A bicycle odometer (which measures distance traveled) is attach0bxz+ola 6ne )7 ahet2ed near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle wxtnb+l )e0 he6oa 2za7ith 24-inch wheels?

Suppose a disk rotates at (9c d dq4,1af92astfa tp+hdq bpub54constant angular velocity. Does a point on the rim have radial and/or tangential acceleratidtfc aqq b9, 94b2dfs tdha1ua 4(p+5pon? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the angular velocity cubz.9culi ih.-:db ,fz o9a ($\omega$ Explain.

Can a small force ever gz)jb, ah l( d 263wwooc dh/j2mrs48rexert 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-2g, cshz onw6py 7a3+aup with your hands behind your head than when your arms are stretched out in fron7spwozg+ ,2hnc ay36a t of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprv d/yl5mkl6+q x*mb 8nockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal,mdqyn8lx5* lb6v m/k+ 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 flesh n: ddc ,;n.f52glghsr,iu /kk and muscle concentrated high, close to the body (Fig. 8–34). On the basis of r,udi5,nhk cfnks2rd gl;g/:. otational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) caubu y8q v(:ps1 mj5/ k ni1etjv4yz1)7j p6mwzrry a long, narrow beam?

If the net force on a system is zero, is the net torqgz zsk :m(-iwh,+(yw4) zgxckue also zero? If the net torque on a system 4, )zhzw(xiw ysk+:-kgz m( cgis zero, is the net force zero?

Two inclines have the same height but make different angle /2 k skzlod.a00xo6dzs with the horizontal. The same steel balzxo 20.d 0/sao kdzlk6l is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start roljg1n1xy ( c mdom3x)bu/hm51dling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline jg 11m u/dcdx3)m51onhxm(yb 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(t r4)64kvt4e r9ho 6wsqkcco mass. They start from rest at the top of an incline. Whicq496)ocs4 khreotvr( 6 wc 4tkh reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular mjt mg p7g,*451k gzk7 qhq/adzomentum are conserved. Yet most moving or rotating objects eventually slo/adqz mkq k1h 7t,75z*gj pgg4w down and stop. Explain.

If there were a great migration of people toward the Earth’s equator, how wou c);mkbw-68hk vu vqr.ld this affect the length of the qbk)m6.8h-kucv;v r w day?

Can the diver of Fig. 8–29 do a so :t 9jky yy17()w+ppzc(tk;c0y iqlosmersault without having any initial rotation when she leavk st lo+c:yytc pk( )j0ywizqy p(7;19es the board?

The moment of inertia of a rotating solid dis d 8yjw zi6+(o:uoov7rk about an axis through its center of mo7z w(6oj ui vr+:y8doass is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg massq9 6(d/owao7oo-jdcl in each outstretched hand. If you suddenly drop the masses, will - do7oo9dw acq (l/6ojyour angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and han b kv-brvj 7i4-0i*lkve the same mass. However, one is hollow and the other is solid. Describe anlrijkb4n0* v vk--bi7 experiment to determine which is which.

The angular velocity of a5gxhie +5 e2n6odc(n z wheel rotating on a horizontal axle points west. In what direction 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 poiz6c5 ox5 ne+2nh eg(idnt at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large frer nv2:5a wzx:sber -65+hsdybely rotating turntable. What happens if you walk toward the cent:ay 5+x6hwzbesb vsn:rr2 d-5er?

A shortstop may leap into the air to catch a ball and throw it quickly.n;lz mnjfc0 ,n16raess )i)m( 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 in the 0sn1)le(,c n;z6sajmnir fm)opposite direction (Fig. 8–36). Explain.

On the basis of the law ofdr gq g.h2xbw(5ev+5u conservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can opge.rdwq+hv2 gub5x5( erate to keep the helicopter stable.

  Express the following angles in radians:n fwzua4gc q lpyh ly 50p/7a58s7jgol2*( i2j (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. C q4tnllq6at9,alculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and ,ltnl 9qt4aq6the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000p 5crjbg5q)8gh5t d u5 km from Earth. The beam diverges at an anglub55 t 5j)cp 5gqh8gdre $\theta$ (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 6p8ea e6hpi;w+u.u(f d qrk7z+500 rpm. When the motor is turned offe a6wzph( pf.r8uie+u ; qk+7d during 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 /qg 0 5a* +p*bqawbrsu3.5 m to another child. If the ball makes 15.0 revolution usqb5b*+ar p0*ga qw/s, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revo u *a3sxin0u:(cvyaq .49 ybsolutions do the wheels makeuia*bno 0u. xyq(9cassv :4 3y?    $rev$

