A bicycle odometer (which measure2b.yz2l b fo 0qf njz4tdusv7,.7ho, xs distance traveled) is attached near the wheel hub and is designed for,l2fn db4jyt.uszhq fo2 bxz,.7ov 0 7 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on the rim hr ae6m h1 irf-ax 05v:+hh,puuave radial and/or tangential acceleration? If the disk’s angular+hr6,mv i-farax :pu5h 1he u0 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 desc(sff,b,le3yp;v c q j9ribed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Explaem 2qt3 pb :cyp 4cw)vs56/pjrhucd:-in.

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 n2lennji 25uyp,4 up- your hands behind your head than when your i5 nen yl2pn,2u- pju4arms are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle hazf-cej an(wq6 i-umges1 h*19s seven sprockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedh(m6*e1- 9nw gc ajzu1is f-qeal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend zinhzuxt5i*gf-e-4 z-d 2.oz on being able to run fast have slender lower legs with flesh and muscle concentrated high, clo tfnx-o2zi-. uz*-i zz5edhg4se to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (ao5 ub9dqn7/ u7.oynaFig. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? If the net t3m-. c v,rrrn7 fw:jbsp;usdk,p/15;su i qfqorque on a syss,,csjp3 .:vdrn/ u1p; rqrmuf-7wk5q; fbistem is zero, is the net force zero?

Two inclines have the same height but make different angles with the h(iyt(*y(if 5ra b l6qmorizontal. The same steel ball is rolled down each incline. On which incline w(liri*6(qt f b(ay5my ill the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously skhh6q h u;+c5t9 x jgqhl(5q;ktart rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom o5u( c kjth;9qh x6;gl+hqkh5qf the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder q l5:; kpo9k+vhjv:bf 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 greater speed at the bottom? Which has the greater total kinetic energy at the bottom?v9:: l f;h5kjqpko+vb Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most movqp,pl -g n. a8fdnd4a*ing or rotating objects eventuqngpl. f aa8d-*4p ,ndally slow down and stop. Explain.

If there were a great migration of people town:ub9 9g;fe/s i6)xiyj5w 5j f5qdro uard the Earth’s equator, how would this affect the length of th/ )djwr:xy9gjq s fnuf ib ie5u;569o5e day?

Can the diver of Fig. 8–29 do a somersault withouo3mdkfko r,,-fi71bm t having any initial rotation when mr,k - di1ok7fm ,of3bshe leaves the board?

The moment of inertia of a rotating solid disk aboz:1ikyz94r( h e bl/dhut an axis through its center of mass r41/9leyhzbz(kihd : 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 h3s5 ir:gwpaa,; qk4w rolding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocit:wg5 p sr,43q ;raiawky increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. Howef h;w*f80tlwx ver, one is hollow and the other ff;8*h0wtl w xis solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points w9uka x0. 7kwl)j,c45 eh5 dp4dqmxpruest. 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 acc9d rp4he5lukuw 0m.7,k5a x d4 xjpq)celeration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turntablejl pip1, 6 l 6ntlqjj 4x*nql)7ora39x. What happens if you walk plr)l x66j,j4atln qj9q 3lip1x o7 n*toward the center?

A shortstop may leap into the akbx+gs0jb5 eul -b59cs2(kd air to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will0bbl x2jaskg-5bsc9d ( u e+5k notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation 10heq4i7;4shhbw ux 2c(r coe +-qqnsof angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate qn04o +-ce7 hh(4cswqu1 qe 2ris;bxhto keep the helicopter stable.

  Express the following angles in radians: (2euwge-a.r t7a) 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 amazzc;5p o+/wjck6v x 930gsp jxzing coincidence. Calculate, using the information inside the Front Covegc9z6x/0v opjk3x jp;sz+5w c r, the angular 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 f,uzq6w2a 0q kvrom Earth. The beam diverges at an angukvq a60,qzw2 le $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender kany(e 4e8qwpfk;e ,* rcs )9nrotate 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 accelw)9 neqkp ne8cf;,k(*4sy raeeration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a oo d) +*ifxgq 83ivaviiax;0+level floor 3.5 m to another child. If the ball makes 15.0 revolutions, what is its di+iqoaxvi); + d 3ifgi* v0o8axameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km(sr*xw h z,k,gaqn,yb er 4+de)(( nmg. How many revolutions do th( )rmy4b nhdq ,sg+ax(k(wge,*zn re,e wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate its anw*e xk;2whv*:kbt :n ygul2 kxeyw: *vk:whntb;*ar 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 revolution in 4.0 s (Fig.(xh,r axr7yz. qnn 78h 8–38). (a) What is the linear sphy, nr hn r7za7(.x8qxeed 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 opl a uox4m 0;-(m6wfhh77ykr rf 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 spoz)0 leaha (k,4, 0dlkbqrv f*eed 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 cen ze/y0 f rsvu hlkd)k3+1ta2r:trifuge rotate if a particle 7.0 cm from the axis of rotation is to experience an acceleratiok+l y1u:3ftzsdk e0r av)h/2 rn of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerate z*:dw,fcnal)q/4 bdas uniformly about its center from 130 rpm to 280 rpm in 4.0 s. Determc q)fld n/z: ,*d4waabine
(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 radiuxfrk9.ap, 9jrs $R_1$ 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 spacecrafvy+dw pg-k;l- jt5b/ ut put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameteyd tvk+l5;- /- bpwujgr 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 rab el4zyt gz/1g(.rp8fe ,7l,000 rpm in 220 s. Through how many revolutions did tg/ylfb1zzr (.8 gl e,7ea 4rpit turn in this time?    $rev$

