A bicycle odometer (which melin s4x pgk+0f,)-e rzasures distance traveled) is attached near the wheel hub and ipr +e)fxz ln-g0k,s4is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocit 4v2 +i uq iefy(t-bszq8t1fg3y. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or qsie82+ t1 u f yg-t(fiq34vbztangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a singjrhpe ja. *. 5q.pwf0a.q/: hpkxhod,le value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force?*fq*yzp/ o+jw 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 headcy lu wb.4dq5lsqs ws6*t:79e 33pf. zyps.nv than when your arms are l9s 3us.v :.sf7wy* y6n35c p4zbp.lwdqt sqestretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the pedal k jb6 6v*o.d 6 wm.oxjmeg18zmcranks. In which gear is it harder to pedal, a small rea1bemj*6z.w 8og.j o 66mx vdkmr 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 lowerkkv s/ *tva+gui9-. ;afp tebavc068f legs with flesh 8/ab*tf f v.c ; t96iskgvp+ak u0ave-and muscle concentrated 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 k )( he;nwr8l6el,v tdqbf y4-a long, narrow beam?

If the net force on a system is zero, is the niu,5q drv0 z4cet torque also zero? If the net torque on a system is zero, is the net force zerz50,vqci 4u dro?

Two inclines have the same height but make different angles with the horizonj x(m+/mdwl;y d +zyr4tal. The same steel ball is rolled down each irdxmw ;+jy m(z/+y4 dlncline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultane1. 0l*eg8jzsua-h kcm loy1: mously 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 j ml zakugo8-ly 1c*.m:es0h1 has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. They start5b 5 sk vj2pnrazu.r+*m9*zmv 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 ton* mzr rj5kp5.zmu+* svab29vtal kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most moxc c;9mavaxebz qt 6 ;i4sk*;.ving or rotax6s;qt. 9*ez4 k;vica; xa mbcting objects eventually slow down and stop. Explain.

If there were a great migration of 7)cjnx51.ih jbl(uvqbhj9,d f p ,r.8xvom6 upeople toward the Earth’s equator, how would this affect rlfp(i b8.hc51d o ,.qu)v ,jj 67hnbxxj9v muthe length of the day?

Can the diver of Fig. 8–29 do a -c)4 skbherq;dx 9c :ssomersault without having any initial rotation when shxker)c;s4 q d:h 9sc-be leaves the board?

The moment of inertix o mpt +3lg.t;edbhnt8+d6( qa of a rotating solid disk about an axis through its center of ; tlme o+b tptqh6gx(n.d38+dmass is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg mass in each outstrem c zxfl..2* dvk0gr:wvd3y4ob:fpy1tched hand. If you suddenly drop the mass43km :yxgfv:ocf20r 1.dbyp.lwzv d*es, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one i kaygb6pa:7r(yac c/1/lns /is hollow and the other is solid. Describe an experiment to determine which6k/7car y/1lbc( agpy /sn: ai is which.

The angular velocity of /s wtn4 p0d6ywxe 0uk.a wheel 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 .0 wptd/e6ukx0s4ywntop of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large fr8vj-gglgoq00 dpov ,4t g lj**eely rotating turntable. What hap8q4p, vol d o*vjgtglg*- 0g0jpens if you walk toward the center?

A shortstop may leap into the air to catch a ball and throbizl u-tf5t / 3qkq51 ibrl)6nw it quickly. As he throws the ball, the upper part of his body rlb-zfn5 qu5/q3tl1bkti 6i)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, discuss why a helic kr7x63v miz8qokxi m z86r3qv7pter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (a) 3 p7i(yxt0lpeh6 vt0jo(2 s1ua 0 $^{\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 amazu7: so3kcba/k, n++n4 zowidv ing coincidence. Calculate, using the information :zw ko7aic+,3 bvn /ok4+dnusinside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directefkar me.y7k4 jr*f*mophq: 2+d at the Moon, 380,000 km from Earth. The beam diverges at an ang*h mkrqf+7*p.meao 4r:yfkj2 le $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotf4:d)l rt2 ta p5zhc46uu:2dtp6 krxkate at a rate of 6500 rpm. When the motor is turned off during operationtux2c)k frhk2t4r6u:dlp:6 p tzad 5 4, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level flooo:7 bhlgxnz1- r 3.5 m to another child. If the ball makes 15.0 revolutions, what is its diazg:-1 7nb lxohmeter?    m

