A bicycle odometer (which measures distance traveled) is attached near the t*2 io1krt6mv wheel hub and is designed for 27-inch wheels. What vk6mo 1*i2ttr happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a poi /p+ lwty4xbvp(z/t* o ip)1nqr4 dpy7nt on the rim have radiaott dlyv/pzp/n)yp*1w 4 (rx+4biq 7 pl and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be descriy g g-+wgpnyi63yzj131ew t 8vbed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? vw2(- pufix5h*ybu5p h;s eo8Explain.

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 heb+qh -i; le s+8v+hn 7i.jpcjrad than when your arms are stretched out in front of you? A diagram may help you to avs7 creqi +h+ijj8 ;b-plhn+. nswer this.

A 21-speed bicycle has : +.k4ubeyf7f2rbsfi seven sprockets at the rear wheel and three at the pedal cranks.f.+k:r by sife 47b2fu In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legs with fj;,1wq(ahe*+t + esdn l9:m(gmskzie lesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of gasm l(k1ij* m9zd(e;s++teew: ,qn hmass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow beam?ksh-2u z04xst

If the net force on 3;z78lx- j f)hmvt2k wb/luj ma system is zero, is the net torque also zero? If the net torque on a system is zero, is the net f8/3)ujkz vxmw2j;l hf 7m-lbtorce zero?

Two inclines have the same heg rk,ic9x:mgz**:vf y ight but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the br g9mx fc**yi g,kvz::ottom be greater? Explain.

Two solid spheres simultaneously start i zj*g/; y089j23waw vrubkhb rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. 2yi9rwb bhj;wg8kj az *30 u/vWhich reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radix8g5 eq ir;iv6us and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Whi;qvr 6xii5g8ech 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 movingb9h 6 *dcbzg/3/rg bgb or rotating objects eventually slow down and stop. Exp3 r*d9bhbg/ b 6b/zcgglain.

If there were a great migration of people towapem rw+ kh0:,nrd the Earth’s equator, how would thiw,+hpn er: m0ks affect the length of the day?

Can the diver of Fig. 81dw loc2vmw/1ug2 c*jb z.8j y–29 do a somersault without having any initial rotation when she leaves the w .j 28lzj w1dvg *cu2oy1mbc/board?

The moment of inertia of a rotating s,nsdq 4 n +m(th :jihghj1(hu9olid disk about an axis through its center of m1j :gqtmu, hs9hhji4n ( +ndh(ass 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 :93ot5w: zg.rtgn/n. c qkwky holding a 2-kg mass in each outstretn9ngcg3 t w.z qr/w.ot5y:k k:ched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hollzrgr8ypv.fn) ;8n04 5p wi parh 1kt3zow and the other is solid. Describe an experiment to determine which is which.3 hnp)8 vniptrw4k5zrzfrgya.8 ;0p 1

The angular velocity of a wheel rotating on a horizonta dn,jktshv,69 sc 17qtl 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 increasis9dvs6h q7t ,njkt,1cng or decreasing?

Suppose you are standia xy*zqhm09j+ ng on the edge of a large freely rotating turntable. What happens if you walk toward the center?x*a 9mq0zy+j h

