A bicycle odometer (whi* + e3m ayiztwwc:qg/:ch measures distance traveled) is attached near the wheel hqgwiwcy3et+ma *: /z:ub and is 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 velocity. Does a point on the rim hav qzw7z man7f k9p,ve+8x0.teze radial and/or tangential acceleration? If the disk’s angular 7xmz9a, v08 7t qe+wnzfp.kzevelocity 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 yw dr0: wshf).by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater+kpcl5ge2luz em2 )i2+,lthl 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 difficult to do a sit-up with your hands b 1t w9p 6m8jcpe*zwb8uehind your head than when your arms are stretched out in front of you? A diagram may help you to answerwbmw6z j*c u te91pp88 this.

A 21-speed bicycle has seven sprockets at the rear wheel and threj qp 8ooc4w *ho8oh5,yqu*w.tt ahy,/ e at the pedal cranks. I ohj o85h4yq*ttw,w pyo/a,u.oc*h8qn which gear is it 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 slend- twg2d(hzyq2 er lower legs with flesh and muscle concentrated high, t-y( ghwzd2q2close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, n.tw f(/;u fmd5i 5hkcfarrow beam?

If the net force on a system is zeetv 7 16p1t o5z)hpptmro, is the net torque also zero? If the net torque on a system is zero, is the net forcee p5t161p)ttmp vz h7o zero?

Two inclines have the same height but make different angles withnz0yo(t9 hfm 7 the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of thnt( m zfy70ho9e ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an incline. Obzp4)v1 kv c* qkt1jqs,x)3ifne sphere has twice the radius and twice the mass of the other. Which reaches the bottomq, k4v xbp1qvji 3)tf)k*1zsc 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 and the same 5d,i0vol1- n udx)nnj mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater lx,n ndv1-di0u) jn5ospeed 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:54c ngpcvif6 momentum are conserved. Yet most moving or rotating objegn5 cc4 fv:pi6cts eventually slow down and stop. Explain.

If there were a great migrati*q-6 s(/pre)89zd4 surjs8 4lhl pasrp6ke ieon of people toward the Earth’s equator, how would thised)uq4jl4eh6i/ ssre 6r9a p(k-spr8z*l s8 p affect the length of the day?

Can the diver of Fig. 8–29 do a somersault with8g fttjw 9;lpu.noik-y: g2o+out having any initial rotation when she leaves the bj pl8wngok ;-+tyoi 2.f9:gtu oard?

The moment of inertiabqe )7t,0ug ls of a rotating solid disk about an axis through its center of mass is tb)g07qules ,$\frac{1}{2}WR^2$ (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 rotak3d z/7 nyu 3i;t)chleting stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular ve hd3i)cln 3/ukte 7yz;locity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. 2d0de xfwv 8j)However, one is hollow and the ot dfw0ej2dvx8)her is solid. Describe an experiment to determine which is which.

The angular velocity of agay 4w0r zg,5z 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 top of the wheel. Is the ,a04zgyrgw z5angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turntable)n c l8p/rhy/k ac.7m,;u spdh. What h)kp ;h8acndpuc.,l/m/r 7s yhappens if you walk toward the center?

A shortstop may leap intonou r+he;dczinr6 (47 the air to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate+zhnur; 7n4c(6do ire in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular moment6ny9-wb zavxl7 r9.j 3uir2ic um, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keerjii.r yux2n -wb963vc7a9lz p the helicopter stable.

  Express the following angles in radia h a nyls65:j*peemfg b*aim x115cu27;rs;pzns: (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 amaz 5plbanmkw-7t+ jvbn l) o0e44ing coincidence. Calculate, using the informatio5lnb 7+p4n)jblw vt4k m-ae0on inside 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 directed at thy yhj7udwh70/kw y 8y9e Moon, 380,000 km from Earth. The beam diverges at an 7w h y9w8/huykjy7 0dyangle $\theta$ (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 69ncjfa c g7 d+c3e07bv l2y+tt,u;o gu500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slog 37y tu ;,ef+9adjoc2lugnb0c vtc7 +w down?    $rad/s^2$

  A child rolls a ball on a level floor 3t pr:xry(p i0+e lak:0oga6 l).5 m to another child. If the ball makes 15 6l+)a(ag:erolkp: pxr0yi0t.0 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many rs o5g +4r ynayjw.v2a5vb)a9 jevolutions do the a .ygs bjr25 9n)v+wjavao45ywheels make?    $rev$