  A bicycle with tires 68 cm in diameter travels 89(xwwmrb i*g *j2 f6vg.0 km. How many revolutions do the v *x ri926 wfb*wjgg(mwheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2 q5ep2bhlx)q f 6.w5zi500 rpm. Calculate its angular vql5 bxiq 5ph e)w6f2z.elocity 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 re.w*sr: 8uef8u f- c7 9ze*rxzita 1ndevolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from t *esz*aur w8-e7ue:1idf9zntf xr 8c.he 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 arounc6ghs.vb,u j (.mx*xtkr5 acy z:ws 1,d the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of 8shpgf 62)cpe a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rod() u( k+avhz3hy lc jwutl;.9tate if a particle 7.0 cm from the axis of rota)y((.hvdta9+3 uwu l zhjl ;kction is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130a5kcw/pg f t80 rpm to 280 rpm in 4.0 s. Determiak 8/g5pw0ftc ne
(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 radiu g-pnn34czr5me-x8m o s $R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard the Apollo spacecraft put themyf3j-0 km ,y4,fiowld: xfd q+selves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft ca3o ffiyy,lwd k,:-fq+0d 4 xjmn 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 re h;w;t et)z bdp/z)xc*st to 15,000 rpm in 220 s. Through how many r; h )*bzec)tpwz xdt;/evolutions did it turn in this time?    $rev$

  An automobile engine slowsoxr +q j(d.x+y.sj:fo 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 hig9 9e)qqgfbrwa 83y,yi hspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reyqir 9y9)ea ,w3fbg q8aching 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 accelerate0 gxcn wn98qe40/lql,0 ajm ous uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edgengu0ncqjlo 48l,mxwa eq0 /90 of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min Ir tc19ye;f,ue -1 en12dz xspxt turns 1500 revolutions before it comes to a stopspy xxt9 fu e2e1rc;1,zd-n1e.
(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 speet1 tl2fqv1 we0d uniformly from 95km/h to 45wq 1t0 e21lfvtkm/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 st8. -37psfv 27jfnhf. ekcsq rpeed uniformly from 95km/h to 47kft73- esh .rnv8f2cpjqs f .5km/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 12si ;2,ens2khvud3h ly9 zjhweight on each pedal when climbing a hill. The pedals1 z22duhl2vy9;n3hijs ,seh k rotate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cm widekq::ipwg*f s *. What is the magnitude p*gfkqs*w:i :of 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 ab2qa + 5ldx6uqgout the axle of the wheel shown in Fig. 8–39. Assume that a friction torquel5 dg+2xqqu6a of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are att loejd0(zbt4mtme)3rd );3kaached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the tao ) d4(ed tl) 0jkzrb3;3emmhorizontal 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 requi cx2t* agvvi09 vpi0/ure tightening to a torque of 380cv uipiv/a90tx*2 gv $m \cdot N$ 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 radl ufv+szxf +6/ius 0.648 m when the axis of rotation is through its center. l/+szv+u6xf f    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diametevt+-o(new /qdn(ij/p r. The rim and tire have a combi dvtwn(/i-nj/oq+ep( ned mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rodw2if(vp7 )euwrvzo; 0 is rotated in a horizontal circle of radius 1.2 m. Calculao7 2vpzwf;)vier0 u(w te
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s whe 51g+* q)uxgbsetz3 fkp-o(vuel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the clay is 15qxpb)u 3g (s+ u*vtkozf1 g-e.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 thep z0bar9gp(7s 13 apkm)uwpajk3p;:t array of point objects shown in Fig. 8–43 ab9pps(ua pr bp3 ma)k;pwk1j07zta3:g out
(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 con+ gpebf5s(g jj3z7t;c sists of two oxygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the correc5na5r6-okd o+zx5 zmcc id3n0t rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and5nddom k+x5acco zi0rn6-z35 a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 20 exwtnjw,i. f4a dj0cm and a mass of 0.580 kg. Cwwix0 end.0aj42jft, alculate
(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 ag1x *631e-l )gxkqakei 1tyyest 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 smawc6 -of im4sa6ll hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm i6ow -ac6 s4fmin 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is eventually /7soxbtc(yf* brought uniformlfxot 7 sy/*(cby to rest by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a 3.6-kgt09misb(i0kd d5yx w v;( 2zqe 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 by th lf,. pqe rhi-b23 (joed)c.y8: tqftze action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly frhjpilty e rcfz t:. )q.bofd-e, 83q(2om 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 bladsd7i0* kwtsax92 oc;be can be considered a long thin rod, as shown in Fig. 8–46.sw02c o k9*ab ;istx7d
(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 consistsl kcxrwgzd9* :n9*hj 0 of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest za2 7j tv9kg179c zt**u a vea4wqk6llwithin four full turns (revolutions) and releases it at a sk9 c*ajqzau7teglv2va*zk w l179 46 tpeed 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 moment of ine.te /z +,c l8tz/izev.a);n sllxf5ylrtia 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 ne*ni9jywvisym +61 a 53b0ce)zmhm+ develops a torque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rollgrjcsp) .: 1v km3j, nqmv,h9js without slipping down a lan19jhr:m,g )pvq cnvm,3jsk j. e at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the 5elk *w0schfk 9uuu8r:j( a*dz8b +ga Earth with respect to the Sun as the sum of two terms,0 r 5 uafjdahw*+:z8uue(9g8k *kcbls
(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 16402(necv4ng( howej 0y, sxy 4w7 kg and a radius of 7.50 m. How much net work is require s hj(yygw0(n47e,ew x24n vocd to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts t- m e:4g8n.r1.xaq ut /qy8bt -szuqqfrom rest and rolls without slipping down a 30.n/u8r- sz8 b qatyt:q-tge.x.qm 4q1u0 $^{\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 star(kqk/ehp4c t0 ts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use chkc(etp4/ qk0 onservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end of* szj wqs)a*u: a thin string in a circle of radius 1.10 m at an angular spe)q ss*z*u aw:jed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grinding tv,9 h0pyk :-haoe)jwx o)qd(wheel of radius 18 cm wt( xhhq9v je,da :op)0w)koy-hen 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 hii46 du zyxspgsf98 mla+my.0 /s side, on a platform that is rotating at a rate of 1.3rev/s If he rzum fs0plg9m4dys8xia/+ y. 6aises 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 i5s;kcj b82 zhf yh.g0e :2cmrin Fig. 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 maker8:5c0m. cgi2z; y2khb hefj ss 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 ;h/l: agp w gn/n a.ep;7nlh-vspin rotation rate from an initial rate of 1.0 rev every 2.0 s to a final r.;elwaa7n :/n ;l hpgnh/-gp vate 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 aroun12qsxchbr4wh.se a3 +7lsb y* d a 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, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel +41shl.b c srhs7 yq3exb2a*w after the clay sticks to it?    $rev/s$

