A bicycle odometer (which measures a*;26rxr o69hi,x3yagb favb*nx9nf distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch r9* nxvfbf293y;6r 6*na i oxbxh,aga wheels?

Suppose a disk rotates at constant angular velocity. Does a pointc;pg9 t/v t4zwq0.poju,yg5 p on the rim have radial and/or tangential acceleration? If the disk’qjuttw, pyz 9vo4c. 5 gp;0g/ps 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 velo1 ;dhsu)wwfg :wjs,f/+ol bx(z,g;3l(hwgmp city $\omega$ Explain.

Can a small force ever exert a greater torque than a larger06kpggwkoos o7; gx3 9 b2ya4n force? Explain.

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

Why is it more difficult to do a sit-up with your hands behind your head c:ro3 ;dvwrf cvc+zw 8no(r xr :,5/kmthan when your arms are stretched out in front of you? A dia ;8 5d c(rr : 3/+nxkcf,r:zowvomvrwcgram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three t(zah-)8h2q3 ggq bi jat the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a l8qhqzi(g) a -3bgj2h targe 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 tvc7d9: s mu/j5,z *p0rjahdg pf ,,gibeing able to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of /j:jcsv* 0 u dtafr9m ,iz 75,dg,phgpmass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow beam?ttjuc*:hr 6e sc8v2: h

If the net force on a system is zero, is thes0)a9me./7q3flg cf g f*xz sy net torque also zero? If the net torque on a system is zero, is the net force y.)fmq3lfaex gs7f/z0*gc 9s zero?

Two inclines have the same height but make different angles with the huwu);z kf t*6vorizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bozv*;t )f ukuw6ttom be greater? Explain.

Two solid spheres simuw.0x:6gdvy t yltaneously 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 there? Which has the greater total kinetic energy at the y0v w:.6dtgy xbottom?

A sphere and a cylinder have the same radiu(f z 5ic5kolxy7he 6t3s and the same mass. They start from rest at the top of an 55ihf 6(c7lykt3ezx oincline. Which 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 hhb 8mu 5ax7syh).t1 smost moving or ro7s).5uhx8 1mthahsy btating objects eventually slow down and stop. Explain.

If there were a great migk-6 c 5my o,rbq+2wljaration of people toward the Earth’s equator, how would this affect the lmj6 5co w ,kaqyblr+2-ength of the day?

Can the diver of Fig. 8–29 do a somersault without hae 0znj* n9gp1ci czw/:ving any initial rotation when she lew0c j*geizn/p n z19:caves the board?

The moment of inertia of a ryptf/e g7 d8*ba0ps) 4u5bavrotating solid disk about an axis through its center of mass is *brp5f)y v7sgp80d ae/t4aub$\frac{1}{2}WR^2$ (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 holdi6ze ixc1vvm9bp+ 2i+cz4.e rong a 2-kg mass in each outstretched hand. If you 1 2riovv+ezm. 9ei4c+ bpzxc6 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 hollowtf mx2yb;d9 jvwb*0v( and the other is solid. Describe an e xb*mb2j ;ty9v0 wv(dfxperiment to determine which is which.

The angular velocity 2 :m4l lnq9qtabv/, )+9ti3jlpml mtbof a wheel rotating on a horizontal axle points west. In what direction is the linear velocity of a point on the top of b99mt t2nqblti4 lllv/3,)jmp:qa +mthe 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 of a large freely rotating turntable. Wsq.06ble- emihat happens if you wisq6mel be.-0 alk toward the center?

