A bicycle odometer (which measures distance traveled) is attached near;m c8sml5m i4q the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24 ;5mmli8m4q cs-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on hn3b9xbhhsfm5 4ocsb(+k d ( 4the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniforml m4ohbhc sfh3nb(sx + 4d59(bky, 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 desl k.rk pc r,-i3f z9s(yx-,xzj0+eznzf/ yg* acribed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert akwi:n+k nm )xvb058i) fi07ke8hncwo greater torque than a larger force? Explain.

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

Why is it more difficult to do a sit-up with your hands behind 76fzguu3ju; 3 axf,9hape2n pyour head than when;agu 2a6f3eh upfn93p x z,7uj your arms are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel ansges/np3e x5 :ha3e4ld three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sproc p5 eexalsen :/4g3s3hket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender low hfaaun,m .5wmt.+ lu;er legs with flesh and muscle concentrated high, close to ;nal5m+u.a t.,mhufwthe body (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 a long, narrow bea,h3r0;2il ha(1u hns 2b2p4z 5hatrxw doit u8m?

If the net force on a system is zero, is the net torque also zero? If tla*bp 1,;laz the net torquea1a *pbz,ll t; on a system is zero, is the net force zero?

Two inclines have the same height buv xyoch:e: 9; m.kbac:t make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bot;:c a.vcxyb :9 eokhm:tom be greater? Explain.

Two solid spheres simultaneously sta+ izvs;zarc ,ee;jv75rt rolling (from rest) down an incline. One spher zcearv;ziej +;7v5s, e has twice the radius and twice the mass of the 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 hafxi;h es2:qarpq;j . -ve the same radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KEir ;q :j;.qp- haexfs2?

We claim that momentum and angular momentum are conserved. Yzqj-g2zre7 , qet most moving or rotating objects eventually slo 72qj,gz req-zw down and stop. Explain.

If there were a great migration of people toward ktqwgyb2e ii2 b*r13: the Earth’s equator, how would this affect the lengtb ieqi2k:w*3 bg1 2ytrh of the day?

Can the diver of Fig. 8–29 do a somersault without having any initial rotation w(piqf25yz7cje q(5 on qz255c7ioq(p efny( jhen she leaves the board?

The moment of inertia of a rotating solid disk about an axis 5 sd/*n.k iudothrough its center of masu n*.o/ d5dkiss 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 mass in each outsgnsb.d8k2mp5do/y j 7 tretched hand. If you suddenly drop the kjd78o.n y5db2/smpgmasses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the sai8+pb1, adtf xme mass. However, one is hollow and the other is solid. Describe an experiment to determine which is ,b ax8dt 1if+pwhich.

The angular velocity of a wheel rotating on a hopr/6tdp3 u w36 rf;gblrizontal 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 point at the top of the wheel. Is the angular speed increasing o3 6dt pbrpwrg/;3 u6lfr decreasing?

Suppose you are standing on the edge of a zw igs5/u7dkjw8i/ 5vlarge freely rotating turntable. What happens if you walk toward the cen jgdwsvw55 8izk/i/u7ter?

A shortstop may leap into the air to catchd4h2ngpyw7j2ka . (rjybb1,t a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in t7jj(bdb,2yn4w 21htya kgpr. he opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular mome /q+bsqz: i w25wqx1mqntum, 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 helic q/zm5+xq21wiwqs:bqopter stable.

  Express the following angl 35 . x(;8.q x3rmsvecxpt* dkbw.myfoes 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 coincidence. Calculate, u,xn5,0qk bmpt sing the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seeq 0pnk 5b,xt,mn on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Evkh lq-tvxd: 9ly39yy2ony;2arth. The beam diverges at ay; v23tln9ly92-y yoh: xkd qvn angle $\theta$ (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 6500 rpm. When fy x+b uc9 a7ew/-a)rathe motor is turned off during operatirf/ a xew79cya)u-ba +on, 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 oyh+yh6,gk /j+1b rqhfh.;0 ptjy5cbon a level floor 3.5 m to another child. If the ball makes 15.0 o;p0,bbj+ +ykjrhy c5t.f6 /1qgh h yhrevolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8. +ib z:o+zyf8zz5 )mhq+l ps)a0 km. How many revolutions do the whee 5boslzay ):h+ q+)zizfpzm+8 ls make?    $rev$

  (a) A grinding wheel 0qwin7j k x. pqf;3g+.y.35 m in diameter rotates at 2500 rpm. Calculate its angular vekw3.+f iyq .;nxqjp7g locity 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 (:vp5)j 5igsd 7p- tlcl6wn.rrFig. 8–38). (a) What is the linear speed opglrr 5jlw5):ip6-sv .t7 n cdf 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 (3 d 6hzuzaps8go6fat b) 8ma85:cq s(ma) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of c omo(h51j9ty:u)6fp 1mrcpr n 9lxj 7a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0 cm from zx:.xoq7 ry;1en h m/tthe axis of rotation is to experience an acceleration of 100,; x:y7ohmeq/1r tznx.000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from ezo .lnxn6 xb9t+ *7yd4*psxn 130 rpm to 280 rpm in 4.0 ltb9 xxx7npn+nds* y e46oz *.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 radiucdqy8 o03gb0o 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 gsimmb//; i/x spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0m/mix /b/i; sg 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 acceleranhwwtm9;d.wncx.9inur ..b 8tes uniformly from rest to 15,000 rpm in 220 s. Through how many revolutions did it turn in this tiww9n8dw mrx;n t.nu.cb9i.. hme?    $rev$

