A bicycle odometer (whichyv0 7y r/6knsntj 5k:p measures distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use vrn y60/j5n7:yps kk tit on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on the r. /ua7n8hw k9xp.ofj gf7q gh)im have radial and/op/ ujff .h ax.kg89)q7w7hgnor tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by oy+7iyo8tfltynacjs1( 6,9y a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than t11 pzg ocu+b; j1rjjel4w27ya larger force? Explain.

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

Why is it more difficult to do a sit-up with your hands behind your head tha5qfvv+jy ovyeg/m8 kj l )o298n when your arms are stretched out in fy/j l59f 2kjygv8 v)oq8meov +ront of you? A diagram may help you to answer this.

A 21-speed bicycle has k k+ akulbc yq-:)rx+9(c lsb:seven sprockets at the rear wheel and three at the pedal cranks. In which gear is rkc+bkau)y 9 qkls(b+ :-cl :xit harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legs 9m f z eq ug*r;31gve oaep4 zzlc8mm64:wbe7;with flesh anl3cm4z4 7o8web eze p1* az;gq:e;mgv u9m f6rd 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)s qbrbq: .+urs556dbd carry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? If thv bf*6 ts09fy7u rac4ye net torque on a system is zero, is 4s9 cu0afytrbf 6v 7y*the net force zero?

Two inclines have the same height but make different angles with the horizont 4nfvx2pl s gtq/:ue7-al. Tsx:ulf2-7 4vntgq e/p he same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start re e.r9x6epp s7olling (from rest) down an incline. One sphere has twice the radius and twice the mass of the esr.ee96p7x p other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius axfr:cf y6mo8 uiuu)x 764j io)nd the same mass. They start from rest at the top of an incline. Which reaches the bottom firi furc4 x:66uo)m y7ijfx o8)ust? 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 most moving or h/e 3.hpoj ,dw:,k kvjrotating objects eventually slow down an3 :vpdehkhjj,o /.k,w d stop. Explain.

If there were a great migration of people toward the Earth’s equator, how woulcm982vsjnc jkql t,n 6pz/ .m)d this affect the length of the d 2jl zkc)v8mt/q 6mp,s.jn n9cay?

Can the diver of Fig. 8–29 do a somersault withox,4aqwz+6)zy2usu 2 ly0gvz mlek q; 5ut having any initial rotation when she leaves the q2y6z4)a ,0x52;u euy gq+zkslvlmzwboard?

The moment of inertia of a rotating solid disk 9ov)f )zcs7;ow q2d paabout an axis through its center of92cvszw opfd))o7a ;q mass 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 rota -ty of9vhajfj-(bw76 ting stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease,fhbw(7 jo fv- 6a-yt9j or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hollow anq, oopq;rsx;q9 h)- 4fth2ms 7a1 5dlbr+m xokd the other is solifomo q2xls 57dt -hhk,or;1 qmqasp9 )+bx4;rd. Describe an experiment to determine which is which.

The angular velocity of a- f0q3t2djsxd 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 the2j -x3dtfdqs0 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 onbv1ao4j( w)lrx0gr o* the edge of a large freely rotating turntable. What happens if you 01xg *waov4l )rr(bojwalk toward the center?

A shortstop may leap into the air to catch a ball and throw it quicku 8y zx5)u(;lqvk0wir ly. As he throws the ball, the upper part of his body rotates. If you look quickly you will nol8k0q ir z(u5;w)uyvxtice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservatiopk 7 jm v -2,mu*m/g;hp.6 dqec(qonvia49e ejn of angular momentum, 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 9kodmecpe,m*uv -a2/j; ( ngivp7hq.j4em q6helicopter stable.

  Express the following anglesi4iae*0d -h.h yep5f0bw k:wz in radians: (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 coincidence. Calcul s ; zkgze;e+m;trr/:tate, using the information inside the Front Cover, the angular diameters (in ra/+;z grez;er: stkmt ;dians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beam diverges t6,l5nm jkalpyw-48 lizq e1,at an 4azp,t q1 l,i5-eky lwm6l8njangle $\theta$ (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 rpm. When the motor is truyzyf552 bg 5urned off during operation,yy2 5zr5bfgu 5 the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another chil()6pi i xbsdr4d. If the ball makes 15.0 revolution()xbp4i 6 sdris, what is its diameter?    m

  A bicycle with tires 68 cm in diam9ubs auh0q 9u7gm(j;oe;*wzfeter travels 8.0 km. How many revolutions do the wheels ma;uugq ua9b9wzo(0h7e; *jsfm ke?    $rev$