  (a) What is the angular momej -ndjo16b 2gjm/ av.mntum 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 the0rk7f3lblc j. 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 cylindu9jqm/pxm: kx; ux-/v bl gy78rical disk of moment of inertia I is dropped onto an identical disk rotating at angular-9 u/qxmxb;ylug k8 m7: /vxjp speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turnss4tz6/ yden )drgfmc0 2bq0:aocom- 5 at 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the cen2thprfu+ s8z rpt7 pmo5r 5*d/ter of a rotating merry-go-round platform of radius 3.0 m and moment of inert 58toz5ptr df+hru 7ppr2m/*sia 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-go3uv(3s fvpxx (+/kb lopl7 9qpn0syz5-round is rotating freely with an angular velocity of 0.8xp+obyvp(5 l/ f7n q39zp 0vk3s s(xlu $rad/s$ 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 eventual hqi* fpdc:wb72kgkl5 k(tyu+1d )e8 lly collapses into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mas 71lulqdeby(gkptd8 2+ w)*:k fhcik5 s 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 ex.p-:m.fei)dnunja: lu.sn u 9 cess 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 swtk b)f /fx048d ab3q$ 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 htd2u1 p+4 yyyplatform, initially at rest, that can rotate freely without friction +hyp2tyy4 d1u. The moment of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diametewhz(as 1 h1xvh,sk4g: r merry-go-round turntsvh 1hg wz1ka, sh4(:xable that is mounted on frictionless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls 1*+g:byr y/so+gtnt*sehjp 0 on the ground with the end of the rope lying on the top edge of the spool. A person grabs the 0pg*/s* +rbs t+:jghyn1t yeoend of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that th1cdkp s7 .xila tr18:9 (ntvque same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (ab (p8 .tv9:lq17dcrst ux1k ianout 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 awo3nim-0fx r3d6+,myi sv yz 1 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 uni( ju5eavqg+ u2u)5thhform cylinder of radius 0.20 m acquires a rotational rahq2e+ )vu(ha5tju5 ugte 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 cylix1 k6of2 bq(ez k/f/blndrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when e( kk// fq l16o2zxbfbit 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 angula/ aa , 5dqph+zakac7mxx96qp5w 7)ginr speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as great, were flyj )dpp;:u )*: :2xzsvwumvrotating 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, and that the lost mass carries off v ;2xmz wp)juu):fv:y:psld*no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use energy wwi;xt*4h 6-3,sljg8mzpi wt /xvm 0rstored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without /zl3*8ttiw;s m6v wxi0,4 hmgpjxw r-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 illustrat 14k) bnklce) lsz9t:ces an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal s 7eu qfm n)p 8wta.21tc8rhoa)urface 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 M1ua k.6r49tbghaqrfe 79/njl and length L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontarg4f1h aan9 69q/ekjbrlut 7 .lly and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of 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 4v7z sni5vk.b:kdq( bR. It is standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it wi .7qnbbvk v4(z5:d sikll climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a turn w/ptg/uy * d.rfith a radius The forces acting on the cyclist and cycle are the nofdg /.yrp*t/u rmal 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 +ksn5td (yma nx2q ;j)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 120knqa +s)yx52 mj;t (nd 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 ashv1oul7/p c i, a solid cylinder and 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 ticipu/lo 7,h1v ghtly against her body.   

  You are designing a clutch assembly which consist ;hvso;o2hx ;qs of two cylindrical plates, of ma ;q ooh;vhs;2xss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 ovwk -cj /t.dq5f Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if iv.cjdw5/- k tqt is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, bu,:ni0 3hig fz lyuj.;et do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolut4o6x-,r izvtn zs2 di4ions as the car reduces its speed uniformly from 90km/h to 60km/h The tirei4 di6xn tz4zo -sr2v,s 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