A bicycle odometer (which measures distance traveled) is a,) pg3otd+ forvknv.3gk48gp ttached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?tp rv)okvkfpog g4+ d, 38.g3n

Suppose a disk rotates a.-qnv3 a ;6pe;n vruab; qbp5et constant angular velocity. Does a point on the rim have radial and/or tangen.;euq3 aeabnbpvvq -p65 ;n; rtial 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 linear acceleration change?

Could a nonrigid body be described by a single value of the angular veg; +w9fze6ay0y9fvh flocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Explain.vjk3c1a *r(-pg djf t7

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 head thanrs5n+ireqy. 9 when your arms are stretched out in front of you? A diagram9+rsnier.5q y may help you to answer this.

A 21-speed bicycle has seven sprockets at the re xa 89ar)nvs*mar wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a larva9 xmrs *)8nage rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being 2(/ bnq h ufgpmsj-,(igu);hqable to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). Oghsq /gb-j,iqm2;hp( fnu()u n the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a ge 0i 2;pckpl/long, narrow beam?

If the net force on a system is zero, is the net t3, *ugdr7).nz 3b-j jjijxhp zorque also zero? If the net torqunrj3hjx- ,uj3z jp dg7*izb.)e on a system is zero, is the net force zero?

Two inclines have the same height ;drw rex:1w7b-s3dijyd +rab 23 xk:ebut make different angles with the horizontal. The same steel ball is rolled down each incline. On wx e:jr -3kbw sb2 ;dw7y+r3d iade1:xrhich incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simulta a:s1j 88zoex 0f,yvupneously start rolling (from rest) down an incline. 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 y e1,uf: oxs zav08j8pthere? 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 from-z(o.ixzap)xjloqijg ,r.). rest at the top of an incline. Wh . ji)gzxx(oriz.q.ja ol)- ,pich reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most movv, o.o.ibgb +drr1et1gp0 tk2 ing or rotating objects eventually slow down andor0be .1b v2+igg, pdork.1tt stop. Explain.

If there were a great migration of people toward the Earth’s equator, how woupqnc *tc: )yy8ld this affect the lecqn8cp y :ty)*ngth of the day?

Can the diver of Fig. 8–29 do a somersasyo1c ukml.k 7i 4)eb 8ljy9;kult without having any initial rotation when she leaky4bji u1ey;c7 m 9s)ol8lk.k ves the board?

The moment of inertia of anngp 3 t 6.7fj9vojki5 rotating solid disk about an axis through its center og3n. tnjp5fkoj7 v96 if mass is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg mass i dg-gmjfq,j md /x0x*g io;hn/y,7c3jn each outstretched hand. If you suddenly d jh,03,g/j 7xodnm /jqg;gy ixcf*-md rop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. Ho5f+zwck.87zbf5y0l2 j +o ig qxa)mfdwever, one is hollow and the other is solid.dk8+wb0 q+if5l )fzj am 5yc f7xo2zg. Describe an experiment to determine which is which.

The angular velocity of ao1 2kxt/n9 qujkp;7 euyw3x 1o wheel rotating on a horizontal axle points west. In what direction is the linear velocity of a poknuqy7w p/;x2k 1xo1o e9j3utint on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge ofds /+f/c59:jz)73e eahe :u7sl pw niggumqw: a large freely rotating turntable. What happens if you walk towardpszg7iueu eaf :s+ /79: 5:3l/)q wedwn jmghc the center?

