A bicycle odometer (which measures distan,/tesv )4mfu7m ts*h lce traveled) is attached near the wheel hub and is designed for stlf4e tvsm)u,mh*7/ 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at cons6r tg3m:g mam.tant angular 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 of either componem:3a .gmtgrm6 nt of linear acceleration change?

Could a nonrigid body be described by a single value of the angular tmy j 7hgs49d6godvi n0nk9:2velocity $\omega$ Explain.

Can a small force ever exert a greater torque thangi :k0bnfj -s 2z8n:ae a larger force? Explain.

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

Why is it more difficult to do a ps2sux9az6*vs p.)hm1) yzutsit-up with your hands behind your head than when your arms are stretched )z mxapty9ssz svup 1)h*.u26out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at e l3.5 jakxo,lthe rear wheel and three at the pedal cranks. In which gear is it harder to pex5j al3k,lo.edal, 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 o46 v j 3gimoz(ce.c3q99nphye n being able to run fast have slender lower legs with flesh and muscle conceng6im evoj c yq93p9z.ne(43chtrated 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 walkers (Fig. 8–35) carry a long, narr(/d dxdtn-p)v 4:kf /jrikf ax5a;9t vow beam?

If the net force on a system is zero, is the net torque also zero+kea9)i l ru 4rbt5hj,? If the net torque oethl )4ir95,bjkr+ aun a system is zero, is the net force zero?

Two inclines have the same height but make d(:e k9 zwp(uekifferent angles with the horizontal. The same steel ball is rolled down each i(puk 9:we(kzencline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rzehyw5/z+2 cbbr.lw, olling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottwyhlz br+z5 2,e.cb/w om 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 s7mei6 txd5f:z1h*snrhy 3do6 r*v6c)zho, l ptart 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 kineth d3pmx*s lt fi,vzro: 1d6hyz75o6he6n )*rc ic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular m0a81 r6vh t 2qjo.h(qg0.upge vv;nczomentum are conserved. Yet most moving or rotating objects eventually slqu t1 vhc(g0z8 re ;v20 np.j6oqhav.gow down and stop. Explain.

If there were a great migration of people toward the Earth’s equad 3hyzss+du:lmfhhnk x b3n.s)6( :*ator, how would this affect the length kn3h xdu+:dhaly.bs*s(ns 3)6m hfz:of the day?

Can the diver of Fig. 8–29 do a somersault without having any iniw7unk )m, xfrj vu9nr9-6ck *atial rotation when she leaves*fxv9wk,9)- 6 nmc ajurrk7 un the board?

The moment of inertia of a rotating solid disk about an axis thp:tl,c5i1o(c es7.xwaxjtg f u1kfp/e e*5q+rough its center of mass is ittfg*oxap 5j5+k,fx7:e(.le1ucwcqp1 /e 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 holding a 2-kg mass in each ou,7o,odf ij;n *u5o eywtstretched 7 yw5, jooiu,;e *dofnhand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the sab( g:pr2kkvfey ybty4) dd1f ,5uod2( me mass. However, one is hollow and the other is solid. Describe an experimentp:,y 2r51 f(ktboe kgvyd)d(24y bduf to determine which is which.

The angular velocity of a wheel rotating on a horizonta gr /zd+jp5wk6i13ked 6uqd v4l axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasidir1w3e+k4v dq/zgp6jk5u 6 d ng or decreasing?

Suppose you are standing on the edge of a large freely rotating turnjche2)+s4p6 nm x7coyn.lz (ei0q) ig table. What happens ()gi4l7mye j occ.pszeq0 2hnnx6+i )if you walk toward the center?

A shortstop may leap into the airxvj j))wb)+mhj*0lt+sq/ 0v3x 1kvij kw+ xuc to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you w/xqh v)j+) juvci m1 *03l)+k 0svjtxx+kbwjlook quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the lawb+iem d hx8(-g of conservation of angular momentum, discuss why a helicopter must have more than one rotor (xg+he-mi (d8bor propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (audhiw 9.m(boc3-ec q 5) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidence. Calculate,ph;oy(i+*kfz 3q -mzf jfp1uv dz-x,. using the in + ,fz31-j(fv p*xu;iy-o m k.hzqfpdzformation 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 be2ekin bn)x 74ham diverges at an axh b2ne4 7)inkngle $\theta$ (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 rql 45cze-r4j bate of 6500 rpm. When the motor is turned off durzq 4-5r 4jcebling operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m tn,-z+s;oa9jak xo8nrn25 sl w o another child. If the ball makes 15.0 revolutions, what 8nj2,n 9 -s solkxznrw+5a ;oais its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 5bx h,m/6c yaod3qb+4i+3z 2rhi hjod km. How many revolutions do the wheels majcmh oiqxa4ry5h/ +,dh b63id+ob3 2 zke?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotatehvy;i/;*)4nctwp k( 7txo ttus at 2500 rpm. Calculate its angular velocity u7pk ct;w(hi;to4t)xntv*/ y 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 revoh35w uk,5fwp.tno +i elution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from onw 5e,h5iufpt +.k 3wthe center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity o(h :odhcuvc-.eop :x m74bzm:wyah;; f the Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed o)k: y1edlte nc o22ya959hhkrf 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 rotatgcgtp8 *,7 tj3a6yo die if a particle 7.0 cm from the axis of rotation is to experience an acc7jtgacd6ogi t, y83 p*eleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates unifoi6yvm7z5 bmy 6rmly about its center from 130 rpm to 280myy66m b5vz7i rpm in 4.0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radiuwe8lu),,twmvt: c;bd s $R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, aw j1st2*bw e0l myql( rg2okmz2q+/m,stronauts aboard 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 12gy(,20 qr1w* mm e skl22 +tjwbzml/qo-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 uniform fo: 0mbruhry;:.sm 1gly from rest to 15,000 rpm in 220 s. Through how many revolutions did itr;1hou gy : 0r:bfmm.s turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5g*u guj3let +gz6zz.* h(328qcx4yt 8qwk bxr s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed jt9-5wvsd8 .ysd za8qqets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before 8vqstd ad89q5. w y-zsreaching 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 dia;blqdu-es e rfn;; 54zmeter accelerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of tb5;4- zu;e;slr fqnedhe wheel have traveled in this time?    m