  (a) A grinding wheel 0.35 m in diameter row y9-azvhe dbpz2o;b(3l q 7u4tates at 2500 rpm. Calculate its angular velocitybh2la 4bv39ozwy qpe -(7;dzu in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.0 s (vt0j08:pkec misr+y t-. 2ris Fig. 8–38). (a) What is the linear speed of a chil- 0s y.triritcskv: +8 je2p0md seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earthvw1v4n kip/ 95 p9s*q1pmeu uv (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear spe /* kvpdm5val-nx;;heed 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 2dxd;; 1qz 9ulb-ug uq7.0 cm from the axis of rotatiou;d z l2u-g b9qqxu;1dn is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center fryp0hiwi 4r nn- ,92ukk gzufb0mw.l;6 om 130 rpm to 280 rpm n 4wb0z l,9 rgkhu-k 0pw2f.6ymun;ii 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 radiuqkvpb,4 (p gw+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 spacecrafwt34r2avciyh;6+ 2go uonok /t put themselves into a slow rotation to disurah3kt2iw4 ny;vo/ +6ocog 2tribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. 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 220 s. Throu1tlu)a,:yy v g0u 2vvsgh how many revo1 2 s:v)l ,yvugy0uavtlutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in1zy1b1, wny qo7t1cqh6yeh/a 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 fa8 :nre9n 0+vhen3sfh,zlj t/ or the stresses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reac,zt /ra+8hfh9: j3les n0 nnvehing 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 3hgrf d 09 +leuq)k4jvuniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have 3f4lu+qje) 0dh gk r9vtraveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It t .ru3pn0d.nu(qeoe a ,urns 1500 revolutions bpenr do u(u3q ,0n.e.aefore 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 as the car reduces its speed uniformly xug9 -qt5x2hb2koxv; from 95km/h to 45km/h The tires have a diameter xg2ov2x;5-k x uhb9tq 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 ijmo5bj2. d;fets speed uniformly from 95km/h to 45km/h The tires hav5defjb o 2j;.me a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight ot)f t24y w.nexn each pedal when climbing a hill. Thew x.e4nt)t2fy pedals 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 forceor-d j,:zvbpd(vs o0 6 of 55 N on the end of a door 74 cm wide. What is the magnizo-6d( 0jvo: br,spv dtude 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 thedw i784ubs m *rtha9vu4p :i0ny g0l+w wheel shown in Fig. 8–39. Assume that a friction to+4gvmuilh7tp a8*w i4b9:nw syd 0 ru0rque 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 enm 9ns )y2.rybcrob:(2+jajvnv ;3tqbds 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 maj3b+ b2.:b9rn( arnvc s) vm2jyoqt;y gnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tighteni750*ofzo )gvby;rwr( 9ky xvjng to a torque of 38 yr9)5g r*0oo7z fjyv;vkw x(b$m \cdot N$ 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 0.64c .w qu3fue4dspt,5 xi k h4uq*t.xg0;8 m when the axis t fqhdu*i .xu4,5x .w gcup 3esk0;4tqof rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia leuz-6yas)u 30r:lui of a bicycle wheel 66.7 cm in diameter. The rim and tirea:zsu e 0yru3illu-6 ) have a combined mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin,cc ,x;z 5xrt mm77et;i light rod is rotated in a horizontal ci cr 5;,t7 mtcm7exz;xircle 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’s whe:n6 l/ krkqb .+e2usieel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the cr+e l2bk/ .ieknuqs:6lay 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 ofb2(dgbhh735ijx+ n r2fpa 1e ( qa/cyp inertia of the array of point objects shown in Fig. 8–43 abouh2p aeqp5na 2i/bbhd+gy1 3(j7c(x fr t
(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 vxcdf9)f .bc),p dl ;tp/ dox.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 c;;1)ern0bvey: wycl k ylindrical satellite spinning at 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, w ;0ye)e1kb ylnr:cwv;hat 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 radiys:9,6ui aw3,dml trx us of 8.50 cm and a mass of 0.580 l,au :,6r my3d ist9xwkg. 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 1,q ptj;4hc dhbb8qlh wb462f s. cc(rbat, 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 , 1rosj/vb5z)t /ukk,-g 0dvqkhjkl6 on a small hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (ebj61h q05gk zvktokd-sj ,,rv)uk//lach 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 i6yj59di 0lj h37 biymts eventually brought uniformjh5yyd6 j0i m97 b3itlly to rest by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a - o hwcslja68d:zk)*i 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 baldbfwvv(/9g0n , mwz bbb66a3sl is thrown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps m b3nw6z9dbs,w bm( avf6/gbv0uscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can bebu v22e 1bwiq1/g9ia g considered a long thin rod, as shown in Fig. 8–462b2gwi1bg1ae /9 qivu.
(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*sj bg.8nngq2 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:4ii 3ed: 1rx )vlaawe within four full turns (revolutions) and releases it at a :e ivla4dxei1awr:)3speed 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 zs;4 - wme5m 1nj:ieqia moment of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine deb 6,yg896k (tr+nn 4dtop.pd3udysunvelops 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 ax2mriv qhfi,jj 6 x58k*i o0.hnd radius 9.0 cm rolls without slipping down a lane at 3x iiohv m8r*.f,kjq 6j0i2h5x.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the 4b7 jd2zr* y ;x;nxkoiSun as the sum of tbzj4d ;;7x*2 yiknrox wo 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 mass of 1640 kg and a radius h45yds,-c+b t8 iuteyof 7.50 m. How much net work is required to accelerate it frombi-d8 hy,y45tcuet+s 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 from resqm df*h- zhu,4t and rolls without slipph4zq hm*d ,fu-ing 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 balancert k5+kd0g fts jc-:0md vertically on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hik:0tfjg+kc smt5rd0 - nt: Use conservation of energy.]    $m/s$