A shortstop may leap into the air to catch a ball and throw it quickly. 0oyjya333tg 2 pbh p1tAs he throws the ball, the upper part of his body rotates. If you look quitt h1yp30y j 2p3oagb3ckly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of ang9 w:ss.fx/ wbleyqkck ,xr jpz .2)a,4ular momentum, discuss why a helicopter must have more than one rot. 2):/pk b jrwc,s,ez9k. f4xy aqsxlwor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angle /nh ,vkm2*hdij.3gnvb h;r3n s in radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazini) 8 w 0vu)qmruu:dz6mg coincidence. Calculate, using the informatioi u8u0))vu dmzwrq:6mn inside the Front Cover, 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 from Earth. The *)nk7b8rlfo n beam diverges at an angle8 nnol f)b*7kr $\theta$ (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,zv ( c,dh8bwn of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in(,w h b8zcvn,d 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a l wm85u 3;ri2cgcac wn8z/hv(cevel floor 3.5 m to another child. If the ball makes 15.0 revolutions, what is its diamcu85 cr hmwv ;nz3/ag(c w2c8ieter?    m

  A bicycle with tires 68 cm in diameter travel*8egknlx .-lt s 8.0 km. How many revolutions do the wheels ma8*g x-.knlle tke?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. l)*k:y 6 yr5tkg* mtw5zm cn1dCalculate its angular velocity 1y wg)5*my nmck* ktrd:6ltz5in $rad/s$ $\omega$ =    $rad/sec$
(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 compl* dtcw h+anf+v1/)m f2m3pug qete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child swm+cu)/n+f v df qth1ag*2mp3 eated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) ip 8s ah*yiwzy ,dit268n its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of i9 x .jwzy d-pa,s;z(sa point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0 cm from s g/ rlk4vb u 1kxr(9/ 9y(ugdrr8hj5mthe axis of rotati /bm 98(jgu49ydx klsrrk/(h1rv ur 5gon is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its cent4gb)l6 ; hvl;b.bnnp l- d qouh2a/pf0er from 130 rpm to 280 rpm in 4.0 s. Determinef0h4q/vpu-l pgod2)nbll6 h ;a ;nb. b
(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 radiukdh 809j b73fui)jplwfvu3d 0a92 sj ps $R_1$ 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 aba6zuk 2w c,x9ooard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval.o9 w2k6uac,z x 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 uniformm3gu+ wf5q322r omv y0s0jam+ *chscp lctu;+ly from rest to 15,000 rpm in 220 s. Through how s2 c0ao sf m*lj3cm+pwcym3q 2th0++g5uvru;many revolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2g:7n.tr9re do .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 1pd mqgwhxbr(4z1 r1it5/4ilthe stresses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its fi1zb td1gw4ixq rp4l/15m hri(nal 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 tbj p; m374gqxyo 360 rpm in 6.5 s 7qjy;p3m4xbg . How far will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it isj9t9-nxu, id6 s,: xfeyemiu7 running at 850rev/min It turns 1500 revolutions before it come6nymjet:uf9es-x iu 79,, dx is 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 revojdav2xz uz; .:lutions as the car reduces its speed uniformly from 95km/h t;.2x udzz:va jo 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 its speed unifoviqhhj/x 9;al l y, tga0u54(xrmly from 95kma,th5ih x uy9l0l(a g j/v;xq4/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

  A 55-kg person riding a bike puts all her wew;qod3f0d(o o ight on each pedal when climbing a hill. The pedals rotate in a;0od o3w o(dfq 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 enc5mlfrchr4x7gmu;n,*n: f w: *3bk f+z4lwqv d of a door 74 cm wide. What is the magnitude of the torque if the force is exertedfnwcwhc mn 3*m; *r4fl+q45:f7k r,uvlb zgx:
(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 thendfdl3;8 qp.umf p6o af a) 9vm5nz3)c wheel shown in Fig. 8–39. Assume that a friction torque of 0.4ud6qpapz3);f)m5v9ndo8c 3la f .m nf $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a mq.m4emhlwp )7)8ajuk assless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude7hp.q)8kmm u)w4 a lje and direction of the net torque on this system.