  A cooling fan is turned off 3 dy uf(0zj2tjwhen it is running at 850rev/min It turns 1500 revolutions before it comes t 3jzjd(uf 20yto 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 reduces its speed uni6u ipah(p. rf:formly from u.:f pa6hir( p95km/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 makexgob 33,s9 use 65 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The s,eso3g9 b3 xutires 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 aor 6lge u3.mzgci))x4ll her weight on each pedal when climbing a hgg3el6.m4x)cz r)ui oill. The pedals 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 N on the end of a door 74 cm wide-(bh:fdxkx5wl s20 )ma6(hduch z 5yi. What is the magnitude of the torque if thf56u(c-a2ly(m:hids hd xbx5k zw )0 he force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig. /xh9wq*84 r*ko6h mxqu ozgj o;w49au8–39. Assume that a friction toz*; u9ao *ho mwkh9gqo486ru4j /qxwxrque 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 tht7*1x2+k+mfjmux jbgc-/ p t 7hbkm d7e ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod 1t mp7/fh*7bk x-kmgjxd+umj7 2+t cbis held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylindebyjet)8 zfx i0b 984mrr head of an engine require tightening to a torque of 380 898mjybx4rbe )zi ft $m \cdot N$ 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 radiul57j1ab5ev ci fv9p*ks 0.648 m when the axi 71fvekc pbv59*laj5is of rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheelqdj71,yqo a)g )opfu0 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignofqd 70o, 1gu)py) joqared (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in a ho w csdy(xv40w3rizontal cis0 vdw3xyw( 4crcle 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 y1+uo krxe +6da bowl on a potter’s wheel rotating at constant angular speeoe6kx+ +r1y udd (Fig. 8–42). The 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 ineo*z sdf:/ho. 8qtu2tp rtia of the array of point objects shown in Fig. 8–43 abso th8pod*/qz f:ut. 2out
(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 conb4;dc6uvqrs4jk:v 4 *id1o 4i u 4yfljsists 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 cylindrical ;y4fewzlxns; 8a as w2fb(nf,+lo2; wsatellite 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 ylfsa;w 8(fsf2n;onxz,+ aw2;ebw 4lforce of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 cm and a m9 ergm-lv( di:ass of 0.580 iv(e -m9: dlgrkg. 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, accelerating it from rest t//huxi9qpp oz12vs f ;o 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merrg2iz)+- d*wa:quqheny-go-round and is able to accelerate it from rest to a frequency of 15 rpm in-wn *2:)zhiq qeu+dga 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 shut off and is kczgf1mpyl 8*y b5 ,e3:e y6*h;wwsgyeventually brought uniformly helz*w85mey g ksw:y3*6,cgy bf;py1 to rest by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a fhsjz-53a 0rh bqm (7xst1fq13.6-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 by x i t14putc6c(nc/,oall is thrown solely by the action of the forearm, which rotates about the elbow joint unyipxc 6,(1n uotc4/ctder the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as shown i 8tf*ruk;;*vei2 sq4o-vwzmj n Fig. 8–46.zo4* vwevtr8ms*-qk;u j2;i f
(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 cons7 vjj)w crq1 .6 bzwz(qu+3bfbists of 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 hamm49 51hmbturg.j8ncn oer from rest within four full turns (revolutions) and releases it at a speed of h8gubt1 .rn49 5ocnjm28 $m/s$ 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 inertwe4x4 1lb1bwwddns 5- ia 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 developseaya,g- du))e 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 y3ipr1x 30qjo kg and radius 9.0 cm rolls without slipping down a lane at 3.ij 1yrq03 3xpo3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the r (45e * dy:pvs1aannlsum of two ters v1a(5l4ndry ae:pn *ms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 /k osi:x9w54; oszimekg 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 8.00 s m54kzo x 9o/wiesi;: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 withou wmku/,*iv q 0n-fpu;nt slippingm*,; /qpuf-w0 knn vui down a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is balanced vertically on its p9le:k*.vwz l1 9x bsvtip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conserv*wpx9z. l: 1lv kbv9esation of energy.]    $m/s$

