A bicycle odometer (which measures distance traveled) is attached l.;pbj,8 -xvolr e6za near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bieb ;ajl-,8r zvo6lp.x cycle with 24-inch wheels?

Suppose a disk rotates at constant angular veloc bfvmbwh7s6ab 9 :)1iyity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angulab yfbh9b: m1sv )6iaw7r velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the angular velc()a;i)a maxqocity $\omega$ Explain.

Can a small force ever exert s ,9+kvh h;e*7 9idlo w4tmjtia greater torque than a 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 diffihq1iyxzhq 1 hb,96hx(cult to do a sit-up with your hands behind your head than when your arms are stretched out i,19(xyh 1b6 zihxh hqqn front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at 2*rorx m0fz4r mrcw2(the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Whz0ro r*r (wr xcm2fm42y? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able tf *aaldp/m80 v3mnpuv9e*:mlo run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fi*nm3e : uflm/vv9ap lmap *d08g. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkd(v ,qsz)mkh;ers (Fig. 8–35) carry a long, narrow beam?

If the net force on a system isxjqizn0v k::0ggo;n- zero, is the net torque also zero? If the net torque on a system is zero, is the net fo0 j z-o kq:gnn0;vxi:grce zero?

Two inclines have the same height but make different angles with the horizonta1+ev7un qh:ov ma50* vqxslk+xv(o;q l. The same steel ball is rolled down each incline. On which incline will the speed of theq0 l+e75;xuhva:vv qkq*(nmv1 +xoso ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an inclh.owqllczoz 1a 6s0am) v0 16qine. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic ene1 lzw ) vm os.z66oalh00a1qcqrgy at the bottom?

A sphere and a cylinder have the same radius awfnh nlxo++ cyo -5q7*nd the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which 5hq xfo7n*clw nyo-++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 momentum6/op qopp)a.r x v:uo9 and angular momentum are conserved. Yet most moving or rotat)9prxv/puqo6o: . apo ing objects eventually slow down and stop. Explain.

If there were a great migration of people toward the Eart +l65atzd )zhqh’s equator, how would this affect the lh5z+)a dq6ztl ength of the day?

Can the diver of Fig. 8–29 do a somersault without having any initiq 92zn.o fsp8hal rotation when she leaves the pq n.zo89sfh 2board?

The moment of inertia of a rotating solid disk about an axis through its h* *ofg:qtc2by5 llb /,ish.dcenter of mass 5 q/ si,yfl*b*2.l :ogbhht dcis $\frac{1}{2}WR^2$ (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 out,n/80 win;nevvuv fmdi-et( tr x5a 6+stretched hand. If you suddenly drop the masses, will your angular velocitar ;e,wnvd -v+n xu5f/8 in0t(6et vimy increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass.2)z 7xowufx4f However, one is hollow and the other is solid. Describeoxx4f)w z7u f2 an experiment to determine which is which.

The angular velocity of a wheel rotating ohychv.laxv14+* w * vkn a horizontal axle points west. In what direction is the linear velocity of a point on the top of tvxh4kvach*y*+vl1 . whe wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standingd dn6z8kl(y0g on the edge of a large freely rotating turntable. What happens y6zl8 g n(0ddkif you walk toward the center?

A shortstop may leap into the air to catck/5q 4errf u7oh a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will noticeqr5ke4our/ f 7 that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momenz. eprw; 7l,uttum, discuss why a helicopter must have more7zr;p,e luw t. than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following . kf9gm .jvwh6eoark2w)x-l ynjl 0/s8bj6 1mangles 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 amlks; za -v.99jqx la:pazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun anjlk a9qa-v x9z:pl.s;d the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,zlp4pq *e j1c/000 km from Earth. The beam diverges at l pz/4qjc1e*p 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 rpm. When the motor is turny x3rcy6tt7+rb6;ms zed off during operat7st6m6yxr 3 + br;tzcyion, 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 j dc d1e8t4 +dzhh :6.oxx*el4nts8zdlevel floor 3.5 m to another child. If the ball makes 15.0 rec: 4d 4st.z xhtz*don6 e hj18ldx8+devolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revoh.rx 9s+m-keavgh2* flutions do the wheels makemefh hx. g2rs+ak*-9v ?    $rev$