  The bolts on the cylinds:q4:58si tjw8gd qjy er head of an engine require tightening to a torque of 38 jdjiqwq:48g:s tys58$m \cdot N$ 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 when677/fg1q ;enusp t+fuy ua ,uunzvm8+ the axis of rotation is throf8u n;eu7p f,7+1 g u6nz/y+mtuqav suugh its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle/w- nhalo18 rl wheel 66.7 cm in diameter. The rim and tire have a combined maswln8l a-ro 1h/s of 1.25 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, lx1t+kk+ 3dx 2x5icape ight rod is rotated in a horizontal circle of radi 3a 5t+21excdxix k+pkus 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 whe*7 yvhb8aiea2id5 vgfctx6lu 8b4 8h 8sne,p1 el rotating at constant angular speed (Fig. 8–42). The friction force betweenet8 65g7 l,d1iuv 4che*8bfpy8bxn aish82 va her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objects sh9g:njio)rq qi+0o ) )(pebed hown in Fig. 8–43 aboj)o:qo) ne9ihd)i q +g0br(pe ut
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen atoms whose total mass tfr)0 rdpn*q(z 1k)j dis $5.3 \times10^{ -26}$ 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)e c+z/ ,sazs5qetw: v satellite spinning at the correct rate, engineers fire four tav+se5 :), cezaqz/twsngential 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 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 a-gu q000g8q6crvmdn lh ;-h ucm and a mass of 0.580 kg. Cacdlh8v 0-q6gq-uu0 ;g0anmr hlculate
(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 fr al8g 04nlog+szpff5 (om 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 m4b2jfxuwv)6 hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in jm6 f2uxvb4w)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 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 snnhy e738yr+z7,e)7bfdpc pbhut off and is eventually brought uniformly to rest 7z7 nbpe7b+3 )8h,enrfyp ycd by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a 3.6:p zv23:(1iz-dzxlfnfrva5 h-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 c9b-ouy;opm 6solely by the action of the forearm, which rotates about the elbow joint under the -;c9ub6o my opaction 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 considekt -b m/6xx 2rv07 z4jdaz wwwdd)ys.*red a long thin rod, as shown in Fig. 8–46d/wm j 4x- d76*bs2)va wzzydx0t k.rw.
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consists of two masses, g5tr2 :ux4cwt$m_1$ 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 hammer ;.udmalv*)*mpyh p t* from rest within four full turns (revolutions) and releases it at a speed of 2 t;ammhl.*u*)y vdpp*8 $m/s$ 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 ( 1ktt tj071 uqgs6asaa 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 de nm; rbm8-x(pbvelops a torque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls without slipping down8*tj)dac k2wlcj7 r,k a lane atrcckl djat,8w2*kj7) 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with r*q xxoyh+-js*7 1mmch espect to the Sun as the sum of twohy sq7mxox- *1j+*c hm terms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 6)xo1obm:2n.t3 g- nw xojkwq1640 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 pn31n2m oowxj6).b- qxotk g :wer 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls wie6 a8 1saskn j8 gznv7uc2akrq ,rs/7-thout slipping down a 30,k7j ssasz6rnkvr8-a 8 g7a1 ncueq2/.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 tiptoq;yu51 d 5.m+)d7qrqn 2md6jdxwlk . It starts to fall and its lower end does not slip. What will be the speed of the upper e uqlo5 d d2j5m;d w)r.yq md1+k6qnxt7nd of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum oft7d8xmnp;h.j a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an angula xd.phtm7n;8jr speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg unif j;ri.olbk(e 2orm cylindrical grinding wheel of radius 18 cm wh.jo2( kbri;el en 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 side, on pqspe z wj0sbz1o0785mx+0xg a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a hw075b8+m xgp1o zss0z0qpe xjorizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce her moment dco;qm0qc*:i.n9hz s of inertia by a factor of about 3.5 when .dq0*co;nim zh 9sqc: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 rm6wlum;i8ddzjj9 u5 2ate from an initial rate of 1.0 rev every 2.0 wm z u6u djm5;i9djl82s to a final 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 it0lij;yv:3 chts 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 plit30y;: c jhvotter 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 