A bicycle odometer (which measures distance traveled) is attac c)mowonv2 7 duk45ae*hed near the wheel hub and is designed for 27-inch wheels. What happens if you w)eko7am2ounvd *c5 4use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a slcvvl.g16 8n point on the rim have radial and/or tangential acceleration? Iv6gs8c .vl1l nf 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 4obe/j:f+atul6yj/zvalue of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger for84c)6qpd,: d 4oklsdxyen n5uce? 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 d+m-y951zkp xi;y)l wmsi e/g:a8e m yoo a sit-up with your hands behind your head than when your armsg exi:5 io+z)w-;1ye / 9kpamlmys8my are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets atcggho; ph f)wii6k .668y jfb n;bqs7. the rear wheel and 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 geqhig6 s6b y b o67;p)8hcfjkw;fg. ni.ar is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on bes)du(lw /9r ining able to run fast have slender lower legs with flesh and muscle concentrated high, close to thwund sl)( i9r/e 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)b c1c,65evl ps carry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? If the netk.yvg5j cz tm.4x y++e torque on a s.ty gyxz+ kv4 mcje+.5ystem is zero, is the net force zero?

Two inclines have the same heig; o1vc ihmefg-.jt-*n)xht5x ht but 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 bottom be );t *chen5h-g-t.xfv omjxi1greater? Explain.

Two solid spheres simultay h7y(eww-t/ uneously start rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other-we7w u/ (ytyh. 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 cylindxwmr i002a 7xfer have 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 7iwxm0f 20xragreater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most moving ippr eyj : :oyv4(cbn;a-/e6x or rotating objects eventually slow down and stop. Explo: e:j rcpyy x4ieb- n/;6vp(aain.

If there were a grea5*ds we6m u/q+u4wg. rd6fqabgn .7uzt migration of people toward the Earth’s equator, how would this afud5qz 46bm sww a.g+ fq/n*e7d. gur6ufect the length of the day?

Can the diver of Fig. 8–29 do a somersault without having any0a(r7 *ss5.nxqfd0t5 nxns c t6ss r0x initial rotation when she leav70ssa5qr. 0 sxtxc srtsfx 6d *nnn(50es the board?

The moment of inertia of a rotating solid disk about an axis through its center0qm3,ds(/ud;it nw( vlxay 8 cj2rwk1 ;t3s aq(/v20i,8xynmur c(1wddklwjof mass 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 holdi. qm0jee37elu gl5r; tng a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease,g;05 e trjlel73q. mue or stay the same? Explain.

Two spheres look identical a;du, q ,esazd,nd have the same mass. However, one is hollow and the other is solid. Describe an experiment z ;,asddequ,, to determine which is which.

The angular velocity of a wheel rotating on a horizontal ax2w*oo 4 b g3lznv0,bn6rj pj/mle 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 or decreasing?olpjo/gr* w,6v0 3nbm24jzbn

Suppose you are standing on the edge of a large frei)f z+i(b x ha*(ci9bc7*zptkely rotating turntable. What happens if you walk toward the cent*k b*zbh (+ixtzpca9)ic7(fier?

A shortstop may leap into the air to catch5m, /2wq 3lv fcgpw9fd a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notfg/l5v9 pw ,f2q wdc3mice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angvll9ae96d p2 gular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discual6p g92v9d elss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (a) 30q n/x/ vt/zrof: .7vvi $^{\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 c*rayq43agf-p juu -s +- drb-poincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth. d-u-rry+qaujs *- -ap3bpg 4f
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed adpro 54 enb/ r;;bl0rnt the Moon, 380,000 km from Earth. The beam diverges at an angln/;o5dre rlnp r4; b0be $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blende(835mt eq6jhc wp mj0or rotate at a rate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the mqw 56jp3 e(mj8o ct0hblades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If,h5((5p jyg;fliadn m the ball makes 15.0 revolutions, what nh(lyi ag(5p,;5j dmf is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. Hore3o 1gilb6- sw many revolutions do the wheels mslebg3-16r io ake?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotat602 ; 3av 59 6m5etlfpy gg)ricsdfaites at 2500 rpm. Calculate its angular)ip5s02 t vgcy6f69 l de3arm5tia;gf velocity 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 revolutioup88tsbz+ ( kun in 4.0 s (Fig. 8–38). (a) What is the linear speed of a childpt8k (ubz u+s8 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 ku9p 2jltkh hoyvp0 ,9g l5tq5e,y24uof the Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a plmk0ek*i . suxm5t .l4oint
(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 rot6k8btr0vvk 6p(kyqq6 29euotate if a particle 7.0 cm from the axis of rotation is t 0kvoq(k26 6 9ery8uvkqpt6b to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 135k y+x yrw*i-n0 rpm to 280 rpm in 4.0 syiw5ky*+x r-n. 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 radius 8 czz9s9tdm6oigv ; l+,ow1dn$R_1$ 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 spacecm 48b2;jw; r k ,qalq5ykescg8raft 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 1r m2q,l;sj58y4bkgec8 awkq; .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 2 p:7pj1ezu l:pbnvhv p3x9+b,20 s. Through how many revolutions did h3b n:vpb7vj l,zx 1pue9: p+pit turn in this time?    $rev$