  An automobile engine slows dow e+t*tp lh*p6ezf2ma0ah.6lgn from 4500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for thob1i6h8 3 wm7n guhsrt5) 3euze stresses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn throughuwutsr85ghen ob36 )37z1m i h 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 rpl c:z7lxf*c7:j5; uxfosd;e;;wko lr c m h/f,m to 360 rpm in 6.5 s. How faro7l,c;;fzl r oml hu:xw5e d;s; /cj*fk:fx 7c will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 150xb7;(owvq eg, zs)pg-a r.i- jx;z+au0 revolutions before it comes to a stop.ae;.p xqz)g7br, v-+;g iuaj- oz(s xw
(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 reyxl b76nhspw;(ajo79 hh k m6m1v uh8;volutions as the car reduces its speed uniformly from 95km/h th(hy6 6j; ;7b m9skw71n xluahhv p8moo 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 revolutii1t6. tf5jgda ons as the car reduces its speed uniformly from 95km/h to 45km/h The 61j fi5dttag. 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 iu9iyvih tjs:len8jk k0py( 9,8 y,)feach pedal when climbing a hill. The pedals rjknej f98sy ,:pl9y ) ,k0 v8hiuy(itiotate 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 a door r sq iqmk8k0;774 cm wide. What is the magnitude of the torque if the force8rq 70q;i kmsk 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 oqbsg-f )l ;qo3f the wheel shown in Fig. 8–39. Assume that a friction torque o ;lsbqg)qf- o3f 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attachekwxz 3+:k gt g9e4;dadv n(k2nd to the ends 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 digd+tnnkev 3 xag2 z49k(; :wkdrection of the net torque on this system.