  An automobile engine : *1mrqzdsit) hkii7+ slows down from 4500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses 7;hfal 7.d :ncc(;k+xl mtc zlof flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 comp. atmlzdkcxc;:(n lc ;l+7f7 hlete 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 uniform*q+kz-q as ,7p doknu9gd16 ksly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel haven ,pu+ 7zqk6gq1o-as9 ksd*k d traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 1507x dhombhg jqhwt./y+6+. u -h0 revolutions before it comes to a stopqox 6t/h.y7-d++u bjwh . hmgh.
(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 uniformi b8p ws7a2-8uiy72vmzql uygu. d 0q*ly from 95km/h to 45km/h The tirbpy7 yd w-q0 *ui qi2 vl8msa8zu.gu72es have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces 9aj 1xb zy( fvek8.lz,iel*7vits speed uniformly from 95km/h to 45km/h The te ejxa8,*7 (llf iz vzkvb1.9yires 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 puts all her weight on each pedal when 8up:t,2 ru9c xkr:ec* 5ulblx climbing a hill. The pedal eru8,c rpb2kxcll 5u9*:uxt:s rotate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a for zl9u4b;h859kzgu )gl noyv.cce of 55 N on the end of a door 74 cm wide. What is the magnitude of the torque if the forcvo84 hgn)9c9 g bull5;zy.ukz e 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 wheel shown in Fig. 8–39. temkz(,h 7ldby9n el h d(7op i5(y8yt41dn j-Assume that a friction torque of 0ndyj79- 1zhltk (4b dmp5nt7lo,y(e yi8(hde.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are at e9 ;0jkgauu8 5mk9xdv4s-t s)vu:lkd tached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially ths ku 9-8 dsam;kj 094ut5l)e: uvvgkxde 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 to a tr7j3exjrr,h+ orque of 38rrh3xrj j7,e + $m \cdot N$ 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 inerti;l .+br 3+qv3 qmxbiw(dzt1ija of a 10.8-kg sphere of radius 0.648 m when the axis of rotation iq+ ;wb.vjd3+ zi3 q1lr b(xmits through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wiax7pt7n 2g3qkcz4:v w, zh/dheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass ofagw/qt v d7i42zn3cp:x,7zh k the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in a hori hh50dl:kw;cxf.v4 d,av,au l zontal circle of radius 1.2 m0 k,dvh a,hw;.avfu:5d4xll c. 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 rotq1yz-/x 1m tu.2by9onf+xc guating at constant angular speed (Fig. 8–42). The frictioyx+c.1u qmb/x z 92nty fu-o1gn 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 o0gsg 1d(gs /iof point objects shown in Fig. 8–43 ab s(igg0dg /o1sout
(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 a ycv 4qp94qpx1toms 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 sb;ui5nenv o70vi 6 z2aatellite spinning at the correct rate, engineers fire four tangential rockets a ve756no;2nuiaz iv b0s 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 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 cm ang2;y-vcq(+poxfgn e7 d a mass of 0.580xg 2 (ny7e fgp+;oqv-c 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 l ooowzsb*ywkpn,c0 l .-)2lz+x f21irest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merry-go-round and is abl-bfx)4vm)c kle 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 )flxm-) v4c kbhas 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 ej pmx il1u9,010,300 rpm is shut off and is eventually brought uniformly to rest by a frictional torque ui pel0j,19xm 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 ball aqs:s8- m w,8neqku v9rt 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 forex6 xhdh d 0w6dlm wgdav2v/512arm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is acceleratedxdd50v26dgvl w 1/x wdmh26ha 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 in Fig. 2pg4d wydp):uwer5 .o 8–46. : 5p.2dwwrpud4 egyo)
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consists of two massesvzpmei 645rpg t;9o91z -m aea, $m_1$ 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 th4bgnu6(s zngo0y 2wj xj -i5k5e hammer from rest within four full turns (revolutions) and relea5n(46wy2x 0bgnjs i5 - kuzgojses 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 mom5 ex,eju(fl 5av 2k-g,eyi-vc3 um8m )xr rew-ent 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 a-,xluq(i/rcr;xa v e 06k (kcy 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 down a epy .pgm3ph 22lane at 3.3yp pm.hgp23e2 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to 3nwr+9njyhpy(uj b6a7mcm:-the Sun as the sum of tmycjna7:y(prhj3 u6+ mnw 9b- 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 of 7.50 m. How muchw +nbv;7i bsl fmbxg3z* ;+dt( net work is required tozx f3i++ts;bwd (b; lv7*bnmg accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls withoutu og(s;9p)pc d slippingpdo g pc9)u;s( down a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is balanced vertically on its tip. It starts xarr41fo(*3 nyhli2 t z1fh(l to fall and its lower end does not slip. What will be th4l(y12f io l rh31f*rzxa tnh(e speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on thz es p08ie(wj2qe36vhe end of a thin string i eqj062eiez wpshv(83n a circle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angu+ x,jnc1*vt b vho75:4pl4w noyuoz2nlar momentum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 conxw * yc74lb,5+nnzu4otjvoh :12vp m when rotating 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 at7w mwzy ww:p(pwg...d mdb 6u7 his side, on a platform that is rotating at a rate of 1.3rev/s If he raises his ar:zwdg.