  What is the angular m io iffu3g*;sstl24p5omentum of a 0.210-kg ball rotating on the end of a thin string int; l ps5f3 g4fosu2ii* a circle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentu 7.i*nw6m amapm of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotatingwpmi.a7nm 6* a 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 yywqe 1 ,6.jmqthat is rotating at a rate of 1.3rev/s If he qyje.y mqw1 ,6raises 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 momenk 2yc 6t/cog0dt of inertia by a factor of about cg0od cky 6t2/3.5 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 hers43yxcpl a,-8rhg: ;+ynj ifx spin rotation rate from an initial rate of 1.0 rev every 2.0 s to a finalgh,ly; yx8fjnxcirap3+-: 4 s 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 verticftg)l-hzg6r uyg;n 7 j;5 al9 (aw4ulsal axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m.ar)sw9ly;( ugn5ujhgf ;lalzg64-7t 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 m( sn,f*3jdw/ h zazfu 2m1.jca 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 moment:ad :xpp h7u k7vhw 2ti6oe(t-um 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 drop(w*fs,hlz 2plec) cbq z;.j8z,q )knnped onto an identical disk rotating a2n)kn w.,z 8pqbqscflz,el*j ;cz()h t angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns atmnryya5lu ; g: 31zqs3 ax-.ww 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-r o wa0.74 9lzawza1eqkound platform of radius 3.0 m and moment of inertia 920 a9zozkwae.w0 1 q47 al$kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angular velocity kftw7vk / n 1)s+io1+bly;g lqof 0.yn+);/qlw7kv kbg +flt iso118 $rad/s$ 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 collaphfzn (ldmy0,8vz 6ws2ses 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 rotation rate be?(rz s2v8 f0n lz(6m,dwyhound 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 exz b o2l0vfah mat69u-4cess 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 c73 p:iu6n1(+f a4olvrbgh mv$ 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, initiallygv:-4t g kfpo- at rest, that can rotate freely without friction. The moment of gv-fpgot k- 4: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 merry-go-ronu9 q. /a( pup07sinrj:ync5ound turntable that is mounted on frictionless bearings and has a moment of inertia of 1700 upqu7 (0ornnc:i59p /sj .any$kg \cdot m^2$ 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 grou84.per y/ ps olz9i*exnd with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of ma x9 .*/y4se lppireo8zss move?

  The Moon orbits the Earth such that the same side ey3sza0e;rkb)7i 5e zn(pzz*always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon e r zz0( ezpanzey;)b3* 75ksias a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 1 mwk wd-:+ haa-1c3h(btl izqx6/$s^2$ 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 theh1lv,9t1fr u o shape of a uniform cylinder of radius 0.20 m acquires a rotational trh,uo 911fvl 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, eze-yqlk6(b+/ ( c-ky9pmkzvk )qa 6d kach 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 conservy pk-qa( 6 z+leqmk-bzyk 6k) /9(cdkvation 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 reac+)0hvt 0meb 3x8svos r 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, we;r(+dbshlah+l1 m l0mre rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the sta+dhlahm l0l+mbs(1; rr is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is fojf3jw5y zcwk6 p539g2k; y 58fjzbbfmr it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylimcbkz9 fp5f 553bw6jy j8zf2 g ;3ywkjndrical 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 ane2ko nark1 52i $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal surface at speedu61vc0fhu oqikhw/fbz e + *83 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)jy ub,sbs(enfn - ws/+hsj+is /o/ and 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 linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See )+,/ ( aj,bb-uif/+ehnss ows jns/ys Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the floor, and byts v*dr r4,fj9p.(n we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (r(9f*v nb ,pdrjy. s4tFig. 8–55). The step has height h, where h

  A bicyclist traveling with speezcgp.p0:5 pe xd v=4.2m/s on a flat road is making a turn with a radius Theg. p5px cez0p: forces acting 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 :avre efbv v 336d/bizs0/*xylength 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm 06bz3vyfsvvx e ba/ 3*/i :edrafter 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 cylpotq.o p t7l*fw;ptz p.8ri /2inder and her arms as thin rods, making reasonable estimates for the dimensions. Then calcul/p*;prlt wtoppi . q 2t.of87zate 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 consistgo h3804ovhl-asc0y- cdzyx) mq24nd s of two cylindrical plates, of m8d-m n0a)cscghh0qy4 ox2 od3z- y4lvass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 ab/ r*i3icox*g long the looped rough 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 leavingr*gi3icox b/* the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assu)je7 pzzqqfk) inr9 -4me $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reducpk)o r )*susl,dj jh6- y-qbg6es its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (y )p*jqgksblhdur6 j-- o 6s,)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