A shortstop may leap into the.jdkmm8 oslu8 tr(5(p air to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legsmtpl(k8o(sj m ud r.58 rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discuss why aznt18y6/z9hixsjx( odc2tb(ebml ,4 helicopter must have more than one rotor (or propeller). Disce/( n ,b i 2htm4z1s 9xtz(c8dj6obxyluss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in raq5khnzl9uf f: z 06h)hdians: (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 amazing coigxyu2r;cym deh.-k(.z6 s 1twncidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and zcmyeh-;sguw6.d r 1.2yktx (the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km zpdul:z 2+p/sfrom Earth. The beam diverges at an angleplpd s+/2zzu: $\theta$ (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 i(3 8x5bx,tx 0uggi/f jwhfr+of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow down? j/b, uigfi xx3+fg8 (h5wt x0r   $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the ball makep)+fi 4kvln c1s 15.0 revolutions, wha fvic+l4pn1)kt is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How nelfm) s;q1jza-j. bjkve0a6* sy /l: many revolutions do the aey 6blq1jssm /) *ejnj-; .:k0vzfalwheels make?    $rev$

  (a) A grinding wheel 0.3vo79ev(eamqh29g b.qgb0 qw6 5 m in diameter rotates at 2500 rpm. Calculate its angular velocitqbe9 g7gaob9eq m qv26v.w 0h(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 revob4lww icoqx.xx.9 x iy2)5y4flution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seat yyx qxobi5w.lxicf42)x 4 9.wed 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 of3xh /f+.z y-shm eab;r 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 of a ws*wa m(el09qpoint
(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 ro2zlf-l*0czqg5 i, gaap +x -kntate if a particle 7.0 cm from the axis of rotation is to experience an ac* ,0gn-gzck-lia zp5x f2aq+l celeration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter whenj obbe6 99 41rg.jlhrel accelerates uniformly about its center from 130 rpm to 280 rpm in 4.0 s. Determine4g9r 9orb jljn6e.h 1b
(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 radius5a r tgl(b25szatu qa0tu 0/4b $R_1$ 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, astronaut.wz/t mb96y d 43xkdhcs aboard the Apollo spacecraft put themselves in9hdd zk6xbw 4/mcy3.tto a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a 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 b i-rwj:oxk( +rest to 15,000 rpm in 220 s. Through how many revolutions did it turn in thjrk+(oi- xb:wis time?    $rev$

  An automobile engine slowswb9junb(677*pm qaqv down from 4500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspee+,6pqyzi+w j spja/ji2; 7 kcwvthu:2d jets in a whirling “humajiwpi+cz2u/kqp sw2y v+j7j;h t ,:6an centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 360 rp 06;z:y xsczy6 mrqk+zm in 6.5 s. How far will a point on the edge of the wheel have tr+smz6ck60x:yzyr q z;aveled in this time?    m

