A bicycle odometer (which measures distance traveled) )5 l98kabeu-nckau0z a 0jfz+is attached near the wheel hub and is designed for 27-inch wheels. What happens if zcb0 0fa)k5zna-8 ual k9+ ujeyou use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular hva5i7to) qq/velocity. Does a point on 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 of7qai qt)5 vho/ either component of linear acceleration change?

Could a nonrigid body be described by a single value of the an zr 7 cypjw05f.,)f1zpgn2cnxou1fl 1gular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than 7q2,( p9 8jazw+2y ycol*xt/bc xlcsta 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 your hechywi-k3jdq ) 1s+.b3 shz;t wad than when your arms are stretched out in front of you? A diagram may help you toh;+cw-b.ydzs 1s3k 3wq i jh)t answer this.

A 21-speed bicycle has seven sprockets at the rearu xr, 5gzpxl9 welq xdj s9c4)h7412az 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 gear is it harder to pedal, a small fro4hl294 g,)drpuxq zx9x17c 5azejsl wnt sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slen6) . ;1r sukqbmr1 pvsbmikhsfg0:4hzu )1a:g der lower legs with flesh and muscle concentrateau 1mr . :s h b10pkmgv;s6sg)f:rkuh4)1bziqd high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walker,9je9/t s(wsuoe dtt23e7) dkiluh;y s (Fig. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is tth4foerm7* * k zy/jh4he net torque also zero? If the net torque j4f h* */hot 7rk4yezmon a system is zero, is the net force zero?

Two inclines have the same height but make different angles with the horizontali64 rqo261; ncnymprv: sc,fk. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bqn corcp2m v:6if4k1y ,; 6snrottom be greater? Explain.

Two solid spheres simultaneously start rolling (from res tzgced72el/4k ,75kr og +crtt) down an incline. One sphere has twice the radius and tw r2ztc7g4,/5tgk7e ro+ kedclice 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 have the same radius and the same mass. They start frm3;mwe;mu3 s aom rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which hass3ame m; 3wm;u the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum arnwm q:ir;614h ypry ;pe conserved. Yet most moving or rotating on; 6phiy:4 1m;q rpyrwbjects eventually slow down and stop. Explain.

If there were a great migration of people toward the k rxig8bs;i/ p2o.v)e Earth’s equator, how would this affect the length of the day?;gisipe/rb v. 8o)k2x

Can the diver of Figgn j *5 vog0x, lk:)kayhw/ia+. 8–29 do a somersault without having any initial rotation when skiya:g*xjnkg )0 ,/ l+5oahwvhe leaves the board?

The moment of inertia of a rotating solid disk about an axis through its cente7 l0rtv.*.ra*q gvoph.z / rrbr of mass i o7r./.r*0hbt*zpvgalr rvq . s $\frac{1}{2}WR^2$ (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 h 75y6g+kxufg+0ssgw ,z3 zodn olding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular vel,x67fksgnzw+ g 03uygods 5+zocity increase, decrease, or stay the same? Explain.

Two spheres look identical and haam,ra973ue/oj k *oc8ntwq +o6 p1zi mve the same mass. However, one is hollow and the other is solid. De,koonacmor89 /w 7upqti 6+ a3ze1 *jmscribe an experiment to determine which is which.

The angular velocity of a wheel ro-4+ qkqt 9,r:ptz +y6hgyodk sv(6 jivtating on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheey p6qovj+kh6tr(g +ki v,zdstyq-:49l? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standin6e; agx scet60qeq/i)g on the edge of a large freely rotating turntable. What happens if you walk toward the center/)t;cqageq 606e ie xs?

A shortstop may leap into the air to catch a ball and throw it quickly. Ahq *8 cx g+;w ,zxvlasis31vd9s he throws the ball, the upper part of his body rotates. If you look quickly you will notice that h +ghd; 31s9 ic*xvsa8l,wvxq zis hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angulaf3hg1 ( q,j+drk odzg7r momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep jhzr,o31q7(f ddggk + the helicopter stable.