  (a) A grinding wheel 0.35 m in diy3 lq6 m-t4ti, qx5axcameter rotates at 2500 rpm. Calculate its angular q3 t-lq,5yxtc6 4amix velocity 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 (Figlr qee.n0i-8 g. 8–38). (a) What is the linear speed of a chigei qnr0-8.le ld seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velob8d93w9ch nrwa (33ip2g s 8 0jixjsdgcity of 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 po .26q hiyu6dkx:u1p n qx8ts4 k.mq sx7vl0c,jint
(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 rotate if a particle 7.0 cm f6ie*19dvkf x kev)sa)x*dj: brom the axis of rotation is to experience an acceleration of 10dx*b))vexk e16dji f a9 :v*sk0,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly ab8dtmp/i )k)wn out its center from 130 rpm to 280 rt)pd i)m w8nk/pm in 4.0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radiusli ru 1d y.u3o08l tiuc6y50az $R_1$ 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 the9e2t psao+ q:e Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they acceleraeotsqa 2e9 +p:ted 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 , *j9 e,2qiafvda* bfsin 220 s. Through how many revolutions did it t, f 2bj9a*f,aqv* siedurn in this time?    $rev$

  An automobile engine slows down from 4500 rp 8c5sf 3ta; szhua.n+cm 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 wllzp/r(kjdmqj / g0a+4z8c0 shirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final(rcdj zl 0l4 +sjza//8qpg0 mk 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 diameterrx fz,7dg5,pr accelerates uniformly from 240 rpm to 360 rpm in 6.5 s. r,dgxz f,7pr5How far will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 150 mzni6o58e 1do:y1.kf)knl ny0 revolutions before it y1)ned:5o8 lkny zik1 m.fo6ncomes 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 uniformu sq w0f- h7f8knd6k )ryq+bu1ly from 95km/h to 45km/h The tires d0 1q6 kk7s+-urff)8qhnwbyu 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 uniforua,:9/:gcep 9ha4c5ziz0n9fq naum xmly from 9,/iaqg n9 n:axz pccfau0eu9 4h:z59m 5km/h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight o ;dxc4t ;8va i0/udcldn each pedal when climbing a hill. The pedals rotate cvd ;a/u08 dlc;td4i xin 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 d4d,zcx5/ c hcdoor 74 cm wide. What is the magnitude of the torque if the fox 4zchccdd ,5/rce is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig. 8–3zoqo7 r0q mk51l(ams7n*+)ps2lm ln h9. Assume that a frictiq mr(akm*077ollsn2ln h+sm51oz ) pqon torque 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 bqgg 52v b qsd10v5-rithe ends of a massless rod which pivots as shown1525gbdr g s0vqq-i bv in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinderhg(pj+ ;hk)drb((v7 nre+pwmh5/tgg3mr)h b head of an engine require tightening to a torque of 38kb;( 3r (whghpg)m(+d+rbeghn trh/v7p5)jm $m \cdot N$ 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 vdm/g nt)-q2d.8-kg sphere of radius 0.648 m when the axis of rotation is through its ce2 /)gv dtnq-mdnter.    $kg \cdot m^2$

