaq9dag :5* hoof 1640 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solof:gqdh5ea*/g9a x 3a id cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from restfq96 he8(;uj8+qe8 xfas glb h and rolls without sqf8eu8l8q hex s + (ja96gfbh;lipping 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 i 1xglsd *)mj2v)0 awxvertically on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of t*xmxa )w2)1dsij l0g vhe pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a joo3:pq1 y9qm0.210-kg ball rotating on the end of a thin string in a circle ofqy:qom 1po3j9 radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentyy4 6(c*my*u5co phz6uw5 db qum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rouw pm6c 4h y5uydb yqz65**c(otating 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 9 )ctaze7qsz a 2l, f+ja9b0ogside, on a platform that is rotating at a rate of 1.3rev/s If he raises7o a, z)2q9zeagt s c9abjf+0l 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 hpnhef 1ejq;7z )45-,isn vgocer moment of inertia by a factorgp)q4v5,es;cz 7 n-f iho1nje of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, 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 4 ro *kpd//r7b2egaikinitial rate of 1.0 rev every 2.0 s to a finagrk72/a 4bkr o iep/*dl 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 freqq4w; py*gae q7m us)4uuency of 1.5rev/s The wheel can b m 4gusqye7w*) pa4uq;e 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 sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skate: a jz1w)(gfnbr 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 moml ddp9xtpv0-w11 js0 is (kz5j a:9gqjentum 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 onto an idew* (sfxic60 cjntical disk rotating at angular s*cif6 wx0 sc(jpeed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns a*8 js.1ywlzwg t 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotating merry-go-round 6suoh2p/0ylgre 3)j +hd.a* vw i3gli platform of radius 3.0 m and igwg*e+ihs6p30vuy2l jl d/)aro. h3moment 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 merry1myc:*8h,z rfrztxcc+ 8d 9r r-go-round is rotating freely with an angular velocity of 0.8 c:y cmh*9t cf+drzx88,r r rz1$rad/s$ 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 collapsv zf,ue *m.kq c7r2gm+ m-m5lves 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 momentum, what would the Sun’s new rotat. ,qgl zu mc+ m7*mmv5efvr-2kion 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 v7*7zb+3li upev54z anon 1 o3ysf5 af j2.zyjexcess 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 *uip j9(2tes t$ 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 rot5 vikx,px gn. wx1drjb-1.pk7ate freely without frikpd11px., .n5 iv7kgbj wxxr-ction. 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 stands at the edge of a 6.5-m diameter fgz za..wf 7u131qjizmerry-go-round turntable that is mounted on frictionless bearings and has a moment of inertiaf71j.iq . azuzgz13fw 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 l 6b96qnf3fvjpn c ;w0kying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?vfb9c6k 36jw np0; qfn

  The Moon orbits the Earth such that the same side always faces the Earth. Deav 2 xa3rek)x ra9:q0eterm0a3 r2:q9xv eax)arkeine 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 Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 1 mk)e : r:gtduv5t:mw0 e.u0qn a/$s^2$ 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 cylinde5gmhyg c70 wx2r of radius 0.20 m acquires a rotational rate of from yg7 2c0xwhgm5 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 cylindri pz,82v dni d-ue29 -d(pdql c7c c86m;vumzbical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.007( cq d m;dp8nv iulzb iz m289vued26-,pcd-c50 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, ho esdc ziu 4w66lrw*r(b*,ls s0w is the angular speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as great, were rotas-gpls y rl(t.pw8ofk;p7t z.hz471o ting at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform spheretyp rwh kpssz8o4tg.z(p.1l 7o7 -;fl at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-polk244wzs3 nkg e n0 kg-dgb*n:mlution 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 travel 350 km without needing a flywheee- gnb3wkgnnm4*g0zdk2:4 s kl “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 1i4eax sm1* /zp+sr kban $H_2O$ 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 oh() mzqrp 2n l6u(u-juj 1cjmelw5i),n a 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 a-xsn;sb9ag uy ew*op2t)x63hnd length L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the lia x3hgse-t ;u6 pybwsxo)2 *n9near 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 3/3yx mate0w, lg-.l db:li wpon the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig.m0w3y gdw, lxlb. p/l:-itea3 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is mak g (n u2vcm (iilow:l2(u,ef9uing a turn with a radius The forces acting on the cyclist and cycle are th(2inm:g (f uuvolu lc,iw9(2ee 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 lev7,3xx 4yrs u*a bhh4fyw(i2cnqy3z+ ngth 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to awxn,2 3r4x hy4hfyc3b*+iv7sqzayu( 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 cylin f-*wp p0 (j.rjwrjo/yder and her arms as thin rods, making reasonable estimates for the dimens0p(y-j.* fjrw/powjrions. 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 clutch assembly which consists of two cylindrical plates,bx lw0(lmd.j ogv1d-4 of m1d-l o0mv4bdw(lgj.x ass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 rougkf9 --:b9 )ewp jmtvdmh track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leavingjw 9ek-pv)dtfmm9-b : the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not asse5.m+lzbz2 cn 3jcr.vf6ctk 4+rh s q2ume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revoluo/1( ,ni xn scx2jsc,g,(ba ixtions as the car reduces its speed uniformly from 90km/h to 60km/h The tire , i ss(axoc1cgxn2n/,b,x(j is 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