A shortstop may leap;t.zk2gx: kn*6pdbe(d+p o t-v rsy *e into the air to catch a ball and throw it quickly. As he throws d;+ zot*:eet6 g d kb vn(ypk-*spr.2xthe ball, the upper part of his body rotates. If you look quickly 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 angular momentum, discussl3( m .xyjxgy):.i jnsi oq4(s why a helicopter must have more than one rotor (or proni s4.3 .(yoxmjlgjsi:q( xy)peller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (zd9gio*9 9-ktu bq1g fa) 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 anscu, nh n+0wf; amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moc+0; nsnuwfh, on, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the uk ,6 2exdje*pMoon, 380,000 km from Earth. The beam diverges a2* e dek,x6pujt an angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotat9;tyhy; plnd8e at a rate of 6500 rpm. When the motor is turned off during operation, the blade thl8 p;9;dnyys 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 to another child. If1 lz3v.hdet2ty4 kv(a the ball makes 15.0 revolutions, what z3.ydh 2l(vk a1ve 4ttis its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revo pjh4u pl.ppg*3 zuy1lq kw+::lutions do the whe4.wl kq:u:p*u g zp+ph3p1y jlels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculatfxwvjtaf3 qe522c5n2 )z*rf hg(4mv d e its angular v*v 4tvahzmej5d5q) x 2wgf 22f(3cf nrelocity 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 4.9ldqkowphd7qrn, 38j m -h6.l 0 s (Fig. 8–38). (a) What is the linear speed of am8jq 3k9oldlh d.67-rh, qpnw child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbi+y csfd 8s(53x-,7aq phxhvzqt around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speev3exvc/6ath1qpd. p ew-q**j ;o7lw dd 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) mus( d+m kffeu/,kt a centrifuge rotate if a particle 7.0 cm from the axis of rotation is to experience an acceed/ ,ffk+um (kleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center fg0pitoyo -*dmv) . 5,a9w onqurom 130 rpm to 280 rpmon 5wdv9ypgt oqiu,a m.0)o*- 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 radius 7 /dvm/hci8w2e3mirs va x/-j $R_1$ 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, astronauyk 1pfdhesx/k.1 1x( wts 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 1ksxwp xd1/1 hk(e. fyno rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest 2ynxty 9 v;l1 ;3 zpsxk,qyki1to 15,000 rpm in 220 s. Through how many revolutions ki3xs1t qk;n92yxp yl,1y zv;did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Ca*k)vxyg e b61zcgs3s-i3b8 jdlculate
(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 4j::a2n)frbo -llcfastresses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.raa cnfj)lob:- f4 :l20 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from)7vvo5wt3 ak l yhcx30 *s4(k+nm pnjv 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in this oysmkpn 34k0c)h (ajwlx7ntv*3 +5 vvtime?    m

  A cooling fan is turned off when it is running at 850rev/mi:auee: w oz(6nn It turns 1500 revolutions before it comes to a z:o6(awen u:e 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 y+ /bo4pz+mcau.px xjsz 593u the car reduces its speed uniformly from 95km/h to 45km/h The tires hp3z+zmc9o x uj5ux+.ap/ b4syave 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 rdf5h3 u2mbqbvm9u3)7 2i6:utxo3z qd9 dxd othe car reduces its speed uniformly from 95km/h to 45km/h Th36z39 fro2ud qix95xdbmu ) vt 7q mdu3hd:b2oe tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each pedal when cl2moc*e;izlw 8b4up+ cimbing a hill. The pedals rotate in a b2wl+*io8m4ce; cpzu 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. sv)y4ye/c,dq What is the magnitude of )yyedv/4 csq, the torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel showne x 3nre:c(v2i v9;e*orpp(p l in Fig. 8–39. Assume that a friction tp ex23(*:(ceovr re;i9ln vpporque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the endsiyj*y lv+ e,ur gxi,q,j m1d0vqn7/r ( 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 direction of the net torque on mq+ 1( nr,u07vi j eiy,rq* gdxjyvl/,this system.