  Express the following angles in radia8)v uz5byq , oewa9vv(ns: (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 Eaooax6;7o q r7jgq8so8 toy 0x1rth because of an amazing coincidence. Calculate, using the information inside the Front Cover, the ang o0 1tqxojxoso;7o7g yqa868r ular 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.itoy2 mm9qrh ,al-+h ia73 c/i The beam diverges aa+r,m2l mi 7qc-t9 o3hiihy a/t an 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 r z*,lefojf hli6. 1-rjca bg4,otate 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 ang,a1ilbz6ej-,h jfc lgfr* o. 4ular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the ball makes ilr sf(mo8 d:.h6 3khm15.0 revolutions, what is its diamet :h m.i6rh8(k fdsm3loer?    m

  A bicycle with tires 68 cm in diameter 9:*6oq +kp. )q okkj c+r/tyswcehoc u/s20udtravels 8.0 km. How many revolutions do the wheels mak oo+ ojh6kcsuudq9y/)0/2 +p r.ct:qkkwe *c se?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate its a z 3 2qmrup(,q .c7vwiuc59jrengular velocity inw,cqp.ju5r(u v czm 32r 9e7iq $rad/s$ $\omega$ =    $rad/sec$
(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 makespl )cyocc hymowp,nt l y-( 6o6)+:pn, one complete revolution in 4.0 s (Fig. 8–38). (an-+,p) (c,y)pyoyl:6 6 tpwo hncm olc) What is the linear speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Eo*(omaeso u43arth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed t/8isuxa 3p ai8y6*all9kly) of a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rr72j1 f(m*-(i rashs qc,mgdzotate if a particle 7.0 cm from the axis of rotation is to experience an acceleration ofzja*rd ,sish1c-m(r72q(gf m 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates u3ei;j 4 wzanppp g42.ahl1-fd niformly about its center from 130 rpm to 280 rpm in 4.0 s. Deter41af w2ag;pp.pl 4 dezh3-njimine
(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 .dx )r;mv wc9r$R_1$ 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 spacecraft pu .qj2 idlvvg)x/1m i7ht themselves into a slow rot./ixqil 27djm1 h) gvvation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly frows;(wf8 q4o mym rest to 15,000 rpm in 220 s. Through how w(w8qfsm;4oy many revolutions did it turn in this time?    $rev$

  An automobile engine slows down from 45*y-pkuelnqu ;4t3g: w00 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 highspeeb.w +hcgcx/6 ad jets in a whirling “human centrifuge,” which takesb gwc+c6/a hx. 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to ke.+2 lavnb :p360 rpm in 6.5 s. How far will a point on the edge of the wheel have travel n+lpbve2.:aked in this time?    m

  A cooling fan is turned off when it is runninyq 4 n)wa ora(6ca1jljj9y;o6g at 850rev/min It turns 1500 revolutions beforey) 4 acoqaonj66(lja;y1 jr 9w it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reducwd y5/:b2rek nes its speed uniformly frbr n2y/ w:ek5dom 95km/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 make 65 revoluttu;uxl 7v+ (ia a(f7cpjfz rvd6j0n(.rx 2x :yions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have f6x x;. a+lp7r(uv: xa j0rj2z7i(uvd( cnftya 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 he6t +bs jmgx82(zu/ff ar weight on each pedal when climbing a hill. The pedalz(fgu6 t 8/m+aj2 bfxss 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 force of 55 clst1zva+6 3-;t 8un pdwv zc)N on the end of a door 74 cm wide. What is the magnitude of the torque if the force is exe8 -)1dvltup wzc6sc+anzt3v;rted
(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 axle9 (mw)ptm2ekquz4t9tw b 63ub of the wheel shown in Fig. 8–39. Assume that a friction tote3b6)9bm q kw p2w9z(utmt 4urque 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 s05b8 :oup;-y: :eiu k,(j2cs,humrnh nnv bfends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direc;-cnri2uh(b m:u n, np5ss8yo jf:e0u v,kb: htion of the net torque on this system.

