A bicycle odometer (which measures di4ls:py5yulf 1 au(ft:stance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheelssfupf 45y: a:1l(ulty?

Suppose a disk rotates at constant angular velocity. Does a point on the rim y 4 -yxqac,r5xb/1t tohave radial and/or tangential accelerationtxc 4ax1tyqor/ y-5b,? 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 venh) .qi5izotz;d 02azt*d ,rllocity $\omega$ Explain.

Can a small force eve1,dol2m iaww1(,6pykhfk m 2xr exert 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 ej()r6wt j4r vdwpn0**rt+ nodo a sit-up with your hands behind your head than when your arms are stretched out in front of you? A diagram may help you to 4 6rvjto prnw()tw d*e+nr*0janswer this.

A 21-speed bicycle has seven sprockets ato;7p )gpq/(gj bkc.ox the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small7./pcgk (bqg)j o; opx 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 ao5 2uhs0:0 us3wb8o tgff* brvqx2 h4eble to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, ex5e03bw :fu24o tu 20or*8sqvb sf ghhxplain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrowjvj o*scf6 wxl c)o:fi:.teux, +k;2h beam?

If the net force on a system is zero, is the net torque also zeffl2 n fpj)p *3(hxz 9f1i;5ttibxgc7ro? If the net torque 5iff i ;cl)fnpjt*9(13fbxth x72 gzpon a system is zero, is the net force zero?

Two inclines have the same height duce7xm7qoe.ol n(f6 *d 79bsp dk6t (but make different angles with the horizontal. The same steel ball is rolled down each incline. On which inclind67c xbodfnu76e7 ke (pqs* . 9mt(lode will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneousl*(a c 332ctyr dgha/e,uh; nbdy start rolling (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 first? Which has the greater speed thee hbh23ca3ra,nd/yg;du(t c*re? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same ma;d ,3bqke4ntjss. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total ki4;ntbq de3kj,netic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are 7qhpr) j0)jqiconserved. Yet most moving or rotating objects eventually slo)h) 0 jpijq7rqw down and stop. Explain.

If there were a great migration of people toward the Earth’s equato*/h5, fv thcfr1iq /gvlle24br, how would this affect the lel/v4 2f1/q gcfrthlv ,*bieh5ngth of the day?

Can the diver of Fig. 8–29 do a somersaurrcxyi hp vx 3pv70 t+q:kj)::lt without having any initial rotation when she leaves )3i t :v:kvqrcry phx0px:j+7the board?

The moment of inertia of a rotating solid disk arzu un199(orde lx:k0+ lx*hebout an axis through its center of * 9or9d:lx ek+1 l(unx z0herumass 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 wz/c8 z /*xvp/oiiif -hjzh8 (rotating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the ( z8*w-hoic hi/vi8zx/zp j f/masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, ljq3:ys x;cylcb+ p2n :p t(5xone is hollow and the other is solid. Dbsj;5pc c:l 2(3pt yx:ynx+l qescribe an experiment to determine which is which.

The angular velocity of a wheel rotating on a ho+va/ ,feoh9pcc+, del rizontal axle points west. In what di c/oh,e +lv+a,9 dfceprection is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turntable. What z sh j98-j3g/a2am.5,3 r kcnqnp gxtbhappens if you walk kzm x3jpnbgcah5 q,3tsn8g.2j9-r/atoward the center?

A shortstop may leap i3r00y vwfysw- rqr92 gnto the air to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly q r2003yfv y-w9gsrrw you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, o;sx 1nskt 9fk/2oq*k qm ib/bc t89b:discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stb*qk9n:bk/ b i8 tq;/to2m 9c1sxf oskable.

