A bicycle odometer (which measures distance trave9*c d7,yfwhc*on h hatkvxx bh81i73+led) is attached near the wheel hub and is designed for 27-itv*+k*hw d3yn8bih x1f x9ah ,co7h7 cnch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on the rim havk u:kbab/ 7r0pe radial and/or tangential acceleration? If the disk’s angulark brp u:ba7k0/ 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 bj0ah8e3/rl9) eah w jqgs7ygeg;ag 5 *e described by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? E 8)g28czhn0an/- f7kzgvdc b uxplain.

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 th kgl8 *qqz z;9 y,cp67k0iccnlan when your arms are stretched out in front of you? A diagram l 9c*l7gcqik6pnz0;z,8 qcykmay help you to answer this.

A 21-speed bicycle has seven sprockets at the rea6 c 9mzs. +aqnnxbz5a3c7kn.nr wheel and three at the pedal cranks. In which gear is it harder t5 nn73 k.+m6a zccsan9qxbn .zo 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 fas6lc vho,p(d1: okc --,ufgt ktt have slender lower legs with flesh and muscle concentrate 1o6tctc u o-ghf:kdv-p,,k (ld 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, narro.+lfb+8) jm1spc hshww beam?

If the net force on a system ip/c c6v- u uct57uqxe*s zero, is the net torque also zero? If the net torque on a system *5vxt/ e uupcu7c-q6c is zero, is the net force zero?

Two inclines have the same height but m19c. frxjm.eie6( v ily202+q1wqwxd wcj:s d ake 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 bott+f j(iyx2 :ev2wiwd9s.w1c1d. jr6 cqqxel0mom be greater? Explain.

Two solid spheres simultaneh n:b/kzqa(7uously 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 hh /q:u 7a(kbnzas the greater total kinetic energy at the bottom?

A sphere and a cylinder have twer/ p7pa;2 5v +vabcehe same radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater;a +5 /w2vpv pareceb7 total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentbsky2)c9qpy gjy-6 *g b*y kq8um and angular momentum are conserved. Yet most moving or rotating objec 8ck g-kyy6qb9) ysp2b jy*qg*ts eventually slow down and stop. Explain.

If there were a great migration ol ( 5k2ene*vaff people toward the Earth’s equator, how would thisk ln 2a5e(vf*e affect the length of the day?

Can the diver of Fig. 8–29 do aevl/ */eu*jdc somersault without having any initial rotation when she leaves teljucv /*d */ehe board?

The moment of inertia of a rotating solid disk abou. urci w07.ewjt an axis through its center of r7i0w. .w cejumass 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 rotatin a y 7in/g6d0.icwddf-g stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explai7-0y fiwn.dc /add6gi n.

Two spheres look identical and have the same mass. However, one is hollo1tr +l,2uq usl76qd,y ctgv;tw and the other is solid. Descrv+ ; y,t6dl lqq ug2c71,urttsibe an experiment to determine which is which.

The angular velocity of a wheelxevl;x)ug 52z rotating on a horizontal 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 25zglex);uv xthe angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turntable. Whu y.eu( uep/ ;+3wvhyvat happens if you walk toward pyuvh3eu;(uw+ ye ./vthe center?

A shortstop may leap into the air to catch a ball and throw it ; jxujl5gm ,-alw)2vz:pgg6q quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hu wpv,jl )m; gg5j-zq62g xa:lips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentqfj blzv 5 h,)n26bd24 vwe+lfum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicoptez l f2n+b h,v2)fd6qwv54jbelr stable.

  Express the following angles in radians: (iyema/)8qoaq g+ j*l3 m74vz/) pm w p5it;paya) 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.z 4c16.phb7anlt vqr coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of tl.q4pn .6rzvhb1 7ctahe Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 38*6 gs nv5rfwn1 gke5e*0,000 km from Earth. The beam diverges atwf gv5* 56grenns*1ke 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 rotate at a rate of 6500 jw .3al77bb/f iux,rwrpm. When the motor is turned off during operation, the w7.xa/7l3ubwb ,rfij 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 j .spr cl7i v .ww5i4r0ifpt(v2x2o1tto another child. If the ball makes 15.0 revolutions, ovi1.fipr. 2pc l( t4 5w7t wixjs2vr0what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revo3if(e)kp o gho 93nni0lutions do tfk e0on3n ipih3g(o9)he wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rota( n5vml1ktq.jsd 3vg;tes at 2500 rpm. Calculate its angular velocityv kmv5 1qt;dl. s(gnj3 in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.0 s g25 *j kq/oejeg0jy 4lska12l(Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from the4le0*jg5kka qoj egy/ sjl212 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 orbit around the Suvityu/ye 2l ;w;d ;s5un    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a point ukzek+7)7(cmw rzwl0
(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 hp(k4nn ks0tqfttu* qe+ 8:*z centrifuge rotate if a particle 7.0 cm from the axis of rotation is to experience an acceleration tqe*:hp*nk(z nt4 tqkf0+8 su of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from jf khw)po.-3-z7ek:ks 2 pvyl130 rpm to 280 rpm in 4.0 s.-3e 7sk)zvy pwp:h-l 2jfkko. 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 f b5 /vunwln(6$R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard the Apolloxojqt(5bg+bszyx2te)59zu j:ht7g + spacecraft put themselves it:7t+xyt)(xzgj5o +q e9bzhb s2u j5gnto 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 rest to 15,000 rpm in 220 s. Through hoivcag)9uk+68n4 lcvc vxwn+(8p f m.ew many revolutions did itc m (eu898f.v i+ck )vn+a4wvxpn6l gc turn in this time?    $rev$