  The bolts on the cylinder head of an engine require tigthrtoa-yg-ykl/ ,doech)- i j9)q25uhtening to a torque of 389ly--dh/a- ogy oe hi25,u)k tjqrt)c $m \cdot N$ 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 radiuo9( 904b;ujkhb0 yo 3cweqlz,8qnk pos 0.648 m when the axi; 0e,obqoj 89nyu9kpzh0 l4kcq 3wb o(s of rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle q w94wt/c x.zxwheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can bewwc 4 .xz9xt/q ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in1d 61/:g3nc wdwane4 akh 4zri a horizontal circle of radius 1.2 m. Cal6kg / :ww4a3dz 1n h1cn4eidraculate
(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 tbb0+ka2i d9+6w x o/8zhrb -o paqp2cpotter’s wheel rotating at constant angular speed p8aio+d2 btwrp2acxb -+06qh obz9 k /(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 inertia osmm- nd f 94p vwd*yer,5/2waff the array of point objects shown in Fig. 8–43 absa ,*w 2y9 md/df venpmr-54fwout
(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 tuh60 ymc3sv 7lwo oxygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the correct rate, n5u*o u7lht*29;nfd; ,)gt rfj4grq,hp wpvr engineers fire four tangential rockets as shown in Fig. 8–44. If thevhdw7u**r,,qro )2pl5rjp4f uh ;9gtg; n tnf 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 wio8 yjf1ch f6xp ju 60; kq,se6wig1j14ody o3hth a radius of 8.50 cm and a mass of 0.58o1h6y 8;pyf6qooj 0uc11wjikjs ,fe64 x dgh30 kg. Calculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swi8 p334zv cu .wz2lczwhngs a bat, accelerating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merry1 tbo25ylpszd rcyrx 9gt:8.6-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s:ybtl821 rr.5tpy6s odgc9x z . 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 shut1zl(ta 8:ljre off and is eventually brought uniformly to rest by a friljr 8 et(z1al:ctional 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 acn1m4 2ss35gt4dwdi s fcelerates a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely byshx2h*v0d*gwl9 g a3q the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is hxwh*sg 9 v3*da20 qlgaccelerated 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, asokq8)s)5 :cb hpni,t5g iex d; shown in Fig. 8–46.k5sd8c) ; hnb ixqi5p,ot:)eg
(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,m/jyoy+ 8 ok:f $m_1$ 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 from rest within foi h;hg;b; x2rnur full turns (revolutions) ai;rhg2;hx n;bnd 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 he z hbo+u8a9vgyes v-58u+; vmas a moment of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque of v fjcc lhkh2b.gb7+ *muc ,/gek o41h5280 $m \cdot N$ 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 downv :/)zjp rfjl(f-h8 u bku6o11chrq-r a lane at 3.3 1oqfj()8h-uk1: / c b u-jvrlfz6phrr$m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth withwj 3 1r5k );pi* chbm/jfsv1a,yfi2 eh respect to the Sun as the sum of two trv spai; 21ck1jj5w 3/e,*yhfhm) fib erms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius p9 akgj1pf6s/3vf wp(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? Assume it is 6fv 93/was(jkg pfpp1 a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls witu6qea-pdr86h *aco, ghout slipping down a 30o6hc 8daeaqp, g*-6 ru.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+7buwsp hrhz g 1t990no 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 conseb1pg h+t9zr w09sn7h urvation of energy.]    $m/s$

