A bicycle odometer (which measures distance traveled) r695k 5cx p.afcv)g8fxjgg orwo983his attached near the wheel hub and is designed for 27-inch wheels. What happens if you use itc8ggh5x6xfj f 9a8vkc 39.gr)w 5opro on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point oj3:mz dm+o/3ax -hfewn the rim have radial and/or tangential acceleration? 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 li- jm+oe3w :/zdmhfax3 near acceleration change?

Could a nonrigid body be desexvbcn ph 6*jqws59 29cribed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exep8e ri,exs5w(rt a greater torque than a larger force? Explain.

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

Why is it more difficult to do a sit-up with yowhm-vivrt 27f6p 3x7sur hands behind your head than when your arms are stretched out in front ofrxw3 vfmh v27si-67tp you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the b ytennf(qu0 (7b e3e/rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocf(u(te0 nb7 bq3eeyn/ket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on b;an l59gews8g eing able to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basi;w59ne8alg gs s of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow b l)ds)zdkhxkt(0p.0y 4t )sl zeam?

If the net force on a system is zero, is the net torque also zero? If the3/ynbbbnt n(*n ,duz) net torque on a system is zero, is the net fob,*(dy/)b n3tzn nn burce zero?

Two inclines have the same height but m ;iy3hh/l n2f-gp6t yk lmck/)ake 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 kg h-3/p;fl2) yc6 /hklni tmybe greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an jp)cdyv.cd1*2 yo :dxincline. 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 there? Which has the greap:y 1vco*ydd . dxj)c2ter total kinetic energy at the bottom?

A sphere and a cylinder have the same 38 2l89f(djenqaq6e 0s ajdin 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 rot(q6d del9n8jsq 32najfe8 0aiational KE?

We claim that momentum and angular momentum are conserved. Yel c2i /ec,nij1t most moving or rotati1,j c/ni ei2lcng objects eventually slow down and stop. Explain.

If there were a great migratir/kwx5qup/w : on of people toward the Earth’s equator, how would this xku w5pwr/q:/affect the length of the day?

Can the diver of Fig. 8–29 do a somersault without having any initial rotation wgbi3,9-tdhl :a x9aaq hen she leaves the boaahx9 lq, da:g9at-b 3ird?

The moment of inertia of a rotating solid dj *bmdx;(+bv1b uy-v fisk about an axis through its center of mas*d(-b+vm1;b bf yjvxus 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-7xr xn 955ag2brob c)z2jnvv in each outstretched hand. If you suddenly drop the mab)rbn-oagn7 z xj5v2v2 c9x5r sses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the sas5f,46 couyjdiv 5ty,me mass. However, one is hollow and the other is solid. Describe an experiment to determine which is whicoyf456vcj u,ts yi5 d,h.

The angular velocity of a wheel rotating on a horizontal axle point3l h52cq:jcsx;; m 1mc9vimdts 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 d3:mtsc lc 9; j1vxc2mdmq ih;5ecreasing?

Suppose you are standing on the edge of a large freely rotating turntabv/+r91 gddb 0 yydame)le. What happens if you wal r/ me10adybg v9dy+d)k toward the center?

A shortstop may leap into the air to catch a ball and throw it quicklqh ,d9 vn3o ;hxlxd5z1y. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his o,3vd;x9 nlhqxh z1d5hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular mome5v,x ddr2nt 9yr5qk e:ntum, 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 stq5rv dtdk,x2e:rn 9 5yable.