(p mwd7.bww.6 puy7mwms to a horizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce her moment of inert2sghq-v6i,odh, ew ljg6o x(/ ia by a factor of about 3.5 when changing from the straight position to the tuck pog go6ov 2h- (d6qlhei ,wsj,x/sition. 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 ipx +pzyah017rg80 nponcrease her spin rotation rate from an initial rate of 1.0 rev every 2.0 s to a final rate +0zynx0o8hgp 71 prap 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 itu1a:i ec*w8u9-kekb zs 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. 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 fr eu u:c-i 1kbek8*az9wequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinniwja o-n wg(8k;ng at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of the Earth / vaea(f6w4 anr4 lw(a
(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 inertiwixm 0;o u,ptp7qv0 m-ke( w(ra I is dropped onto an identical disk rout ,ioe-vkwpxmm00 r ;p((7qwtating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.q6yt*7 ns3ct7 idqmw n-;(g ia4 $rev/s$ 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 thpab 3u ug+dz*;bq)l2o e center of a rotating merry-go-round platform of radius 3.0 m and moment of ineb;o+z3u)g *adbpuq l2 rtia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angular velocity o01 wxy6 1vwjjc/ws4f uf 0.8 4yfj 10s6wwcx1v /wju $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, losixjy j4m9sz7y6 ng 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?(round sj6 jym74 x9zyto 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 wi2f.e 9-1 55gqf mrhy e(sd1uvfo9lrbcnds in excess of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass x0z)- itu pwtkdv . z5m0o,i*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 p+ sr8g4-zjt kalatform, initially at rest, that can rotate freely without friction. The moment 84arg-k+sj z tof 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-round t.mfxab8g:oa,s m g,9a9s fyr4urntable that is 9gaf8.9bmsxf,r gm 4 oaas,:y 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 the ground with the end of thdvl.lrq 0*4.bd ak/sxe 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 be.q4rl.0 b* dxsldva/k hind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the lana( xmty+ldy;htx/48c+ +usame side 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 laatl/nx yx+l4m;ctd uy8 +a (+htter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates frl tfy )8tu a8ebzk.7s/ nct/8vom 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 th/lr4f 6 payt jlt8z:s0e shape of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular la/js :68l yt4f0 pztracceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical diskshu8/lv+yzwiy 39u 6 k7acckw5, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculaalw5uuvzc6/yck39+w yh8 k i7 te 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 rea5b rl:swxg m3/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 Sun4d6y82 yc .wnr::a v hts9qctgb 6rcs;, 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 t ccr2y8nh:4g6s tt wr9v:saq6 ;dc.b yhree-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile isjcvgh f)a8a 8,+riw *p 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 withpwir,*ajv g +)h8fa8 cout 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 illustratesdh cb xz19otbi9x8, 0t an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling oni f cwjus.f)j3 bsa:un90b*r4 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 p:zsok;s e )kf: g;cxg6ivot 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 acceleratio:6osckk)g;zes ;x :fgn 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 vertically on the f-.+e qiykg9q9u.c ttf3 c ki*qloor, and we want to exert a horizontal forqyi9k .tkc*u q9 .-ce 3i+fgtqce F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling wihdwf:w -wjvjny 3dlq g(k6u+c,z7. 9q th speed v=4.2m/s on a flat road is making a turn with a radius The forces acting oq6 nzwwd9vgucfhwj l - (dy.7q:j,k+ 3n 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 s p38h f7p1xgcon nj+s,1kh(x 0.50-kg rock into a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head8xp3s,hcgfk nn( op1x1+ j 7sh, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a sol:4jo mzn+.ux0oa o; hwc,cuc (0s,wwnid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio .j 0:x ,mcn;uwzouoh0cwwac(n,s+o 4of 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 ln8xz)m4lgb 9, of mass nb84g9zx)l ml $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 aluhvzf ;4pji 1mm93 6km-.iqzbong 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 highes ikuq.ph;fmm 31-b vz9 zjmi46t point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assumejh7o tg5x:rx3 $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces im3 g2)uxo/ xahts speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What wasm32/uo) gx axh 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