  An automobile engine slows down from 4500 rpvc0 a.)y axrylzdeh(4+qn17c m 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 highspeed jets in a whirlifvtta)r 5 uc/i:fo0m. ng “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its fin0. icoaf) vm:trt5/ufal 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 accc-x9co.mea4uc n, tjxw(f25k elerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in thiwjek c,axto.m-cu4529n x(fcs time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 1500 revlsb0y muw )m 5.6dxhxjd-x:s4olutions before it comeum .lm4-)0x: ds6b5w xyhsxdjs 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 spee 5 ;mjh5iaiobk,6ny137 owmwt d uniformly from 95km/h t b1t3m5nik am o5o;w,jh7wyi6 o 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed unifoakoyw w0 2nl07rmly from 95km/h ton2o7 a wwy0k0l 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each pedal when climbii gzrs 7cu0kh1 tlxqc*5; 6e:gng a hill. Thzh*;uesc rg 5 lk6gx7:1t cq0ie 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 k131kpleh:75au:ogrkq l:eg * /eifz 74 cm wide. What is the magnitude of the torque if the force is exerted qk*g :uegzilkeo h1 /15ar:7fkel p3 :
(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 ax02q 5osd70mo1a4qsf yf ruoe )le of the wheel shown in Fig. 8–39. Assume that a friction torquedf405aue0o q7sq)ryomo1 fs2 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 zi;6uy1f mxj/-ux )lrof a massless rod which pivots as shown in Fig. 8–40-mj)xr;z /f1iux6ul y. Initially 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 of an engine reqwao s(9 j*qr;iuire tightening to a torque of 38io;*ja s9( wrq $m \cdot N$ 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-kk xzu uw(7 rzili-0r-+g sphere of radius 0.648 m when the axis of rotation isuw - iuz(i-k0x +7rlzr through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a 7v- ktu/ ;yr9wgj-zs gbicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mas-vrg sy7g9/jwkzu t -;s of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is reuhs (ie1 ,nv9otated in a horizontal circle of rad (e9usv,1ehn iius 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 wheemm kov3- 4cxmh .l75jol rotating at constant angular speed (Fig. 8–42). The frkm3omj74hm cv -5o.xl iction force between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the z; k tk-;q*avpxh6u;.zo88kkt, ,dw hq bw1uoarray of point objects shown in Fig. 8–43 az;*q h ;pdvhto -bx aq8;ok.1zk6t,,wwk u8ukbout
(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 whoa: ydp i7ogzj-1e-+ opse 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, r ac16xl0 ay0hengineers fire four tangential rocke00 hlxrac6y 1ats as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder wiy:z+gn, myizqz*gq5*th a radius of 8.50 cm and a mass of 0.580 kg. Cal+gyzmiz*zyq*:n, qg 5culate
(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 r hwfglyql2/* 4 nyt-h7est 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 merry-go-round and iqvq:,(9eu f ynm*vlh:s able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round i(v u: f *,:lqe9vymqnhs 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 bunu4. rrz) g(q6g gf0z ixi*9brought uniformly to rest by a frictional torque of 1 qx ) n(iru.bgui 6zz94gfg*0r.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 accelerat.r 6qss39ptd0saf r 21c7smzx es 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 bym7qq h 8bly+pd i4.ks; the action of the forearm, which rotates about the elbow joi;i+mql. hkb 84ps7yq dnt under 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 in F4b-qqdfi:* ghh3:u cnig. 8–46 4hq fb3c*h-in:dgq:u.
(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 oyo ccnzh.6t4 3/oo.x3x-rry pf 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 hammer from rest within four rjj8 zg l20wxsjhmaee6dol80 z7 2(7tfull turns (revolutions) and releases it ozwld2age0e 8xh78(lrjjz s2 j 70m t6at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a f d /0eojp-n2)zcja ,cgr31jwmoment 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 280 bl0o;o+r mkgs 3swlb7 4f ayefv2sbx;7o. a.5$m \cdot N$ 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 withounwj d*sholhvn-:7 9r4 t slipping down a la7-d:vho whsr4*n9jl nne at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with res - 0 wcwwi5me/qn xbx+p*ng30epect to the Sun as the sum of two txx*gp0e 0 e+bqcww nm/w 53-nierms,
(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 rayg2je.gf9i8s dius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.9es g.2gjy8 fi00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from re+ ppznm6fqk29*2 js u))fvrphst and rolls without sl*r)jk 2 mu vzf)nh2pspqpf9 6+ipping 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 vertical,a2sdl z8/l mely on its tip. It starts to fall and its lower end does not slip. What will be the speed of the uppdzl8me2l/ s ,aer end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of nmg 2tq +0jaemft ad+d79:5 yga 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 mg 5gj2tq9 am:0ndf7+y et+mad at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum ofr1.mzw9 wmtz5 a 2.8-kg uniform cylindrical grinding wheel of radius w1.mzz9rt mw518 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at h(i:qb;bi g, nbq g(-ju 8)liosis side, on a platform that is rotating at a rate of 1.3rev/s If h)iil8osg( bj; :qung-q,ibb (e 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) cq: , 6m xjz6k qqfn)bai0kqu:3an reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck qkjuinzf:)qm ak 0:3q ,q6x6bposition. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initiw/akz5 *-hnyaf,yomnf 7uz6 5 w28wkoal rate of 1.0 rev every 2.0 s to a finafo7ah5, n/*8az u5mzk n ko26ywwy-w fl 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 ac7vbz5fbuy0 2 ec;b5a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approxic 02y7f;u a5bbecz bv5mately 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 sa2 e-blv7i(fj-- p3ahx*sj u hkater 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 mo)d3 x tcifts4r-2o0yumentum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$