  The bolts on the cylinder head of an engin:ct2vt4g t pc7b24r( 2ly (w kakjz-gne require tightening to a torque of 387((c z aggtl pjkwyrtv42kb2ct n-2:4 $m \cdot N$ 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 sphdvwvv5kh+h5)-:k( dbj p9yb mere of radius 0.648 m when the axis of rotation is through i 5pwbjhky - )kvvb+vm9dd5:(hts center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheelq g2, 88vhlxsr 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignorr 8 q2vxlg,sh8ed (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end os. c-k.bhu1oc f a thin, light rod is rotated in a horizontal ciro.1 uc.bs-chk cle of radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotatingi )vd lb)9*t v lcjbq8-sdk2u, at constant angular speed (Fig. 8–42). The29dlujlc,t sbvi v-d8 ))kbq * friction 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 (f 4fc xgw,udlasabg.3o) 8+o the array of point objects shown in Fig. 8–43 alugb4 c o+(aw3sx8 adg,f)of .bout
(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 consiststnqwbu(zn/a gi / vud 21t+;hk2/ql 4y of 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 cylinkk u;chry 87sktx. 4v0drical satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 mi7 vxk4tyh 8u kksr.0;cn? $\approx$    N(round to the nearest integer)

  A grinding wheel is az*jwjrx g,5s.kb; o2q uniform cylinder with a radius of 8.50 cm and a mass of 0.58gx5;.2 jkb*q sjwzr, o0 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, accelera3t udr -keru98ting 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-dri/qgufjca j wvxr6:v 0998r4zvven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is gzr a:8 jcfvrvjxu qw46/09v 9a 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 rpm is shut off and is eventually brou ujxqyc9bbwh 0a2kme hw0:gs ,rw ,/arw7(9i4ght uniformly to rest by a frictional0sarwwjmrx:h97u/g, ckyw4,h a w e20(qi9bb 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 3.6-6ts5vu; ps* wykg 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 the for63-vk8wkrkpdo :p3 qb *ile3iearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to k*3woi l 3r e6k q3:i8d-pvpkb10 $m/s$ 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 qt4 v fafs,:7adh652kk*, fafm*hdwj, as shown in Fig. 8–46a*f, f4kj dv,:sfwt6h 7m5d2*ahkqaf.
(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 9h:j0 zcc vg5aof two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hamme;me h ,bn6h0nxr from rest within four full turns (revolutions) and releases it ath nh0bxm;e6n, 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 w1yl 0 p3/hlhktwcj 2:a moment of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torqudq.amqsc g fa0h:8m(. e 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 rvcplr-63( rc hadius 9.0 cm rolls without slipping down a lanl- hc6pv3c rr(e at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect t ybp/1:c e;a+omz9pbu o the Sun as the sum of two :+c9;boamp yu1/b zep 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 mavq*7 8tcrxyy2q.qb bco*wz1 jg*gk36ss of 1640 kg and a radius 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.*17**wrg6t bgky z8obq cc3 xv 2.yqjq00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts fromm0gzux h5jw u2,c+:b,rl h,ki rest and rolls without slipping d+ zmcwg kj,rx05lu 2hhu,bi:,own a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is balanchr6ah,x;f,edaf)n4h sj .sd6 ed vertically on its tip. It starts to fall and its lower end does not slip. fd6fe;,rhx ahnajh.sd6s ,4) What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular m*-2;j m1r ; uuxpub)3gox wdanomentum of a 0.210-kg ball rotating on the end of a thin string in a cr ;3nx j)1xpoudub ;gm*w-2auircle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg unifor dk+guh co.)2om cylindrical grinding wheel of radius 18 cm when rotatdo)g. u+kc 2ohing at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotatik gk )yzfb 2y+3o5zki+ng at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the spey3 z)oyk 2z+5gibkkf+ed 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. 