  (a) A grinding wheel 0.35 m in diameter rr e( eldle t)2s*t2zn; +hgsv4otates at 2500 rpm. Calculate its angular vesel4e )(nt* ;d2 2etlz v+grshlocity 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 comple7c)bi-sr f6me te revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m f bi7fs6re-c m)rom the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular v +f( kgjk2j)nuelocity 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 q+ fgl+tdc/9s 7q6rrjpoint
(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 particlhr (3ocfvim: 1e 7.0 cm from the axis of rotation is to experience an accelerar 1fi (3cm:hvotion of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpm to 21 p lihd84xjpw8dfs:9 8hf8ddwxp1lpsj89 i : 40 rpm 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 radiu p.lbs00:35j)wyak* oq fha93ym obz js $R_1$ 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 - yk(dski/d xsj29pi)aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerate9i-dss) pkd/ 2 (jxiykd 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;e+ hqesl)t4qh-gn yi7 (/ffjl8oc l/ how many revolutions ;8eo)lqs+el7g qj th ( ifh/-/cnyfl4 did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calcul7jkfhoamfu kde2a/h 5z; jm+c,:64tsate
(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 whirli1xoav kal / 89fky4 3tx-*2eq)nyvfiang “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching itsa- *f 4yn 1)8olq9 ix2vfvkextk /3yaa final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates unifocn , cqy9 kf+8(0otqr.v3hqdvrmly from 240 rpm to 360 rpm in 6.5 s. How far will a point onn o,8v(qqy9d0 t + ccqk.frhv3 the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850zf4p1i626wa/huq ksu u cy43arev/min It turns 1500 revolutions before1 a2uq6 f 6i/uu34w zsha4cpky it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65o 0fe:j 1 .wn2xw3/ixswbpo b) revolutions as the car reduces its speed uniformly from 95km/h to 45km/hs:.wp032w/oxeoijbx f n)1wb 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 / qcm(/e3ecmok ckl4z33 4ha4zw7nlwuniformly from 95km/h to 45km/h The tires have a di m/l3n73z ce oh4e m3qkckz4 (/alwcw4ameter 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 c0 znuv w(66sxky6aha7limbing a hakna6u wsy z60 hv(x76ill. The 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 74 cm wide. What is the mj .:gqels+ x 1;sch xf9wg4w/uagnitude of the torque if the force is w:gjhg+4ls fuscwx;/ 9 xeq.1exerted
(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 ozg;*e:zj;p0 zl vqz7emm5j* xf the wheel shown in Fig. 8–39. Assume that a friction j em*z7z e zqpjml0:g;5*z;xvtorque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a massles ct5 cq )bcrxs*xs8.6ms rod which pivots as sh)6xxc * .bsq 5cs8mtrcown 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 cylinder head of an engine requ.2wipmmj- ,gn(e x3d yire tightening to a torque of 38 gw m.2,ji(y epdx3n-m $m \cdot N$ 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 momentdqv q*kn ol.04 of inertia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is through itsdl .*qnq40ov k center.    $kg \cdot m^2$

  Calculate the moment of inert5py/pbu aaw0 k8a jr;1ia of a bicycle wheel 66.7 cm in diameter. The rim and tirep0r;kjaw 15b auy/8a p have a combined 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 thpr1y)liv .-0xqjr4pet k am; 8in, light rod is rotated in a horizontal circle1k4 tjp08emar p )rqxviy.-l; 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 bwbdst/qs 14a/s p05z.q bbs;bowl on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The fr5b 0ssw1qzsab4 t ./d;qpsbb/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 array of poin7wws;;a c) nu4j,qdfo t objects shown in Fig. 8–43q) 7n da4;jwsfcuw;o, about
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen atoms whose cpexcp;f t hv0)i7.r; 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 0ur kvb s7od2;3hxf,d cr0dj qa0e*u1the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a*crb0roqdkj 1de 0;s3 h 7xu,d2f avu0 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 unifoqrqbm 3na5w+ 4q 0lbpd)6 +vefrm cylinder with a radius of 8.50 cm and a mass of 0.580qq e+ w5vraq0m6dn +b34)bpf l 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 b 5)b gkddrbed3bswef+ w;i, s(y0 (r/uat, accelerating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merrebb b j5px23topv)2-:b;rwk71z rncl y-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 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 prer2nbb2jk c)1bx3o:z5 p-t 7l wv;pb roduce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rot*) h;b5gnyp7wl ezd w.-zkh.s ating at 10,300 rpm is shut off and is eventually brought uniformly to rest by a frictional torque of ..;-zywhn l*w)5pe7bdkz shg1.2 $m \cdot N$ 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(g9qg1rv37:6yg wm3rr( b vxsp .r cej 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 by the action of the fore, 8/q/m lqfbm2)vdu w-uzwr0g arm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated mqqfwu8) 0 /2g wdb,zml/v ru-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//2)tx au ei 1nnlucc5 considered a long thin rod, as shown in Fig. 8–4 tuen/nu2)cl/ax 5i 1c6.
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consists of two masse;al*xax.k f b:nmo41es, $m_1$ 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 resvznqv1 85g2d*8 xj-d0- m njhok ny1tnt within four full turns (revolutions) and releases itj d88mnxh*-5tykngnnq 1-1 2zdvj0vo 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 moment of 74xhd sg)xnc )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 develop rvklm uhc )*88bqcv8:s 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 rolls without slipping d9e) uy4 ( fr4x+wviyaif6 /ulzown a lane at 3.3 vi y/fu9( axzfy+w64i e)lu4r $m/s$ Calculate its total kinetic energy.    J