  What is the angular momentum of/he 21c/5n)k z oqxegn a 0.210-kg ball rotating on the end of a thin string in a circle of radiuseo)1k//hn nc2eg qx5z 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 bod2,8nikfv*. kgqbhul f42s /nd ,:b uniform cylindrical grinding wheel of radius 18 cm when rotating at 1.doignskd b8*v bffn22k/ l,,:qb uh4500 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 at1:1iorw87pmnu e* 5m,mteaqk his side, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases t*u e71mo,m1 qwpe5m:rnat i8ko 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one showtq5/ vqcec iejqq n6/m51yx*0n in Fig. 8–29) can reduce her moment of inertia by a factor of about 5 y/qv*eq n6j tmq5q01xeicc/ 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin 0r+u l8b-kv lvrotation rate from an initial rate of 1.0 rev every 2.0 s to a final rate oflub l vv80+r-k 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 i .uqi-h.wdaaog/ 9-zmts 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 p 9q-zdoga/ a.w.miuh-otter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinning at 3.)plr (/l1bc gi mvey7,5 $rev/s$ 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 ost f1zqg8*bz-f 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 drop2.8n5kyeo r tsucm ;/2qshvp.-t f3cy ped onto an identical disk rotating at anvey.ps 3kch.2tmr-uc 5osqnf; 8/ 2ytgular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4or ze+tb ,k*:x $rev/s$ 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 merrle(m+bg btkw4 hp(/5 oy-go-round platform of r+(h bop4(e5/wbglm ktadius 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-round is rotating f,x yq) i.g-znureely with an angular velocity of 0.y,xng.) zu iq-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 coll.iar1 tt9g, ( 4+(nsc 0eb*ytqybnv)tix xl.vapses into a white dwarf, losing about half its mass in the process, and4(syx,9nv *y )cgr iebt0(.itl1tv+.a q bntx 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 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 g vu1ri.k:nt 5of 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 hay;,rii gto-gt-yb8;q w5m/+a e1e t$ 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 platfex3wbm e0r7iqas/ r/5orm, initially at rest, that can rotate freely without friction. The ea70m/e b 3rr/xswqi5moment of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m dia,t .2ijy ) s-1kc7 wqjdvoalh5meter merry-go-round turntable tha 2)l-oja,hv71 s5.kicdqtj wyt is mounted on frictionless 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 themtzv0 k,j xz8) ground with the 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?jvzt,x8)0k mz How far does the spool’s center of mass move?

  The Moon orbits the Earth such thaeki4 ( 0qrpsoqxv;o 2uv) 7gjy x,91-tqdtby /t the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbiyxru kgt;ov7 0bs ,y9pi()4d2q t1-jq xqo/evtal angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist acceleratesv6rd lb;-eg;ye(b;ax;pw n5w from rest at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of radius q8ad3t m5tx-qno8lbyb.7:mdh y- b3y 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque d7q8 -qn tydmd .ahbx-oy3 bty 3:blm58elivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each o3 k h)lgk-6pd(hy;my .op1vxo f 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.dh. gk)vo op;16h-y3 ( myxlpk010 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 thez +3nrr 9qko2v 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, were 5oi;6vniaw8 drotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collaini;vwd5 o8a6 pse 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-pollution02pjhm cpm g50 automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 2402 jppm0h5mc g0 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 illustrateh7ocwe xg s45.9* c xsiquu6u+s an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal surface at puxr hvly37k749ot+v 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 fricw; n c2l(q,srwdgldq w jj6) zu)+j0,ttion) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of,jjnw+swuw2lq)0dt r) (cl gj ,6 ;dqz 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 verticaq i2.+7w yz e0rrbuhiy zc8;7slly on the floor, and we want to exert a horizontal force uhszy 702.7qrc z8e+rwib; iy F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat rogdy8wc *wd :y3ad is making a turn with a radius The c :*y 3w8dwgdyforces 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 roc19 gbx z;ucf 06( uk.piydgz+5xe:xslk into a 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 tor 9.6 f;5uu(g:kpdy x+x sz bgecx1i0zlque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thin kv bfavdamz zk f04d24csi .2a83*k0e rods, making reasona 20vffkc2dz.m8k eka4 bdai0z3*vs4able estimates 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 assemblyc+p2i)wl l2t/x+;rvw oim /j k which consists of two cylindrical plates, of masi kljipm //xw2); covl+r+2wts $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 c1dp7c xor1.8.wb ui dlooped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point of t1c7ix8rpu1d bd. w.co he loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assg,j d8n wx01ogume $r\ll R$ h =    (R-r)

  The tires of a car mak4a 1ry)g-gumw e 85 revolutions as the car reduces its speed uniformly from 90km/h to4a uw rg-1g)ym 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