  An automobile engine slows down from 4e( 6+vps;;k0 cr5cqapo 7u3mktrmpx9 500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed jets 9qhofejol 21:8n oq8waum) t. in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its finl qw mfho8o2a9outnej.q )8 1:al 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 rpm to 3,j4 rlfnxz 1)m60 rpm in 6.5 s. How far will a point on the edge of the wheel have t)mnrz4j1,xl f raveled in this time?    m

  A cooling fan is turned off wh sb ,6gq8+zx55 ,yqxrlsvv- xyen it is running at 850rev/min It turns 1500 revolutions before it comes to zqgl-5 8 y,+56x rvvsyqx,xsba 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 rg. sg/ 8qhq:fsbu)6oqeduces its speed uniformly from o6qsqh8f:/ gsb) g .qu95km/h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car makegml9 3*(2u4 qr emptpx 65 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tire4 lm (tgp2 urq3ep9m*xs 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 riding9zzso q nps gt) agubo,1dgkx0:g;)d;u- *ke: a bike puts all her weight on each pedal when climbing a hill. The pedals rota,;*ag 1;e:sdo9uzgq )s: -dbu)t0 pnxgkkgo zte in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of ah6 w3;:pqfp nt :,h p/pjuip5k door 74 cm wide. What is the magnitude ;p hf6:t/q5n pu:hpwjpp k3i, of the torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig. 8–3,x1g3wbx 2qvha qoj1b dwt7;59. Assume that a frib q tav32bq151; wx ,jxghw7doction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of d rg2 +q/2san9kyfjm2 mass m, are attached to the ends of a massless rod which pivots as shown in Fig. 8–40 /29d2nfa my s2gq+rjk. 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 o i/p :lr/.wplrf an engine require tightening to a torque of 38 lr/pirw./ l: p$m \cdot N$ 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.jn/h4u tt opvx0f6) j6648 m when the axfpvoj/j 64) t6n0 uthxis of rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicyclml meq2-pum: 4e wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of -um2 :e qpmlm4the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated ixyugqn)c . valhz)fx)g4z3e6u (.ri* n a horizontal circle of radiuz uv4.rc)z. hg6xqf)na u* ge3 i(lxy)s 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 wmap535tkva emv +n3*w+h5y6b(nap jkheel rotating at constant angular speed (Fig. 8–42). The friction forc35tn+ y n 6pm +bka a5*mp5a(hwvkev3je between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objects shown in Figmw l,7wqf 4t36s,etp* xqryzbv8 qm7 w+ ,-nnl. 8–43 about lpbtq6*z+87w-3w,,mn q7lxt 4n fysrw, vq e m
(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 whosex,wh+yr* p dz1pjx77g total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite sping;jlz fq ; d/ewmkvg6)1 jf.i/ning 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 ste);z .6e ml/vijg/g1;kjwffd qady 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.y m(yf3hm a:yw4tp8ztt 7w94x50 cm and a mass of 0.580 pwh4y :f3y( 4az 79mxw8ymtttkg. 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, accelera1njha / 8ul:tating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merry-go-round and5b8fh qdb tu0(;hz4dg amy. 5j is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume;hd at h5 u4ydfb0m.(5j b8gzq the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 o7 mv tlsmo.p,*. fv8srpm is shut off and is eventually brought uniform.m* ots,vlpf7mov.8s ly 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 3.6-kg b f(t8-epi yfssk9(/izall at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the action of the forearm, whsnc5 brv:bnx6n8o*46gozn g(ich rotates about the rbb5(nz:x64nsgog 8n*6v o ncelbow joint under the action 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 aza f6j8byj7ic756qqn long thin rod, as shown in Fig. 8–46. 7b6f 8 nj5q67qazycj i
(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 masicpyjgj6 2 :m7db6 q;uses, $m_1$ 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 accelcn(9shso a 51zol.r e0+ czcb(ss),ry erates the hammer from rest within four full turns (revolutioncsby,9( cseznrc s 1oa0os()l5hzr +.s) and releases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment of ineow 6zhd4,e9kq+ xt:a frtia 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 oywb :i+5g1sd+1u y-qfw*2oorq j-on develops a torque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls wit8+obwk,btso f+l4tgw;y y*m 9 hout slipping down a lane t48y,9ftowykb g ls* bw;m+o+at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic entl lv.i;;o7kq s0z (oaergy of the Earth with respect to the Sun as the sum of two teri lt.;kal(o0 oqzs;7vms,
(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 r/:-dpj3bg xb gadius of 7.50 m. How much net work is required to accelerate it from/pxdb 3 jgb:g- 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 max*(ivyrn . pj.pl)pw7ss 1.80 kg starts from rest and rolls without slip.*j(x)nl w .7 vipppryping 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 verticalpnc 7 nxx+qou 8m*vnh2o14/ vlly 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 nx vhu*n7xo18oqpmn+/v c2l4 the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end,-9p qct v/ i*ga9vwmuioex/* of a thin string in aut ax-/ci*9egmvp*qv o wi/,9 circle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angulld g8zzbue+k:wf-:/q 6r9u t oar momentum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cr+zlq:6 u-kde9t:f gw z/ou8bm 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 at h2yta57q hbta8is side, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, hta7ba 8yt 25qthe 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 mw5 o rq4vu dw9ry6wd/q1 wrp .97cfh(boment of inertia by a factor 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 isq wwuchf./y b9r7o(5r6ww v dq1d4rp9 her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial rate of 1.0 vok6jxb4r4 zowc 5(x++dgpr5 rev every 2.0 s to a final rate c5+o+xb6j p(xwvzr5g 4d4 kroof 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 verticals;(r :axed ob5 yor)b5 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 diam s(doae ry:;bxo)5 5breter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinning ouvc;f6nj+-z xfczs.x+.ub6 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 t7mms /e82zr1jdkkf 9ug +s rm+he Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$