  A cooling fan is turned off when it is running at 850rjfr vk/d3k*m9 n 8xzn:ev/min It turns 1500 revolutions before * :mz kk9vdjf r/n38xnit comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed uniformly frfcy /z)dw i-e6gx.i;flmcj0sx 9 fy;2 om 95km/h to 45km/h The tires have a/m;dysifxi2x6;0w-ly.) c 9fzej gf c 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 ,t -q*xk74x zm 3shxuno0: z31kjvqdu65 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.80 m.q *jdosxutzx10,k hxk3:mz7 u4q- vn3
(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 wewm: qn)p+kg 9;u(r, ak eg4gjpjsw9 lj/hz5w*ight on each pedal when climbing a hill. The ped z aejg5gqpkwukhpr9(*),m/g9jw;s +n: 4jl wals 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 4)hy)( h.7 wqex:;dp old9klesb 4osd door 74 cm wide. What is the magnitude of the torque if do;9lpl(y d:oeb7x h) eksdw) 4 qsh4.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 of0 nzhb:7pu, wg the wheel shown in Fig. 8–39. Assume that a friction tb:n,p 0g7hwuzorque 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 ends of a massless roylb8ksc,);dgzmb 7mq g * 8h)od which pivots as shown in Fig. 8–40. Initiz bm,)d)lch*gg my 8bok7qs8;ally the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head o7y7a)etmernq3er 4 e -f an engine require tightening to a torque of 38 an3 )-rtre mqey7e4e7 $m \cdot N$ 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 sphu4xx) ( ktm,b8bv ,bufr69jld ere of radius 0.648 m when the axis of rotation is tvbtx,4 k6u uf8,b r)lm9bjd(xhrough its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bn) ec k1zj54n 2v3ngd1 jrfg9ficycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignn53 cg4fj 1k)rjfd zenvg12n 9ored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod: 7efvcjizy 6q 58doy1 is rotated in a horizontal circle of ry56ciydo8q z:fj17 e vadius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating af38txil ,98ic9e qvj1 3 wrjlct constant angular speed (Fig. 8–42). The friction force between her hands 81itj 3e8lf9c 93qrcilv ,jxwand 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 pl*snl* y8egt4pz/pr . s5sw*point objects shown in Fig. 8–43 about4.n*lgpstp*ez*ys ps r58 /wl
(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 mass4vxz)exb 2 oz3 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 ratetmovqe;--ok2mjp .l3, engineers fire four tangential rockets as s-pjm 3mvk.l2ot- qo; ehown 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 unifwp y g0l5)kg xbvts/)*orm cylinder with a radius of 8.50 cm and a mass of 0.580 kg. Calcuvg xs)t bwgkl/5*p)y0late
(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, acceleeq us ci*1l;:jrating 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 /3hjujqc5 *hla small hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in l3j *ju 5hqh/c10.0 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 eventually broug1:nblxd 8pa *ss-a5zjv7ohb 4ht uniformly to rest by a frl *n4sx-7 havp1b s8:abozjd5ictional 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-kg ball at 74 p+qso+-guxn1-p *tp+o4ht a mc gy*w $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the action of the forearm, which0js a1nl7zog -rl5 b w/eo8go3 rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accew3 8oooz0s5jnlega b7l g/1r-lerated 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 in Fig. 8–43c6xzsx fpp 376xcpp7s3z3 x f6.
(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 consistsx4 (chsy.c32f0evmlu 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 thlu ( ccy064eone hammer from rest within four full turns (revolutions) and releases it at a sp6clue(4o0ny c eed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has aed( ri5yw8uzt)0u )vo 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 dy1bp*(zzjdeyh6y 3p y+-r hp 8evelops 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(zmz6iew;clpanql 4w f kl 50u35 j;1t.0 cm rolls without slipping down a lane att5w 04jcf6i3a; l(p5zq1 luw z;nel mk 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the sum of *,8 dxgv wmlt/j)bz p/two tjv8/wxb dmt,g * zl)/perms,
(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 of 7.50 m. How muccs;pze1ui) ix1153 j +bfte ach net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder. c)c1e1+us1bi ea5 ;f x i3tjpz   J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls withik(l6(ni 1gj iout slipping doik 1j(ilg6i n(wn a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is balanced vertically on its tip. It starts to hot 7:z:wl kxnv19bx ;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 conservation of en;x :tk19bhl:7 zvwxnoergy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end of a thinb zdtbz ynr* qub63 88h,jgt*32nsn4d string in a circle of radi 6h 3bnd8rj n3t,qd* g*sz2ntzb8y4ubus 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 grindin9r 8+wvkrri0agtf5sw rb6 4q+g wheel of radius 18 cm when rotating at 1500 rpt5+i6q gfbr8vkr0 r4 warw 9s+m?