  Express the following angles +ldzvy h ))mw cmru41+in radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincp7pm -q3ijd1k0kj5yj idence. Calculate, using the information inside yikp1kj q -30 pj5dm7jthe Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directedn: (t a,fccu)v at the Moon, 380,000 km from Earth. The beam diverges at an angv (n:ccau)ft ,le $\theta$ (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 (h f)fouqg1c/ ,- 0aehxbwoc4fi/gb -rate of 6500 rpm. When the motor is turned off during opeuxbac 0 -1e/gfqwg-o4 ,( hhcbo/)ffiration, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the1+-wdagat.kc5pw 3w0e o)q vt ball makes 15.0 revolu)qg1aw+ktp wt wc3e-vo5d0a. tions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. Howj-jb+lirbj kz,a g6:4 many revolutions do the whek :a,-4l jjrgz6bb i+jels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 jog01g+fi1 dmtjck 7,rpm. Calculate its angular vel0k1gfo ijjdtg7+m,1cocity 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 4.0db0t,jpv s-(19oh zej s (Fig. 8–38). (a) What is the j 1, bevjp0o(- hz9sdtlinear 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 of the Earth (a) in its orbit a,(kocfvc4m,;9u,q7z y rkit uround the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a 7vmg1a8 q.ust 32gsyypoint
(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 cfgw-wjltzihrl*.zt..k ap7 qt6l y;1 ,2yp+a particle 7.0 cm from the axis of rotation is to experien6 fi7tgpyzch r, ak2 qw.l+;-j1l . y.t*pwzltce an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheqh(cq oov-+.iel accelerates uniformly about its center from 130 rpm to 280 rpm in 4.0 s. D qc.oivho+( -qetermine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radius:yjpmg0 fi2:4fyav0b* -pm so $R_1$ 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 Apollon5brci*nl;r fm(x 1;s spacecraft put themselves into a slow rotation to distribute th(il5bxr ;*nfc1 s; nmre 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. 4trddp zn51oytwy,8, ffdv,2 Through how many revo tw,rof 1n dvzpdf84,5,d 2yytlutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5,l 79jr*ccg -btc8fqt 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 stressesx9vizwt7x0e*v4u ( d,hghgpt, 2 l,qc of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 cotcw9zhv ui,g2peth x*gq 07,v ,x4 ld(mplete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpmfavzt z2gx66g dr-;4 s to 360 rpm in 6.5 s.gx z z4td;s6f6grav-2 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 850rev/min It turns 7/w w2tp rqx *rn*e-ur1500 revolutions before it /t2xnw*7qrrw*e-rup 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 it8,drthj; u :ats speed uniformly from 95km/h to 45km/h The tires have a diamete tr;j,dat8:uhr 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 ne13yrduq5d+6 67cwij-g lea9dz vq 9uniformly from 95km/h to 45km/h The tires have a diamete q-53eucqi769 a + lge9zd6 j1drydnwvr 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 eao8g:ddx ap)f/ ch pedal when climbing a hill. The pedals rotate in a cirp:/fo a) 8xgddcle 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 wi7.-cw k q4;b(nb j4ex qszegs4de. What is the magnitude o4e(.gb ckq 7q-zjb;4xwesn4s f 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 a3-7acav)c+2fy3 bi,e axozo f xle of the wheel shown in Fig. 8–39. Assume that7c3yfab-3fz+a io, v2ecx)oa a friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a massyid iw7n6xt1mr uw2hzz t,n 9(/ xwd*r3q(f6dless rod which pivots as shown in Fig. 8–40.htw3 wrxr nd(7t69y2 1/iddu,xq 6inzw z*(fm Initially 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 requir+z bi, 1z;ubnye tightening to a torque of 38 n+;zb 1ubzi y,$m \cdot N$ 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 radw nx ,fh1wiqn3 -iehe* +bu;k9ius 0.648 m when the axis of rot ,ih3n;-qbnkx9e +1 euwwhif*ation is through its center.    $kg \cdot m^2$

  Calculate the moment* n2i1gxt-snb of inertia of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass gi s1 bxt2*n-nof 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod i42sd wwkr2+:hcx f ,vis rotated in a horizontal circle of radv+2 f:s hc,d2rw4 wxikius 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 cons+j e;4n zmq2)u5lg bwztant angular speed (Fig. 8–42). The frbln;zu gmeq4)jzw2+5 iction 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 sj*ca.zw7 4n3usbn0j .gf3gklf o 55aof inertia of the array of point objects shown in Fig. 8–43 0k gsl7oujjw.ab .53n faz5g4f *3csnabout
(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 werp (gtag *f1b b99nc;hose 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 satellmbtq5p ;z+q c4ite 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, what is the required steady force5b;q4q p +zmct 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 cylindj h60kax )u(pm nysj9qlq0 0p.er with a radius of 8.50 cm and a mass of 0.58jxqp)s0a9lk( h.6 un0pqjm y00 kg. Calculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerating it from rest gd9-/8 ghyuylto 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 ableqw6*p+dhfe e1 wcz)*v to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25ef zwhc*)w6* dqve p1+ 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 ifiqm u;1 aag4:s eventually brought uniformly to resa1ug i ;4q:fmat 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 3.6-kg v(vf7pixa81j+ xc:+cxg6woc 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 kp 3i+plos6e1 by the action of the forearm, which rotates about the elbow joint under the acpko31 s+lip e6tion of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as shown im(db.yy48ydps3lw :xn Fig. 8–464:(dwysy lxbd8p3 .m y.
(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 og)l -f, f/ejl* la.i-x+ c.:n qbkbfoif 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 fr)3 39bxu htlw(ino(t ull turns (revolutions) and re33(lrxi)t t(nw h9uobleases 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 momentaz;5 4p f/ipmh 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 developi4oba7o mc70 ws 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 and radius 9.0 cc2 9nc2wz wxm7vmt(h. m rolls without slipping down a lane 27tcn.2mww9ch(m z xv at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the sum of 8hpjok.0ocjr o6 m6( ftwo terms, 6ck8p hj.oo6mor (fj0
(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 o/adk 7+ ru2*ow6up.d0qwe rqv f 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?rva/ p7.owq62uue+k * q ddwr0 Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm 4kf(x2q3iifw and mass 1.80 kg starts from rest and rolls without slipping dxi324kfi fwq( own a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is balanced vertically on its tip. It starts tic/)troool48dab)5(9n w hptp9vt 5go fall and its 5 5l4 dt /pv8pw()o9bitg honr)oca9t lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotati r9jbgevk k3w-4 b16 l8bijri(ng on the end of a thin string in a circle of radius 1.10 m at an angular spejib biegl6 r(j9 -k84k 31wbrved of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grindi tywu2t4p+* bdng wheel of radius 18 cm when rotating at 1500 rpm? yt2 +w4bd t*pu    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is 5r h cp0ut5ckwbmm8 04rotating at a rate of 1.3rev/s If he raises his arms to a hcck m h00r4u5p 5mwtb8orizontal 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 reduc*qh;f/4hui zyp:h. kwe her moment of inertia by a facthfq/yp k*z u4h. i:wh;or 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 ssm ; 7v baukoiw20vzn4p6.yr* pin rotation rate from an initial rate of 1.0 rr2myi4vaz .so ;knvp*7wbu60ev every 2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis through its center t cv6o k.vm17 *dc4brnat a frequency 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, apprcnm4k7cvrb .1 *6vtodoximately 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 angue .g+nv ,ol86vr-p* ) aoxatmllar momentum 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 m7.prgh 5p, ztjtul2 il5w0g7 ce)bd3bomentum 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 droppedca+/)zlne ,yitua/f. onto an identical dz yl +eu.tan,)a/cfi/isk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns2mxm(ck-hzs 5ap*v ,p at 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-rdu3uh4s u:h-dgsk*9t x eix) i 1fh92jound platform of radius 3.0 m and momen sgu4u2jf1d-de: h 9h s9xt3i kxu*i)ht of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freel.u:nyeoho.x o : r-iwhe pewn :,d )b95z7c9jby with an angular velocity of 0.8ch:eeir.bep yow: -79. nwh: 9z)b5uj o,dnx o $rad/s$ 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 dwhn9 d2pt:6alt p*4yuwarf, losing about half its mass in the process, and winding up wita6 *udt29tpyl:w p nh4h 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?(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 of bphz 8yi pol,/j nt;33120 $km/h$ 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 w. cy8hoo kl07;(hl;gd+vq8vvq, g3l qtyw-p $ 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, f3/6nqqr n5 zuc9iw)o that can rotate freely without friction. The momen6 q)rcwni39u/fnqoz5 t 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 i/fi( s) kv/shr8+*n -ou abvmat the edge of a 6.5-m diameter merry-go-round turntabl uvr 8) k+*hsm/avsbi-nfio(/ e that is mounted 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 thcnz:(dfg3*2jdb m+hh e ground with the end of the rope lying on the top edge of the spooljn:(c mhh23 *bdzfdg +. 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?