  Express the following angles in radians: ,(qd8kr80 j:mlw ixbn (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. Ctc9)pht ar r-.st1dp+ alculate, using the information inside the Front Cover, the dtrtt1-ch9sa+ p)r.pangular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beam dk ws5/ys5o;z biverges at an asoz 5y/5sbk ;wngle $\theta$ (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 b5kh1j3y6vi1nw tu 7xthe motor is turned off during op vbih1yk tjxu75n6w 31eration, 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 6waq+3jug6 hh*/fj zqfloor 3.5 m to another child. If the ball makes 15.0 revolutions, what q6u wa +* jh3jfq/h6gzis its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How m a(x gfu.5vl ;c.q3pxiany revolutions do th5xa 3l u.q(gf.cxvp ;ie wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate its anguqag6 s1s/y0 ploa)az0+- ivsg lar velocits0-q)61 ayo /vlig 0gzapa+ ssy 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 mak kbfnrs+6h+d9m 0wuwe.0)xfles one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1. 09hl6ne+mf+fw dk .s uxwr0)b2 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. fig5z*-y i:7x ortfz its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a uiv,t nba4ih 5w5n 2;aea1u2u 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 7.07g.n vqc2ul.g cm from the axis of rlg.q.2gcn uv 7otation is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpm iq w.brvh*k59 bjr1qk(ra)n ,to 280 rpm in 4.0 s. Determine*bqrbk9r ,ivawj n)r(h 5k1q .
(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 radiux. wpf o4 ,s4cqkdby.: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 spacec3 7(sospzqog j/7s6 hfraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of thei 7go7(3/fspqss6ozj h r 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 accelera6t ;9lrv4wkqaf :wk ;jtes uniformly from rest to 15,000 rpm in 220 s. Through how many revolutions did it turn in th9fw 4lrvt;qkj6:w k ;ais time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calculaaf, 0pawjklb p/w79* hte
(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 dc.,d emi9y-a jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revo id ,a9cyed-.mlutions 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 240gj60st0yk+(84 sdksvhp/ s yz rpm to 360 rpm in 6.5 s. How far will a poinv 4k/d8yp g6t0s+s s hyj0skz(t on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at ;6qjjm6q+ azpq4riuaaw5 : : b850rev/min It turns 1500 revolutions befo;w qamj6uqq5p:zbaar6ij + 4:re it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car maelkt-,/qw 0)e5r w utake 65 revolutions as the car reduces its speed uniformly from 95km/t /w, rektew5a-uq)l 0h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces- yysrn; .2mvi its speed uniformly from 95km/h to 45km/h Th ;yn- m2ri.vsye tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each pedal when climbingj/1 ly 3foyhs3s9vszk-(0p , iylxt +h a hill. The pedals rotate in a circle of /o z-pf+3(v ssl hyyhxy3tj1 i90ks,lradius 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 czcz+ o4yvy57r1z ne r9m wide. What is the magnitude of the torqy5yrer 9zzo7ncv 1z 4+ue 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. 8–39.slbrv xq35t/n*(pt9 v Assume tvns5 pvlbq x(* t3r/t9hat 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 massless rod edw.h- p*y hi3ang-9w (0nmla which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of thg(haew n* ai-n-093h mwl.ydpe net torque on this system.