  (a) A grinding wheel 0.35 m in dicy6)a. 5lklc47pyfna ameter rotates at 2500 rpm. Calculate its angular nak.75c6fa )pyll c4yvelocity 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 (Fig. 8–38). (xi- (mha5775lq mirih a) What is the lin (a ih7xm5qi5rl-m7hiear speed of a child 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 velocity of the Earth (a) inyh +z ( :;(cryxqjz0yw its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed onc76ixp+ )duloo1 zmn u 2r31af 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 rotate if a particle 7.0 cm from the axis c2-nosr1-ztiof rotation is to experii-s2 z1or t-ncence an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpi* 9.:dcw phxem to 280 rw*dhp 9.:xice 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 radius i. d-5s*swyvb:s o,m t +4tmsa$R_1$ 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 spacecra/3hl2jkni uz zet* 6+:jwr)xat 8n9 lkft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the s6/ j9uwhit el)nznr ka*zxtl+kj82 :3tart of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm k,,* myxj.0ssevc6iq c(z; ey in 220 s. Through how many revolutions did it turn in this time? c,ec.zv*e qmsyy0 i;j6x (s,k   $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.fu3g 7s-rcz.n5 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 p)wd-v6 0lnzj si,m 0lfor the stresses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20wn6d-lm0pis ,0lv z )j complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly fromb3 (mgr hl5swv.r/e d- 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheeg-/5 l3bvhrwre(.dmsl have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 1500 reve s3uoqv:4mt +olutions b t4v+3eom usq:efore 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 65 revolutions as the car reduces its speed unifonmg;)0j : xdo xb87dudrmly from 95km/h to 45km/h The tiresg) dxmx ; j78:ddobnu0 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 )po3 yz iis)*rthe car reduces its speed uniformly from 95km/h to 45km/h The tires have *osz i)yrip)3 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 aiyct 67 t,;ddo/ eeuaji25m/a bike puts all her weight on each pedal when climbing a hill. The pedals rotate in a ,u /5it ;djt/7ieea ayo2mc6d 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 doox : fclvfxd he530+yhcdxk*j71 o*2 osr 74 cm wide. What is the magx hfl xd 0od cy*hj2c31e:fs +vk*xo57nitude 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 about the axle of the wheel shown in Fig. 8–39. Assumhk ) 2/a;qsks0u+ )yhp5rkrb6t6tsd me that a frit+sm)skkdk)2ybr/thu6s q 5 0 pr6h a;ction 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 the enekdfa;3 r(eu)dum q1 :ds of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal posak:dq1uu)e re 3md;f (ition and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an enginer.26dzu lvop;3/ildv20s opm require tightening to a torque of 38 .2zvi26rm o3d0lsplp v/ d;ou$m \cdot N$ 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 radius 0.648 m when the b yx+ xisd owvs3--*k.axis of rotation ikv- x-. bw*dos+ is3yxs through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diami;gu;hwb),t 6nbm ac( eter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignored (,anmu) g(iwhc; tb;6 bwhy?).    $kg \cdot m^2$