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at hirgi rd e6rdg c694p7x ibl*0 6nkm7f6os side, on a platform that is rotating at a rate of 1.3rev/s 4xroe c glm67n6ir d6g6r 9*0kp7ib dfIf he raises his arms to a horizontal 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 of l63jhp4 yvihs9db + :iinertia by a factor of about 3.v943+jdysb:h ip i6lh5 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 rotat2;38xh n p07ntiq/fdyzhr d )nion rate from an initial rate of 1.0 rev every 2.0 ;)/7nd38h d2rpnyn q i0xt fhzs 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 its center uw2j65ea 3kjqk s0un n sy/:v/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 5: 23 n6/kjuawkej yvn/ss0quas 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 figj8j*.c bydj qg;9bo9j ure 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 s; *g+ahnnhc. 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 disbaj;k sj/zug 86gi(3yk of moment of inertia I is dropped onto an identical disk rotating b/gjs ya(8;gukz3j6 i 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 - k f9nmev vm+ rf(krgx:-p9m/ddp-9d$rev/s$ 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-r6d,h p+ cbh5ssound platform of radius 3 dsc 6bp5+,hhs.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 dfgbxz2nn; hjs9k1m.8 q v1auor,p (xp1.k8 lfreely with an angular velocity of 0.8 2a.v;x1k(8nxdsp1p 9b,n.og kfj81qrl m uzh$rad/s$ 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 ii83 s g1ui myqvqt):v8nto a white dwarf, losing about half its mass in the process, and winq8 t13y) viu miv:8sqgding 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 o 9x abxig+q i/)gfa jd*,lh690jetw,af 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 n *b-w,ye8uuf ymws mp-5 uj:7$ 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 platfo dgi6tz0l uan bz3htd9-(r w1,rm, initially at rest, that can rotate freely without friction. The moment of inertia of the person plus z1,ndr(3 h9l-60bztdw gi autthe 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 diameter merry-go-roun1gdn- hk6aoa, 2rqk u0lew-vne :65 gpd turntable that is mounted on frictionless bearings and has a moment of inegu ha 6k rvaon kw gnqp0e52l:-d1e6-,rtia 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 the j.t*uj pf4p5qpnb5 3jrope 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 witn.ju5b*pj p5p3 t fq4jhout 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 :klnj iq8h* 6wsame side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital *qjn86w: ilh kangular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a(,;ecegu(v x uz j1ofk(g9 zyp z)z(wo3+b (ia 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 0.20 ma2(ouwu2 j992 cxay z-dt:q)i 2tiyd b acquires a rotational rate of from rest over a 6.0-s interval at consq2- 9 a22ctz dwt:)oxiua ujidy 2y9(btant 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 0.050 kgsa0fxd hy;0snn(c 9ni+xi x92 and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 ki02sxxafx(+0n i99;h dsc ynn g 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 wh(h tqochw+3i.n3 (vx am; 8z.8x lpqreeel ($\omega_R$) 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 timy4u,3p1a6gwbc /azss* 5ysyb es as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere a46,pgs1 bzyc3wbas5 *yua y/st 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 0irf 1x15zw)f m vihm2 e,:mutit 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 mfh2fx)i,1izm:0em 15vwu rt mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates ak,;-q qttog nru1 mq42n $H_2O$ 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 n 7rx-uur -,)m e;skhn+nkjll 9q lob g;u*24chorizontal 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., w:fs.s (ykmp 4nbi/uc8e ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and thes8fk:y(mi4/ bsn pcu.n released. At the moment of 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 ia8l.th /knyjh/c b;7g2h 3lfes standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step again g y/ .bh83/jah7ncl2ekl thf;st 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 +bdfs)qgzi0yji/; ybl hg/ 19lkr; + gmaking a turn with a radius The forces acting on the cyclist and cycle are t yfl)ik+bz /bdg0;lg +91 rq/gjys;ihhe 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-k,p aibmapr* y/vi p+-qcr6v.:g rock 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 ip+*rc6rmv: pviy-/ aq ap.,b120 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 skater’s body as a solid cylinder a 35)cka g:afzdnd her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly ada): za cf3k5ggainst her body.   

  You are designing a clutch assembly which consists of two cylind u m90k39rmtz,h zc)5vr h5oja2nji, lrical plates, of mas)t ij0rn9, h35mlvka z,2zjmo 5urhc9s $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 l:fpjo; 5h *zvyooped 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 the 5p* o:;yhfj vzloop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assujjo2xl ww7f,j, ba*.bme $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed unifxl-j1y evh7a(4ij-sn ormly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the aesjx-vyni1l a4 h(-7jngular 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