  What is the angular moehj 3wv7l(uv p6 +t2ukmentum of a 0.210-kg ball rotating on the end of a thin string in a circle of radiu l 3v2jwh+u(tpuevk 67s 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 cylindrical grind4 ouset**dgo55,wmv xing wheel of radius 18 cm when rotating at 1500 rpm? 5*mws5 etx,4 udovo*g    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotating at a q(i2jlria zq)w9rmor4hgrh)r(-9b lm ; *x 0 erate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotariahr2eml-*rgw) oh9zj()l m(r4 ixq9 0r bq;tion 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.z5sa (qp5:.uqt*ssutl ,c u* b 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 sub,z st *q*pu.q:t5u(sa c5 ls 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 frl/14)bohr sw(pc . nmage0js:om an initial rate of 1.0 rev a :/bopsl1c)(mhn4g sw0.rejevery 2.0 s 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 j2 - hi6wc4xk2b5tzet 90 q*kdfl kew9through 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 shaped as a flat disk of radius 8.0 cm, onto the center of the rotatin9l-4w2eh kzqe 9 ci kfxk*t2b0w6 jt5dg wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular d cw- *vrh ek/e12sb)rxj.-nsmomentum 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 t)lwomi, o0kh-e. 8cj. jhuv3ihe 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 inertiw: 1bmkxzfut+87h3a n a I is dropped onto an identical disk r+k17xmn wfa8:zbtu3hotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4 hg5je8ld2t7 wf6uw m lu42,ug$rev/s$ 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 mqaeq/ df7 (eso/bb5l )erry-go-round platform of radius 3.0 m and moment of ine/ a7/ )5ebf(qqoselbdrtia 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-cgenyy n6lo j5+lhevy1q*()9go-round is rotating freely with an angular velocity of 0. +g n9nl*1eoyyj( l)h56qevcy8 $rad/s$ 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 co5j7fn.i hriv.llapses into a white dwarf, losing about half its mass in the process, and winding up with a radius .jhii5r.v7 nf1.0% 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 wh 3z0f .ilcu.oxqv:0 winds in excess of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass ufx j7d1 7n:w2tbsy3b$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, that can rotate freely without xre+u vx8;87 hslnd1q1cna(o friction. The moment of inertia of oclsqxavh8rux( d; n71 n8+1ethe 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 dia1zkzt*j4 xi b9pbqg( b0ga6o7meter merry-go-round turntable that is mounted on frictionless bbij61*gx(7a bzk gzq94to 0 pbearings 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 en ob xpy-wd4)5w kh3j zt5fb2z;d 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 tpx w o-fj25k5zt)3hbdwy; bz4he spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always far3d,nd0-u(wp l:enmn yv o,*lces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particley m*nnenlp,uwo ,l 3(-vddr :0 orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rests7 5t9hfg*c:b4il jpi 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 shapeab 4crs*v:m g7tmh7ia :. :ioy of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval a i.mbr7*yhaiv4c so m:::ga7tt constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.0v7.c+6n9vv1crfso :uegro p/49ylh d 2/ksux50 kg and diameter 0.075 m, joined by a (concentric) thin so k vvhcg pcvsx/.r2sy fd9 u/7n9ou:e16ro+4 llid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular gc 73q0l(k drmspeed 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 g2-pobjz,, gdan3n xw8reat, were rotating at aodabz g,3n2pxj8w ,-n 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 (tt*uii; x;.ys tynk )low-pollution automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total maiu.;)s(tt* x ;ntk yyiss 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 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 illustrat4s; :ot2yyq k n.t87m z6weqgles 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) i;re+dm a2ur+ fs rolling on a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we ignore frictionxz 9 +r7dikx4vdv-r .c86qcbs) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of -s+ v7dk8 c4x czrdq9.ri6xbv release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standi lr s.7ijdys/9 d 1m/)imuhnq.ng vertically on the floor, and we want to exert a horizonjlr9m.ni .sm)i7/d qysu / 1dhtal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed vb/+ovjda ydcex 5 21;6sicx+m =4.2m/s on a flat road is making a turn with a radius The forcesdd2sx e+ocbxiv;jam+/6c1 y5 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 z9ll66bi fw/tdd 9 h7qsling 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.f7dwq 6/99zld6tih bl 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 cylinder7gosvd;+ h (i5ww;nir and her arms as thin rods, making reasonable estimates for the dimensions. Then calculatg( i ;vih;d+sww75 rnoe the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindricalp5bx1i u5: tjx,ib4 zb plates, of mab4x1uxj55 ,ibb: ipztss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 Fv tyq:18wxe3ini(s- qig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to s3w-iexq yi vn(:q1 t8reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do notst;np3h2 5k*uooy6z4 5unuzb assume $r\ll R$ h =    (R-r)

  The tires of a car m3nqfse)os5h ck g*blg38x)j 8 ake 85 revolutions 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 ang s*sghk3lo )8bf8cjxq 3)gn5eular 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