  The Moon orbits the Earth such thapmd*u g7cvp5 d 9ky 8vhd.v*i-t the same side always faces the Earth. Determine the ratio ohkmi.py d*g8u-vvd 7 d59vp *cf 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 acceleratesb1s0 +k3l lvns from rest at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of radi ;jf:6wycq4y bu qi e:j8wzo16us 0.20 m acquires a rotational rate of from rest over a 6.0-s interq:jyw8c41qwz:fio j6e6y b;uval 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 .8 bb12y0i+gd jawni1u0hv dpi o:n r-mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long 1u:0.j210 n gbib hov8r-nwi dpdi+aystring, 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 whe*zplph) 6 2rytj)ww,vw5wb a3el ($\omega_R$) 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 Sun5y iqspb4r6v8dv,lh w .r,sj./bu(nb , but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the processb d4pbi/wy rnsj.(l 5 .rs6,bv vh8uq,, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for izililftck+py, :6o(e+ 3,u3txevmv 3t to use energy stored in a heavy rotating flywheel. Suppose suleu(vxm p+63tt c, o :ykf, 33+ilevzich a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates2 vzfo iy;cy5 rh*5xfx p-(n6s 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 rold 1c,jkcyv ml749 4ssgling on 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 and length L can py p:um9vmwlcs,c5v9)c)h79m3x e y d civot freely (i.e., we ignore friction) about a hinge attached to a wall, as vc y3e5 h:cd)m9,9lxmmpu9 ) scvyw7c 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 Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertg1h0g +2um;g zyktb7oically on 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 ug12gyht0zkg; bm7 + o(Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is )va3v85r 8nr,ii kn9in c+fxtmaking a turn with a radius The forces acting on the cyclist and cycle are+ 598rt iaf3 vxkivc)8r i,nnn 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 slinh *ypxff3. oat:1:pu fg of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and w p* :pfau:.fxt3 1ohyfhere does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thin rods, m j0ae+z7w99c r-6z:ynthp bw6 be mfc+aking reasonable estimates for the dimensions. Then calculate the rt 6+z909eabz m6 hc:c jp7e r-ywwfbn+atio 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 co n)gk+mm ,zd lv4.fw(ansists of two cylindrical plates, of. kf)ngz,dam+(mlv w4 mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the loo 7p:6i wh q,lp5pube3yped 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 leaving the track? e675qp,hu b yp:wpi3l Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not ass n(.i.xd6 kypoume $r\ll R$ h =    (R-r)

  The tires of a car make 85 rev (1s-jqbhc: ynolutions as the car reduces its speed uniformly from 90km/h to 60km/h The tires (n -by1qchs:j 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