  A small 650-gram ball on the end of af 62p vkn0fu2 zid8uh/ thin, light rod is rotated in a horizontal circle of radi2dff68 k iph0nv2uu/zus 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 wheel5fdcl4yr2yzx3hxyp)w+ v 13g rotating at constant angular speed (Fig. 8–42).d 3 3hyyfrv)zpcy5wl 2xg41x+ The friction 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 pl:g/s gx/hod; oint objects shown in Fig. 8–43 aboutdo/;lhs/g x: g
(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 b3 w j13 j at5ti51vlxrj6tvj*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 sa3p j;b3+ qj lmoz2qvm.tellite spinning at the correct rate, 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 required s3o m+jz;b.jqm32 lpvqteady 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 wiz5 oz 56 gi7ilhmqgr: :42zuksth a radius of 8.50 cm and a mass of 0.580 kg. Ca7z osqh5gi:2k:uzi5g 6rl4z mlculate
(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 fro9lqq af;qagj ,q 3n--q95+zqa ntdb1fm 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 zt73 ww-(fja. n36kxr+mjskg 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 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 aswx-an7 .rtjkkg3f6j3+ m z(w cceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating atmy7 yk4 74le/+z0y iwufsxyl2 10,300 rpm is shut off and is eventually brought uniformly to rest by a frictional tofw0/key7ix z+l2yluy smy447rque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a 3.6-kg bszmj cq(50c8yall 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 foreaxct py:-ew/chm /u *5orm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly fro-/ hpu/ot:m*5 yecwxc m rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as shown in Fij9ixxn7g du(+ g. 8–46xj9 +di x7ng(u.
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consistsof) zd1s-2ymz 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 fromdmrkj.w904:lux 9ht:7 zwmq m rest within four full turns (revolutions) and relemt9.zkjmhu rw4l:m79 wd0:xq ases 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 momenw 3h)tm ckujmb(v;-p n7:hoq3t 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 de;v ,cg8ai(yo64eez7bll g+l f velops 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 withoybe9,* f9h mdhhca68lxr e7 4-)rloc eut slipping down a lane )rbdmey hel*a-7 hce6l9cf rxo849,h at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect 9;)e- +uue1aw74a1ppq:ygzjyx le u rto the Sun as the sum of two termspal :yw uqpex4guy7e9u;)a1-1 j + zer,
(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 7.5sove)q9 xeb/a -4 *mgw0 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 mve g4 * qxbw9o)-/easrevolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 x0zq:2nqy z( wmg5+ qmcm and mass 1.80 kg starts from rest and rolls without slipping down a 30.wq gmyz2q(mn0zx: q+50 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is balanced vertically on its tip. It starts to faldd /m6v0 sc9a-fw qcb;0hnp +hl and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.] fq9 6h/h+0cbvam;d0ncd w sp-   $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on th9-1df zt0gf fte end of a thin string in a cirgf1-ftt f09z dcle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular m lcmq9 wz,7phs dnk- d9k:yj2,omentum of a 2.8-kg uniform cylindrical grinding wheel of radius c k7 :-,kwn9sl9mdd2 ,jhzpyq18 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 tpuf 9fzo,(3r side, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizon,tz(o u 3pr9fftal 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) cajwsgqawf)+m.4db) a,n reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes aj)m .+wsdg, wf4q) ab2.0 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 rota-x;5,yte aefc x7u+yztion rate from an initial rate of 1.0 rev every 2.0 s toft,ax +5cyxue; 7 ey-z 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+/rfg0 j v46g f/mu:vix4 fbri)wuih / 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 diamet0gf)+ihvxii/4muu/j 4wbf /frvg 6 r:er 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 after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skaq+ pen.uhf r8:gvb+b (ter 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 the Ear mgi42a wj9to))er0zmth
(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 drr5qd: xi a.pqrw88h,eujc89 qopped onto an identical d8,9crpir ejqa5w 8.hxqqu:d 8isk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.p ,w82wf7 tc+v 7ojz mc9lesk14 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotating merrcs/-gw; v 3wm-xv0wm ny-go-round platform m/3ws g;vc0nw m- w-vxof radius 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 v6 egviokf/ bjhe,8 20iv4s1+scg 9sjrelocity of 02vs1soih6gerf09b cg+jjs/ 4 ke8i ,v.8 $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a whittzx(f d qfd7rv)996s3f xwnjdq: l;-te 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 angd f(l96w:t9;zvd)f7qfrx3s nx- jq tdular 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 windskk6 .5ft(bv qc-y+q21 pe mob db(zzq/ in excess of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass e* ,47xbj;k lnwwt1 o;6k /mjz ;ui-ygqt(iev $ 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 canp9oj mj*)cqg2oem g+ + rotate freely without friction. The moment of inertia of the person gg pj oo2q9m) *m+ce+jplus 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 diameterty87b z57xrbhd s x(/e merry-go-round turntable that is mounte7 rb(besx5 h7 dt/xzy8d 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 on the ground with the end of the ropemdvcn+ g 4oh2yvy5*:w lying on the top edge of the spool. A person grabs the end of thnvo c25 4wvyy*+mhg: de 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 q(by/xom h/6ev.ex5kftvxx (x f;:rz1+txc. side always faces the Earth. Determine x5x xyete.6fo/ vm1(xz h bk + :tx;qxc(.fv/rthe ratio 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 from rest at a rat p;6g+q3kqpat e of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform c2o o,pktquym0(bx qnk * n9o40ylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the k4tx02qqop90omn (b,*kou y n motor.    $m \cdot N$

  (a) A yo-yo is made of two so1 )cgyt;n+up+6voar.fw y rv,z) et2glid cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (conceog+u,v.rz2y)t n)ywg+rtpv1a;c 6 f e ntric) 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 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 +7u*8)xy z msbld1nky 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 tio4 c4vi6jz1ihrn0cdqmdn3 :i9.f f h+km )yh5mes as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo g3f4 1:miy)dih.0ohc jfkv+mnzhr c5d4q96n iravitational 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 , drnad)b2,z (v0q5ezjts t drz5 z9v7for it to use energy stored in a heavy rotating flywheel. Suppose such a cars5v 9z ttr7(b n)dr5 0qv2ezd j,azdz, 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 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 u8v;duwn4ba9es 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 horizon8l-z .jp. *1mzelrcg/n-;ig- tqx zmk tal 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 fr5u4lnqrg.8s hzb0b/c eely (i.e., we ignore friction) about a 5ghlu/b8 s.q c04bzrn 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.]

A wheel of mass M has radius R. It is standing vertical iqdzo*+ ++ur1mzn5d0vj xb4uly on the floor, and we want to exert a horizontal force F at its axle so tqmb1z+o v+0z ji4nrx+*du5u d hat it will 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 makin ru+ts6a; teu:g a turn with a radius The forces acting on the cyclist and c ;6aretu+ u:stycle 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 1.5 m and begini1wt4 j mvdwgo9o5m- 9s whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 9d9 ojmi-4w5 twovgm15.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and hedc, dm.dnn j-z 5;wo6ir arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.w; ddj n5o,idc-.6nzm   

  You are designing a clutch assembly nzr.: vi5h3mx which consists of two cylindrical plates, ofv3 zx:nhi5m r. 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 loofnq+ar:f ut 0m8b)r4wped rough track of Fig. 8–58. What is the minimum value of the vertical heig bfwn4u8)atr+0:mfr qht h that the marble must drop if it 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, but do nowj avc8 :7h2,* 0qrywbnv:xun t assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uks,-ps ,xw cl:os4 /tnniformly from 90km/h to 60km/h The tires have a 4, wkn,xps/o-csstl: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