momentum2we qam /c+n9 kn+)hht of a figure skater 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 momentum of o;ntw) jspz.xr a 7n*i0tdya489 )fq bthe 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 inertihjb1yrm,r/)xt go ) svo8, n)xa I is dropped onto an identical disk rotating)x srx,htb8)/n, om vjy1ro)g at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 238 avfp nuksa-ej;19v .4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotating merry-go-round cc, 0b-x6 aedifp0bom0rc34b platform of radius 3.0 m and moment of0 -a cfrcop46ib0 0 dmb,ecx3b 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-round is rotating freely with an angular velq6+fub) 9e gutocity of 0.qgu bufe+ 9t6)8 $rad/s$ 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 abocgktxo.8h .w; ut half its mass in the process, and winding up with a radius 1.0% of its existing radiusk.xt8ch;wo g. . 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 excess o gff:w .i2.4pbeabjf3 f 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 zd2a9 :p1y:p/a;b dqutk idg3$ 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 rr v 3vp; 8 )q+ 6+0dng,focbt3mbodgfrest, that can rotate freely without friction. The moment of inertia of the person plus the platd gg mv06or qd3+;3 cp+nb vf)rb8t,ofform 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 me pu.53rsttp zzzn+yq rt ls0 5p9+lf7+rry-go-round turntable that is mounted on frictionless bearings and has a moment of in+zr+tsfpt ls557 nq9lu rp03+z y. pztertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of t)p3ooj)vk 4one+8n tghe 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 theg no +tn8ejv3okpo))4 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 al +7c3 fqomsbq d;k-n7eways faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital a;s kn7e qmbd7oc3 q-+fngular momentum. (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;d6z*5xn3 ests 7ehxb 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 radius 0.2s*vp6k).7 -zce:2uc4mcqyy za q0 iy)kc fb 7i0 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered i7ike0m.c * vp7zu-z 4 kcy) s2yq)ca q6:ycfbby the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.0oreuzdjie:f-7s+ :p+d-f 2jw 50 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 2:df +7 jjzdue+rofi -sw-p:ekg 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, how is the angular speed of the rear wl(q7x5 qe:c2ej m jb5qsey 0:yheel ($\omega_R$) 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 mej6 g (3 ky8mem38bemzass 8.0 times as great, were rotating at a speed of 1.0 rev3gjk ezme8mb e6(my38olution 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 automobile is for it to :u-l*/3.htfv cw hlyw 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 should3.ywtlcf /:-u v h*lwh 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 illustrai5l* /x5y-(uk 3lajjm/ ozabmtes 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 onoto.lr*,/g dhj 6gwop5 f33oo 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 cau)2g:b2 v a-i tccgs:ln 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 ava -2b)tl2c cug:sg:i cceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the floor, and we nah 0jhvaazq8 7*ve69want to exert a horizontal force F at ihahzvn e*890 aa j76qvts 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 speed v=4.2m/s on a flat road is making a turn witqc5y8e) dc 8u(ijv2srh a radius The forces acting on the cyclist and cycle are yu2 8ceqjcrv)(8s5 idthe 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 1aarz7x/ /tfx..5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating 7z /r/aaxx.t fit from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skatezeh wtli(2*zzn 3(rdp/td5f .r’s body as a solid cylinder and her arms as thin rods, making reasonable estimartp.z (fzni*dd3w e/ 2( t5lhztes for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which cfgkc3k:7i99qfw zd) eonsists of two cylindrical plates, of m )9wfqzf3gk dk97i ec:ass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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. n + +5inarj5ku1r 62x2pfqk ,o e7hqnw8–58. 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 without leaving the trane 72fn,jq 5 6 urkpi5qoa+ r1wh2n+kxck? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not a8*4a ii j6fwrlssume $r\ll R$ h =    (R-r)

  The tires of a car make 85jpl-m/dum* je u3- 8cw revolutions as the car reduces its speed uniformly frcje mj- u8m/-pdl3 wu*om 90km/h to 60km/h The tires have a 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