862to ssivws7 jec:z/ ya v8zr*:j m29r–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight positioz2rj76vjey9 2 va os:*:8cs/swtzm rin 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 from an initial rate ofq0(6 i u64 1wjzpkpuip 1.0 rev every 2.0 s to a final rate of 3p pqip( w66i4u z0ju1k $rev/s$ 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 cw,zz ib 4)o y*gn,)bgyenter at a frequency of 1.5rev/s The wheel can be considered a uniform disk of massb iy,)z*zng w y)og,4b 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 spinnile+wf5*ea :rcg vi1ydg -8w0mng 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 m : r2ahs y qzme1y9pz7a1vv83womentum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I s1s4;ium8l mwn k+c i(is dropped onto an identical disk rotating atu m;kn18m 4(wssc+iil angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns f2fzuef/)p6 /m5uep uat 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.n -m* 73zd8s cuhs77uyapvoh stands at the center of a rotating merry-go-round platform z oc7 7dhpv3 .ahsy s-n8u7u*mof radius 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-rouc jcb)sa ;5,bnast25 znd is rotating freely with an angular velocity of 0.8sb;sna ,at5b)j c2cz 5 $rad/s$ 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 c5e +h4b nxax5 vl1n*dmollapses into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s nelv5 n hx*+e54nb1madx w 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 win6b,vgj .lijdsc)sr8tpc (b22 ds 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 l2 zmw4yn987t ip+gfyyliz2/$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, thah o/+et2r oni sk (vn+(zz2jt*t can rotate freely without friction. The moment of inertia of the person plus the plvk is zn+ r+nhz2j /e2*o((totatform 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 tu id0 *a c6p+e:jejhft)rntable that is mounted on frictionles6djtejp)*: +e ifh0a cs 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 eui(93 zdilgd)h ae4a + (mohp+nd of the rope lying on the top edge of the spooluhia)(e4z(dp 9igm+a +ohdl 3. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces the Ear(phwk . e.;fp ze)iz1zth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the lat(zfpe;)w zh.ep. z1i kter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rg 1zyrs)rj 1)t. )j;qrde3eu nate 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 thefjl7 qt863o8v ehq*t zxw )7:ohiaox63rc l7d shape of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 738dzrq hx3v f:77)h l xeo8tctlo*qa6o6wj i6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical ddwz-0qtn 9/5dsxwh wh +g y)1jisks, 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, if it is released fros0nd-ztwwd/ q5 )wjx hgy9+h1m rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of thep1f6the* )fcgxkefx+ 2fht t64a43 aa 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 size of our Sun, but with mass 8.0 times as great, werei k i5.l up)7az6nd1-dma1pk y rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the1 . 6ualmpd)aykz5k i7n1 di-p lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a lowrki(wr- 1x.zk6 dja p7-pollution 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 flrxridj6- a71pk k.zw(ywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates anlj(ti ra /lsh96gpr.d*- ni 3g $H_2O$ 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) : s: bjmi 8(m)jnoo:op+kgxl)is rolling on a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can 9hm)eu8hb1v- xq(ivz 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 8x(eb)vhuqmv1z9 h -i 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 ve8 oqj (;dxfcp;rtically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). ;odc8(q p ;xfjThe step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road i7:ffj2qzf97vobvxg na5-bfx 1 c7 6shs making a turn with a radius The forces acting on a1sfvx bgfh obx7q-z9 f5n76 7vf2cj:the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into rucwab7u(s.kd8w-i 3a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achievd87sba.u3 u -kwcw( rie this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid ,udf4o i5:v dxdih l.i3 2ej*jcylinder and her arms as thin rods, making reasonable estimates for the dimensfx j 3l4jiidu d2 .h,dov*:e5iions. 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 cylindricsp4ggdg ; 8ek*al plates, ofkspg4 ge*8; dg mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped rough track of Fig. 8–vzf 5wcm*.gx. 2gz:e n58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving w.mz5*.zgx:cn g2vf ethe track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not5i;jol krm)-vq 9xm8f assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolu8/rcd b55d v mp+2lzjstions as the car reduces its speed uniformly from 90km/h to 60km/h The tires have d s+mr2j 58p/zldc v5ba diameter of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s