(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dropped on7l aejv. .xtihli (4/mto an identical disk rotatim/i7i l aj ev.4xlt.h(ng at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turnsc0emrobz: hgo l08/wph5 3wv, 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-round plat uognc k3)r2wl c0f z3v:i,(mvform of radius 3.0 m and moment of inertia w0 rg imvkcv:)32, 3nf olc(zu920 $kg \cdot m^2$ 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 ang,,(lt y s:wv d uhfr)m;3aldn1ular velocity of 0.8 (tla,,nsvy ):hldu;fmdw 1 3r$rad/s$ 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 zwd1)7ss, s hntgt k;j1r1+ yedwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assumn )wds+ trt 1kj1g;szh y17,seing 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 wi ffi,)pt6 .de8wn;a7l)dvjlqnds 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 )btg a-4 dkn7k$ 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, i 7;tjs/6h:vk -+jcbds /r nrronitially at rest, that can rotate freely without friction. The / 7rsb6crj/jt +kov-; d:nhrsmoment of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edfw9vy txv3rc-, z 8)bmge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionlesvbwfvmc3y 8 9x),z rt-s bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of the rope he1 u0t-ft5b mlying 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 rolls5f tuhm t0-e1b 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 Earthii 3u:qb/ /+ gbxpvt1l such that the same side always faces the Earth. Determ 3 pbt/i:lqbuiv 1+gx/ine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest / qh)on crowng 3jw5j) j80b3oat 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 *lj3:dmeb00rlu 2 /t dmidok5uniform cylinder of radius 0.20 m acquires a rotational rate of frod dij l5:o bmu0me/t kdl0*r32m 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 cylj*:year y2 ,3pp8rxar)a 3 rqwindrical disks, 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 calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, :ary 3p*jr rp3ry8w)a qe2,a xif it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angum/k(o 6;do58 1gj xdgwh9m rcflar speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the sizpti, ;+sjm aur)6opis hz gs01;w*;rwe of our Sun, 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 ofmr 1;t*p;siuzwhpars;) +6 j os0w,ig radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-po/h2 ifpjw fp h9qcfjq,gg- t x.8sc,/.llution automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.” -qfg9sh.t 2.g8x/jcfj q,f,w/p ci ph
(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 a jzyvz1/q./p ,4 4 m5pqbczrkzn $H_2O$ 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 horizo3+oi prfb2 d(nntal 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 lengp +msd 97uh2:gy(pslu(e)y*rw* extith L can pivot freely (i.e., we ignore friction) about a ht7p:px gy)su ume *s+hedy*((r9w 2ilinge 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 Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the flj4l-zjri witd7 1d/1 qoor, and we want to exert a horizontal force F at its axle so that it will climb a step against which itjti 4z/ird7jq- 11wld rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is o0v:pliet.eqf5u 55syq,m -p making a turn with a radius The forces acting on the cyclist and cycle are the normal fos-yeq:uf .,tpp5oq 5ml5iv 0erce $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5 m and begins whih1 raz 8a7wbcwj 9;oxiu6d+, irling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm aft rba+i,wwji c91dhx;u6 ao8z7er 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 cy1en kkli ,,h-klinder and her arms as thin rods, making reasonable esti-,k lnh,ek ik1mates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindricalvo ,acw3(t i,wv q d5avln.w0* plates, of mass*wa,va(,vl0 tic3vq w nwo5.d $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped rough track of*rkhcrt l58 8r Fig. 8–58. What isr t h*ckrr858l 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? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do 2b8huc c:k5ug not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revy,vkv /:e wew:eyl9g 5olutions as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 9 g ekwv:5yeyw,/vl e: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