  Estimate the kineticd)(:k..h s)dgyxubgb 6qd. yy energy of the Earth with respect to the Sun as the sum of two tbd 6q sdhb)xy..k)g y:yd.(gu erms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius ofn 1 6er4h6qikz 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? As n 64hqie1rk6zsume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls without sovg9dttn kd1 0mav b1 3al-u*(lipping dow0* ak3vbltua m o-dg9tn 1v(d1n a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is balanced vertically on its ti)8p1fkrt ta, r,hpz/ gs vzxw :5y,/rnp. It starts to fall and its lower end does not slip. What will be trt8y1 n)5p/,,f: tkr/vrp xs z w,hgazhe speed 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 tr72elxt8e p 0jhe end of a thin string in lxpet70 2r8je a circle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angularppuw(2kxu c 69 momentum of a 2.8-kg uniform cylindrical grinding wheel of p2(uu9 kpw6cxradius 18 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 his side, on a platform that is rotating at a ra d1tn-zj zl820x+ lhhtte of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48,h z ln8dj0+ttx12l-zh 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 -hwo n ,4+z(;kbxqyvzher moment of inertia by a factor of about 3o yzq n;k(h-w,b 4x+zv.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial rate ok sn4tz80 8eio5x8suqf 1.0 rev e4izsk8ones 88t xu5q0very 2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a verticaleh 2+o:*vq:gcvev eqq-(f4a q 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 o -:*f aq2(eqqe:vcoeqhvg v+ 4f 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 after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinning at 3dm7dcra2m5s ln h .x j0..oi,u.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 angularrg y:u zga;r :,: y-7pp9rsair momentum 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 hej0 hsmv gur:;ev(9o+e(;yhdisk of moment of inertia I is dropped onto an identical disk rotating at ang;vh ey0+jeh(rgv(s9ouh ;m e:ular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns atiz ) n vzdo.10t8,po,uyq dvh (idg2h, 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 center of p-rdk gqo m it1.(2*laawfm6 ,a rotating merry-go-round platform of radmki,f62p.rq altm1ogw -d*a(ius 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 rotating freely with an angular velnu4fyw., y t r+or.9hhocity of 0.8 rnot ,.9y.urw4y h+hf$rad/s$ 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 pzynfsuqw)1:uzp hhb6(5l.fk7t/v(liow -,ly collapses into a white dwarf, losing about half its mass in the process, and winding up with a rp h-)p:n,z6 st( vwuylfulbfh/.wio(q 71z5kadius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in excess of 12ngpz. ,if(in8 0 $km/h$ 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 tr mvs g7rrf pj1l;:s040d(ft$ 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 w0ewr5 xj5 pqmh bv+*+t tp4cv*ithout friction. The momet *h qjtb*5e+w 4vpc vp5x+0mrnt 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 diamevt ed2gprh9ys9ga 8g5+7p uc1ter merry-go-round turntable that is mountedgsa 759 1e+tgprypvgudh8c 92 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 onxy t(i7pgvag1 r):*p o the ground with the end of the rope lying on the top edge:gx7i grt1)v*p py(a o of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always facesqg-t en;g2m2on-pq a5 the Earth. Determine the ratiqoae-2g2p-n tg5;nqm o of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates fr1ip s(3s-q s+e/jy wddom rest at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the sgon.bu1j.ax 5hape of a uniform cylinder of radius 0.20 m acquires a rotational una15 j.gb.oxrate 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 cytsrzs ,ay6 w*(y9v. 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 it reaches the end of its 1.0-m-long string, if it i.*s 9y ,y6yv(wrztascs released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the anguqx h x5b8a yw 583a1-enbkbem0lar 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 sigd;7 gd 3l+m2c9qcdxgze of our Sun, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to ungd 9;dccl2 m7g+xg3qddergo 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 it to use energy stored izlkdtt olwd d-2m1a(9x7 8t 333jvrbon 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 kgdlmx3 d 8z3vl97-w2 arttjb tod(k3o1, 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 illustratk 03zazjq otn-)5 a0dg;y;vyoes 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 n).nrmo)s uay 2 sbw5r;4u)po 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 pi2o i2m-iq(ygv jb5e3 fvot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and thenq (em2igvjybf2 i5-3o released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the floor, and we+yzrl,u+g 8s3bjq(h c/ fumtw5p 5k9x want to exert a h8 9zl,5gxcyu h sbr tp+qmukj35(+w/ forizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed fy/+ 02ditvjpv=4.2m/s on a flat road is making a turn with a rti/2pv+j 0 ydfadius The forces acting on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.af,55:gb,vmf aqvif7xpiwvq-99 bd ,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 wgf xd-5 mp5 7,ab,bviv9 v a,9iq:qfftorque 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 and her arms7l rzr7owe3d5 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 owlr7drz53 e7tightly against her body.   

  You are designing a clutch assembly which consists ofxxwgm o5kpbdoey 2 3r ,b,2d;db374ch two cylindrical plates, of xb,7 xhd3 doo3 epg;25y2bc4b mrkdw,mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 r d1z/acen y3h/ lmttq09 36nmiough 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 a z/niy6n1 9/30lmqm hcet3tdthe loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do ; 89w+z. r qlx3rjbnvknot assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as.z7sacj. , gmarwh)(n the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m ch s..a)gr(, 7amwnzj. (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