  The bolts on the cylinderpy2 n./ 4jm+uyfvsey: head of an engine require tightening to a torque of 38f+ y.my:2ns j/yue pv4 $m \cdot N$ 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 yw.f8tdc(z)hnme 9v-5 ; t ydiumg7c;a 10.8-kg sphere of radius 0.648 m when the axis of rotati8u mf;5hnvid)7mye; - .wctt9c dg( zyon is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicyzy; )m3vjmn. icle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25.m3;i)jz mnvy kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod icrl,. nd a.+lfs rotated in a horizontal circle of ran .fc ,la+rdl.dius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating at po9f6ovxl47 z0i3km2r t)wf p constant angular speed (Fig. 8–42). The frxk rw tf7opp6m9lizf2) 430vo iction force between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the ak3 h.,j *drvavlis7o 5cx6c1krray of point objects shown in Fig. 8–43 ab j v*s3 6,hlc17ck5 kvra.xoidout
(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 con s1.jnw v)j,it4d7a vrsists 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 cy;k 6dt denp)z9lindrical satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellitkzdp d;n)9 6ete 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 cm7 z-0r7a mzmao,yi fx.qc:o*l and a mass of 0.580 kg. Caqmz-mx,r7o.0aaozfc*ly :7 i lculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerating it f5lv/vdprso0l /nb51i)q* zmx rom 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-drivenyq;0 3aoukj4 o 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 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 requiredqako30yoj ; 4u 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 evenzacca:a7 :kl -s 7n-phtually brought uniformly to rest by a frictiokaa:l7pcasznh c--: 7nal 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 accelera emsc k*8og,wnoi-(s 8tes 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 th*2aknzm opn5v +(8)tzr4hl xtr)t o0srown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. Thel+8t 0 nr*tzt5xo 2orpz h(4vm)s k)na 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, (,z,y a8rx uojas shown in Fig. 8–4ajr,yu 8(,xoz 6.
(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 twsib a3q) *e:b kut6/-gjgx5swo 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 theim 3fhjb56,37 eaxby o hammer from rest within four full turns (revolutions) anbj3m b65xy7h eoia3 ,fd 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 inertia qs )j3 6pvvny:)pufjt1n*w ,h 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 torque ou9 (sf ((ra-u1qnuej hf 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls without :/f -hoa hnyki--rpj9slipping down a lane at 3.3 -oaih h/:yfnp-9-kj r$m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as th(m1)i)r85 t ph nouxsyomj)7ke sum of two te57mpok n h (y8s)m1oirxutj)) rms,
(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 za8jhqd3uos y/rh:0ze .qq ,6 l bd/d5a mass of 1640 kg and a radius of 7.50 m. How much net work is requird3685sqh://qj,y.bd a 0qherzl o zud ed to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts fr7x a37x6zrond9a4k zvv2 ;e moxr, dl4om rest and rolls without slipping ,lv47r4n e2;6ra mz39 zoxoxd 7vdxkadown a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is balanced vertically on its tip. It starts t-z6, volefu0twf6 7,l egk4ieo fall and its lower end does not slip. k4,ol ,e6w7etuz fvg60- eiflWhat will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end of a tpzxku8yp4 z zc.;0x6p hin string in a x4z0 p up.6;c yk8zpzxcircle 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 uniform cyl. g+esgpq1 j(pb 5nrd+e:xs2tindrical grinding wheel of radius 18 sj(egb p e12npg.+s r+5dqt:x 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 at his 7m*o+7t/ okp7c tq 4cwtg8sn iside, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48tmi4k7post 7 /7o8cg+q *nwt c, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce her monzo 0sl u-8if ,qnx8-bment of inertia by a factor of about 3.5 when changing from the straight position to t0zui -bs8xnq o8-fnl,he 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 fu9so 03 :h3;pbgg lvean initial rate of 1.0 rev every 2.0 s to a final rates g9he:0gfpu 3vol3 ;b 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 rotati vd z5-d) tsgen,cfs/-.kd 0ppng around a vertical 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 diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately sha.z5 c-df) s0gdkn/v-sep,pt dped 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 ska,ot2g zgm -za/g-d7mhter spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular n3 /m d/cth4 x y9d,2injpnaz.momentum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dropped ont-;y y)sqj xwcac4;9 xpo an identical disk ro9 cj)cy; a- ;pyq4xwsxtating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at (pdc: myk-fv29tsqm +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 a l5)vrs)kesld f5)6 vjt the center of a rotating merry-go-round platform of radius 3.0 m and moment of inlv))ld6ej 5k )fv5rssertia 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 mer: 8)eys,q nlben l83q22nropery-go-round is rotating freely with an angular velocity of 0.8 )el8 ry n2: 2qpl8qon3beens,$rad/s$ 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 eventualk9;wusoy m nofqjm:91 f 6bk,7ly collapses into a white dwarf, losing about half its mass in the process, and winding up witqy9s9u, f6om : ;w jokfnk17bmh a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in 91qme 99; g zq-goeimj*v8ge mexcess 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 4*1gzt us qg3c$ 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 avfs ,g49gw dszyu/ r3,t rest, that can rotate freely without friction. T3 grwsv 9dz/fy,4gu, she 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 aqamisk9 o( f8*66 ulzj 6.5-m diameter merry-go-round turntable that is mounted on frictio z(l96*ofaqjki8 s6um nless 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+9zv9u1fvqge*v8 3rww uv0j q 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 of rope unwinds from the spool? How far w39 1*v rw8gfvqve0zu+q9vj u does the spool’s center of mass move?

  The Moon orbits the Earth sucpy:3aap- u:kno :hp w*3yiu j.h that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (:noaj:3 y* :pi uy.a3-phpuwk In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 1 m/mbu m;/mey)h :$s^2$ 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 radius 0.20 m ae-4n*+ginunop :6p 3kq1 t lekcquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.qnpk-e* 4ktgni+1 ue l6op:n3    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.0505vzx 0(q8 -iuft x)gou kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.00q)(zu fxv-8xu 5 io0tg50 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 from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, haet*oq /8j2o vow is the angular 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 size of our Sun, but with mass 8.0 times as great n7.+hycb+0b8skzo zrxvbi( b8v/s okjzn*79, were rotating at a0/(zbvz+cbxr zyhon bv n7+b.s*k 8kjos789i 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 the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution autobxht)q9b m6pwdu,9nij 3t- 9i mobile 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 ipb9d, 3 niqx-h6 )tmwj t9u9bflywheel “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 an 3(7ett0nzvep $H_2O$ 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 kznh04vi xvknux tkqc5s z3, (,9d,ks+ u2 k:xhorizontal 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 pivot freely (i.e., we ignore fdg y+1js,/pfue9k0 z;il 8 huuriction) 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 thezyj8uh1/,9gid;fk u+sple u0 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 floor, and we weuhzm: p gaeq74f./3iant to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step :pu4e .g ih3aze/ qfm7has height h, where h

  A bicyclist traveling with speed v=4.2*q4u:f wwf,-od.6i zydz oj j1m/s on a flat road is making a turn with a radius The ,* jz f-o:wy4 jzo6fdw.q1i udforces acting on 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 a sling of length 1.5 c-ac y ).rnur7m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm afte7rr-.n ucy )car 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 cylindern, m7av7ert 6p and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and witpm7tervn a7 ,6h arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindrical plates,a1trk 4fyy w w+vb9a6; of masskf4ryv; 9aw1b6wyat + $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 Fig5 6ik;ur,osn0ic o, ub. 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 thu ,i k;56si crboonu,0e loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, bu,hta9xpg wgx+ y 15(81jsjb lgt do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutionh:p ry .+en4y+ sgnzs2s as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the an yznr+s:sp4+2 yen gh.gular 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