  Calculate the moment of inertf rjxk.7g8;n i/a1oy(hmhz ;tia of a bicycle 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 ignored (why?)am;z h o8y7j1ikhn fgr/;(tx..    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in a hvlb3e3z)4 s.gjv gcqxdk(+.x(h 0 vvoorizontal circvk)v.l. b(vjxs +e 3h covzxq4(dg03gle of radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating at ctn3pg-k e8z( zonstant angular speed (Fig. 8–42). The friction force between her hands and the clay is (gzt -eznkp831.5 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 point objects shown in ( pjv*;x0 utdyFig. 8–43 abovd ;j*xytpu0( ut
(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 c8hcn df1hx/( 1u spw 5*usajf)onsists 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 correct ratr4 3 72ilqpvknn /xlj(e, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the requirei3n v/4l q7kp2xnj l(rd 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 auka6k (dibm -00 2vb3tb 6z,ejgpff*a uniform cylinder with a radius of 8.50 cm and a mass of 0.58 ek6ab bm2zk vt*f6fj3 p ia0bug(0,d-0 kg. Calculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, acceeq;*kftq5 hu8smq1s- lerating 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 tangen)xca 5*z9cub *x eo;qctially on a small hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform d*a ;obzcqx*x5u) e cc9isk 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 rota9p *ka cjnx 0f()phd4cting at 10,300 rpm is shut off and is eventually brought uniformly to rest by a frictional torque ofpkf*a( pj )dchcxn9 04 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 bn4(rh4kw hjj+xj.5 bb all 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 the action of thd6 fp d7do-js,e forearm, which rotates about the elbow joint under the ac6ddsp f7 -jo,dtion 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 in2 rp9v00thip *pm+vni Fig. 8–46.v *p0iihprn9p0+ 2mt v
(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 consistiz5+b*vgqhcpn (ae 3o n1)h+(facrn4s q; l8cs 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 within four full turns (revhet2,.nh m)ohod h+ c8*n1 fpcolut 1op*hcc8 odenm.nh) ,+f2ht hions) and releases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a momaz d alqm49:z+ent 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.ctjd (wxrc ,a85m5hu 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 rolls witdflw 8)m 0q0:xqs+t k bklboi,:3w. p7wuv9khhout slipping down a lanqlw+m3,h0pw k ti7kvq :)9:k.dl ouw 8bsx 0fbe at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to thdt j il493qxb1e Sun as the sum of two djb9qx 4 3i1ltterms,
(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 mhj1, 1te)tscnzyb y +w. :war.uch net work is required to yn:eh ,bzsa1wj ry +cwt1.) .taccelerate 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 from rest and rolls u n)pkjzs)hm(lz1f )/without slipping1 zs plnf())mj/)h uzk 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 balancedgxyoyug+iow7 ,( (n2k vertically on its tip. It starts to fall and its lower end does not slip. What will be the s2g (yowg7n( iu,yxo+kpeed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end of iq 4d j.ab,he+rsqmfps a- bv+93x)h:a thin strin d+pe qbrh3,bs4vhm iaxs:+9q)-ja f.g in a circle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2vgbv /- jna8dci87ey/ .8-kg uniform cylindrical grinding wheel of radius /7vacg8v y/-jbeid8 n18 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 71 (kx1m nxb:m:cb2h cm6njdeat his side, on a platform that is rotating at a rate of 1.3revnxk x(n b6e 72chmjc:1m1b :dm/s If 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 kutw(/h q -xojkp2 v9;dja 27smoment of inertia by a factor of about 3.5 when changing from htq x72o;-9 aujjsdk/2vkp(w 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 rotation rate fr/ 1d8sqc/czmm om an initial rate of 1.0 zm1cc 8s/qmd /rev every 2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating aro krocdb ;0-k.ssm+ ti8und 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 freque.+kdb8o-si s0k m; trcncy of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momenv 8k/fxrg7zji0w8( iatum 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 mzr5*t:j*.q q g s d(itlfus;1eomentum 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 inertia I is drf g)iiczt,0* s )n/supopped onto an identical disk rotating at angu)i/,nuc0t gsfz s* ip)lar speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns)qcq 31n)x,4ry(bn rl tkzaxbvm gu97/rj3w 7 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 sxojfdk vb 2s03u9 r6j :p* 3apcalqj,1tands at the center of a rotating merry-go-round platform of rj3*qs ppcvuxa3rb 91aklj0foj62: d ,adius 3.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating6mj ;vnbu vr6,irzg .1 freely with an angular velocity of 0.mnbv g61z6;i, v.jurr8 $rad/s$ 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 into6 rqz /huiiz4hqv*5uce 7 v0c( 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 mass carries away no angular momentum, what would the Sun’ eu z /5(vr0*hhuz4qcc7 iqv6is 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 eialbs91b8 u6o xcess 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 s-f:;qi t qz)t$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, that can rotate freely t 6zd( w+zmn:qy+j vn;without friction. The moqzn(mt+vd:n j wy6 ;z+ment 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 diameter merry-go-roundrt 0 2mag.1xeyqs4h4e vjoir/r 9y79j turntable that is mounted on frictionless bearings and has a moo2v./m qy j ts1exh70e4g9ira9 j4yrr ment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the enamj1okdg p es8 i0 sz2 m)n)q65p75gqkd 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, zs2gsa 7 q16qmp0pk o)8jd5 im5)k genFig. 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 Eyv,) tne; m77gwbe4fcarth such that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat,b env4 m7)7e gfywc;t the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate 3) 1 faxi/qvdpe3m ,zhof 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 unifor:nklny-8; czp 39rv/f +usyftm cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calc8uct3y-n ffv n+s kp;ry9 lz/:ulate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two -yx1x) z kmqq+nree2tnp nq:m2,d(kg, q st)4solid cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of masqx mekpm k)1-nn n,r) gteq+xz:24sq2t( yq,ds 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of the rear whe8pkr 6tz:7y2.b0dc ejvuw)s lel ($\omega_R$) 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 ourje35; ch nooa1pnud*ip z::8d ;i re6n Sun, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolutioanupi;3j5dz;r8 oe ::e6noi nh1d*pc n every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for itbdn3wuy)6 o,mq-mqk7zi za 31 to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, usemi,wy671n 3kb)zoda mqz-q u3s a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrate/f -alrlm: ( s5fvu7vds 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 (hor:c+ xt)6igg8 la+xusop) is rolling on a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e. +nl4:4;cqodz 8fhmzx ,t3bw*h kq) pd, 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) 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.8wmb zc+ ;f4:qdn,t*hq3z4 )lhpo kdx]

A wheel of mass M has radius R. It is rz1g8tpoh5o6z 3i9aanw,x* e standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a stapwo*9o 3,gz1en8z 6rhi5 xa tep 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 flagy-+moaz dt21 t road is making a turn with a radius The forces acting on thm2g+ayzo 1 d-te cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of leng- nltr 8-lc1q65in givth 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the gcn-1rqn il8 lt 6v5-itorque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thin rods, d +edse+;.;tcmy4uq smaking reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms hel.sqme+ 4ud t+;dy c;sed tightly against her body.   

  You are designing a clutch assembly which consists 226qhcf. a8hsh 2rusk of two cylindrical plates, of mass.s6 c2a hkh2f hq8r2us $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 .idk i2+ 4 ctd/nrzfo0rolls along the looped 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 loop witho2.dn/ko40r+fc z ii dtut leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assuwufd4:j/).pxe t hk(14xgqdz me $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as ec+ sc yk,4zju7e j+c/the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acc++ z,c/7c4usjejc ekyeleration 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