  The bolts on the cylinder head ovqz9mqe*)cq(g 41xyx *ow mt(f an engine require tightening to a torque of 38 cz9 gq )qemqwm*( 1ovx 4t(x*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 radius 0.648 m w qra0xipn./: 2zqcx*t hen the axis of rotation is throughp.ri* t/2a z:0xxcqnq its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameteqasr2yy 076zjzl qo 66n c6lo:h)n lf0hi0jo8 r. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignored (wh7lq 0:6y i08)n o2c z6srohq6fjlalj6o0z nyhy?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rot5l4r+e sai7dgsxh eo; htk v(4s( (/kt/:tugdated in a horizontal c/(g/ed4 o;t5iuagr4+vxt ss7kh tel hd:ks((ircle of radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a pottpfe 703m8hge69dteqn er’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between h 7ed0 3t qphge8emf69ner 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 sho 3h46zqijtpn1cu gn );b1p 9srk3gay2 wn in Fig. 8–43 abzph9 n gcs3 11uyaqp njgb4krti2)6;3out
(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 of32ffq:q k8xz .6jixwh two oxygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinn4ok shybkv;w 2+*vm6ding at the correct rate, engineers fire four tangential rockets as shown in Fkbyd;wmv26*vos h+ k 4ig. 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 and a mass o;f /k)t wks6nf/wo f-pf 0.580 kg. Calculkpfw6wt) k/ff/;s-o nate
(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 lqb mudtt38-(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 merd n fbug66/b*. vbzk3lry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on tubzg vd 6lb/*b63k .fnhe 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 atnnz j-0dutm4cf8+z:um n l+k 8gm( -mr 10,300 rpm is shut off and is eventually brought uniformly to rest btkm-f j nzz:mgu8cd(+mr-n +0m n 84luy 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 accelebs/9nw-t2 qn qdtp;5lrates a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the action of cb n/ z;zl:3gg-s * -bspdkgw*(c;bwcthe forearm, which rotates about the elbow joint under the action :z/zwbc* 3;gs-w (s; nkd lbbgcp-gc *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, -)b hnpl7tri kr (u;s-as shown in Fig. 8–46npu ri-h-( )l7;srbkt.
(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 twz-1fp 6;mqgva; v sy;so 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 accew tr8,vo.vwdjvkam: a,x(- o /lerates the hammer from rest within four full turns (revolutions.avdw-kvv:,omxtwjo r , 8/a() 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 iner5vl*/big y,;dsu,b )g n mmw3ftia 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 develokxb.ucp5fczbg n3fd c1p;tbdo **.3m- e: ry*ps 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 c ,a,tj m;xts,j*+dqm 2(p;kb 1tjklezm rolls without slipping down a lam t 2( a;*pj, l,sqdtjzk; ,ejbkm1tx+ne at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as thi27ujcu:wif7gbd5 wyc.2/ y qe sum of two terb/7gwwf 5 cjcy u.i:dy2 uqi27ms,
(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(rbe-c eny+pt+qz8*v of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 zqvr 8p +e *n+ec(-ybts? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg sta p5dbkqh-; cb1rts from rest and rolls without slipping down a 30.bhk1p; cqdb- 50 $^{\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.nmzb/ e6v/f5jj96tzw It starts to fall and its lower end does not slip. What will be the sp6t w6 /bjzfzje/m9n5v eed 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 the en;zo, 35tpx* 0 kzjowpm )btu :gg6vg7rd of a thin string in a circle of radius 1.10 m at an angular speed of 10.4 :jg0o6gtbru,g 3*;)pkp5 vzwt 7 mzxo$rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg unifory)au9 kjo93ag m cylindrical grinding wheel ao 39 u9jay)gkof radius 18 cm 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 6 rqx;dpq,a1sat his 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, the speed of rotationp d1sq6r,xa;q 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 inertia -dn3,duhn. 1i kzq.amby a facto1,mhinu.d.- 3 k qzadnr of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate 2qw8cjta xa 3ck9:i, g(cg8ka from an initial rate of 1.0 rev every 2.0 s to a fik:8 9 tx,j8aqgak3iwc2ca g(cnal 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 p-f0zr;r/o8x inwpr9at a frequency of 1.5rev/s The wheel can be considered a uniform do -fi prx/wnpr;980rzisk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater sp3kc b*0wtf ,tlinning 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 Egdn4 4 b)/t9ucn6.g3grep *f o :zikrlarth
(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 w4,mwv84mxj sof inertia I is dropped onto an identical disk rot sw84,4xwm vjmating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at *etus /hs uq,on* t,b62.4 $rev/s$ 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 st5 pg+fy*k9f8wx j 6ffhands at the center of a rotating merry-go-round platform of r fh6g*fw8kp5 +y9 fxjfadius 3.0 m and moment 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-round3v.,j q 5oaved is rotating freely with an angular velocity of 0.8 35a.vvo,q ed j$rad/s$ 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, losing about half it2h /q 9xlu x 0mx:xr5cbxkv9p.s mass in the process, and winding up with a radius 1.0% of its exis2qvxl9xcbpmxxxr: h9 uk0./5 ting 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 windsy4b4 tiyakjgat;/q qlhdegw*(d)+ 14 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 t0 9rq65to o4irls2wz j.5rmr$ 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 pl6p jwlctuw dc1d n(-*mc186pfatform, initially at rest, that can rotate freely without fri l(w8*wuctfjd1-c1 npdp6 c6 mction. The moment of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m dwdu(g pm pqvaw 9)bj73/l2j2 m+sqw1eiameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia o7lwp) s je2p wauq(g3/1w dvjmq2b+m9f 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 tb3*n fa89 beoacg8 g2;w;j kyche end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length ofec;wjg8y2kbag3 8o9b fnc a *; rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side alway-a zvk8)v inq0s faces the Earth. Determine the raak vn0i-zq )v8tio 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 acceleratetx ,f,v 5/y z.-g*4-bksven2g zyay jrs 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:,hjyp8 u +9hrs 7bcy-- rghyius 0.20 m acquires a rotational rate of rr:c9yhpy- ughijs y 7 -bh+8, from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solxnag:c s1ho w*)y2js(:sy q+wid cylindrical 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 re xh q(asygyj)wcows+ s: *2:1naches 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 k89ketx gmw39rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, buwr9 5q5+exvi:xfs ts/t 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 process, what would its rot xqx5sit v s5er9f+/w:ation 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 it to useqb,.t2 3ry cel energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a u,q.b c23ylretniform 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 illustratd6 wjb16 )3rmzweha2 ves 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 on a horizontal surfacc.xyo cx jt7j 0.c 9uvl(ca;0xe 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 Ltfe:i 5whfrdub1.3g z6 gw,z3 can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass ,3w 6bf:fu1grzwd.3gh e5 zti of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the floor, and w2s5qhza u.ca p* ar57le want to exert a hor5 7ah2p.s5ql zau*arc izontal force 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 with speeds)(qoj51 *um 1guapbp v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist and cycle are the pj1(quo)p u1b5gmas* normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5 m and b 8e9xy-vq -3nq);vyr1tauz rh egins 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, aqy )1;ye-8v hnt -9vuzx q3rrand where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylm k(n(xf1vfo9 qtow3y cnv)).inder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ra) nyqw (vcotfxkf9o (v1.3)m ntio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindrical plat*g e;9oi2pvvltoj+ f:x3ohn *es, of mavt ix l ofpn:+9oe*vh*jo2g3; ss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 radii6(e r*au rml)us r rolls along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point of the loop withou6imeral )u(*r t leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not as/is3h/i0ke xv, 9 d92ykgxr+wnnp 6 mcsume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car ;/vv r01g hsjf(v(m) ji2gttr reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the ani/fr svg20jtj1h(v(mvt;g) rgular 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