(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inern x7a69q7yww ctia I is dropped onto an identical disk rn 9xwycw7a 76qotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.447b (zwfi 3ucz $rev/s$ 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 4-w m)cy 8mrqws8i0sz stands at the center of a rotating merry-go-round platform of radius 3.0 m and )8 m -z4mw 8qryssw0icmoment 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 rota eh3z2 g;w rf,hec0wibfw7*5k ting freely with an angular velocity of 0.we235h fg0zkb iw*c;,h r7f we8 $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a wh) k0od:e 6zrlvug.5ls0hk-ip ite dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming th5zksl6pog l-0irh )kd:.e0u ve 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 ingy 8cq9 zgav;n8oc e1m*y(1 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 9e s u2gb1;msa 1apv2i$ 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 platfpakud9,i,nq g k79t.c+vn 0 dhorm, initially at rest, that can rotate freely without friction. The moment of inertia of the kadvp q i+k9 0,u,nnd7. ch9tgperson 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 4pxx 0exny: 8nwmb63 xedge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment xb06x3x4enpx wy :mn8 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 o (t;tns83jhfi n the 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 rope8nt;3f(jhsi t unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same7,etd vhlnrf1 3t-4dl side always faces the Earth. Determine the ratio of the Moon td,-31 levrfd hnt47l’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 1v b wa9uk)8tse 8dn*4t 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 radot6;)gk. mip7gt a:i060ny cygv2a 6 pwwvht5ius 0.20 m acquires a rotational ratea2g.:taink ytv thi w 05c6wvy70 ogpm)pg;6 6 of from rest over a 6.0-s interval at 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 0.050 kg aow;li xe;pvd qax ,51r bfrw61*1 dul3nd diameter 0.075 m, joined by a (concentric) thin solid 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 qu,;rdd w 3*lfp5 ix16xvl;abw1ore 1reaches 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 angupqm em( 4cb6qskb3i ) kh(bgb2h+p /w8lar 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.qt2qah4u vs 270 times 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 at all times, anhsa7 4vqq22 utd that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution autm1mln 6l dfwhi268x ;tomobile 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 cylindric;fntwl6 m1hixld2 68mal 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 illustratedv6w- xnbrw ,+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 y+h9 ftdrf s9 6w249jsb/wt yh8 girj1horizontal 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 free v-vf n 7mza+f5mt(jve3,3cgn ly (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) v3 g5-vejfvn n+a,(mtm3c zf7the 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 standingtb,ifhqku5-nee h: .) vertically on the floor, and we want to exeu bn,htfhke.- 5:iqe)rt a horizontal 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 em ;z7n2lm*kx )mno c(with speed v=4.2m/s on a flat road is making a turn with a radius The forces acting on them(2lk*7n) c;mzen omx 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 si*f/h3q1tu v4f ypf;1 n7haa 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 120 ;a*qfntyh vi3p ff /ua17sh14rpm 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 a48y5adhp 7 n0.h, lsw*l;r sjg qdif:pnd her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater wig hs 8ld*4; nh pyfir l05j:.sa7w,pqdth outstretched arms, and with arms held tightly against her body.   

  You are designing a cluhg 03cp15l uz2. tmo.vy (zrtftch assembly which consists of two cylindrical plates, of pt1.g hcu3(.yr mztzf o50lv 2mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 trackv06 2l sw:ar+;(f :qm ikulciy 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 cf;r:m ii2ywlsq0(uv:akl+6highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not asn9,x ;w lauk-thpl,-jsume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revol)*n f:bi ml66mw/yy7tzq v 2swy 16zreutions as the car reduces its speed uniformly from 90km/h to 60km/h The tis6:*z1 7v eiqyyt2nw mwrz)bf/m 6 yl6res 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