A bicycle odometer (which mi e c +a4sns66claa88seasures distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bieca8 ssil4sa6a cn+68cycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on the rim havjqryu1* wlg+4q, fr4e e radial and/or tangential acceleration? If the di f4,1*gqeqjyr 4u wr+lsk’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 a single value qme;y oymz v9,z;u6(vxo obld 1c11d :)rjg4mof the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger forccxek.jw1o p55m n4;zt r* h/ooe? 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 sdg).);vlca6oj :sl66l7ao edd amyo0 it-up with your hands behind your head than when your arms are stretched out in o )dj)mooall0a c;a66sldy7 6v.d: ge front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the 9q/ yhh- -u sa8fhj2ocrear wheel and three at the pedal cranks. In 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 sproc/o hs y 2-j8h-fuahq9cket? Why?

Mammals that depend on being able to run fast have slender low7w5v.jp ;zh46y ,e.ccd9or fv ofmai/er legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution4cdofha,ozi5p/;7m.c9 y vjw v.rf e6 of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a ldw 9e pwbn45 -s 5r-.zhul(ejaong, narrow beam?

If the net force on a system is zero, is the net torquem k 0.dt-dz)*xtuvn q/ also zero? If the net torq*nqdx.k0dv z/ m )utt-ue on a system is zero, is the net force zero?

Two inclines have the same height but make different angles with the horcd:5a, mxwq.j9lplx -izontal. The same steel ball is rolled down each incline. On which incline will the speed of the balllpcxjaw5.d ,-lx m:9 q at the bottom be greater? Explain.

Two solid spheres simulk7 0ysf)c ck0hwd6vk ,taneously start rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at t0kkf,c6ksyc d0w)7 h vhe bottom?

A sphere and a cylinder have the same radius and the same mass. They sgzw/o*s5 3 4ogp:zoxdtart from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has oo3z*w s4 pog:x5g d/zthe greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momenzadfi ;-)r-gmtum are conserved. Yet most moving or rotating objects eventually slow down an-md;g r-izfa) d stop. Explain.

If there were a great migration of people toward the Earth’s equator, np(( qzd6:w9inf)+v t m:lnsehow would dnnvt+iq mnfs :p96zw el:(()this affect the length of the day?

Can the diver of Fig. 8–26krv ;y7ix- d yt8i,hh9 do a somersault without having any initial rotation when sheyrtih;d 68 yvkh- i7x, leaves the board?

The moment of inertia of a rotating solid disk abif881fzo4b qmout an axis through its center qfif8obm418 z of 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 haqy7v*a5e6 t-hrd(guuo7l 2on a rotating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, ddro5 7l 6uahuay*q -ve 2(g7thecrease, or stay the same? Explain.

Two spheres look identical and have the sa qd-nsn 8*l0wkcjp3wd9 iu (n(me mass. However, one is hollow and the other is solid. Describe an experime(n(q8- njku3ndp wci 9*dwsl0nt to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points west. In qa3sfafh9a8q51 k 2dowhat direction is the linear velocity of a point on the top of5d81sqkq2 ha f 9oaaf3 the 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 standing on the edge of a large freell6pfuz2 ske-1yi6je1c9e)a c y rotating turntable. What happens if you wal2 1kysezifa jpel 6e1)c-cu 69k toward the center?

A shortstop may leap into the air to catch a ball and throw it quickly.2k x3tb116n n2v bnde9 4scstm As he throws the ball, the upper part of his body rotates. If you lobvnbs4 213tt162c9emnnsdk xok quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, / r(yh0vs2nhq discuss why a helicopter must have more than one rotoq( /hhr 2n0svyr (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians.x4 26cdf-qksnxjupc5 ttz0d-ic ) n7: (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 avs7a h5ms9r; amazing coincidence. Calculate, using the iv 5sa hm;r9s7anformation 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 the Moon, 380,000 km from Earn-bfborsy ;;.t rb6-sth. The beam diverges at anssy- o-tr6;bbnbr;f. 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 tuzd51 ub9h1a 4tutw32.scy pfnrned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the n4y 1swcu 1 9.afu3t5dbzht2 pblades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the ball makes q3/bu+fn pt5q zwwul t0,g 1cyi.f-m215.0 revolutions, wha qctpuwwuq2- f3 nzf/.1 gi50 tb,l+myt is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutio wf*ndx8r 9)8kqv8uuwns do the wheels maku vq8nu*r8x wd8w9) fke?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2*qh mebne/p.n+tv6 -b500 rpm. Calculate its angular velocity inten-mp+eb.q*/ 6hbvn $rad/s$ $\omega$ =    $rad/sec$
(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 complennbb0 dr +s5f*te revolution in 4.0 s (Fig. 8–38). (a) What is the linea*f +nrd 0sbn5br 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 z(f hf n/v+ ywjjek(1h:qi*7m(a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed or0 9j,/av hyjlf 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) mus. 6wz co7-3gh 5huqdhp0xvq(ot a centrifuge rotate if a particle 7.0 cm from the axis of rotation is to experience an whdqq5h-7 (p.z goo30c hv 6uxacceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel acce49 g *s: vu8rmiti qc.snvqe41lerates uniformly about its center from 130 rpm to 280 rpm in 4.0 sssg1 c4v*. i 8em 9vquti:nqr4. 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 qjf+dumu l awn)e/,1ik-8, 3t j7niqz $R_1$ 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 aboar 9urt ;s;kqv;td the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they avturkq;s9 t;;ccelerated 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. Thre(p,f (j(rn2wtasmvw:3 af pby7 o51lough how many, yv71smw2f(w ej p 3f(o(l:t pnraab5 revolutions did it turn in this time?    $rev$

  An automobile engine slows down2; , bagaui9dmgwq,4c aezn/ 0t13hy j from 4500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed jets in a whirling “hu0tmro9b pp ,eb4i5 mc.man centrifuge,” which takes 1.0 min to turn through 20 cbitrbc90 ppmm,.4o5e omplete 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 uniformly2pv0 :-uebf uy from 240 rpm to 360 rpm in 6.5 s. p fy-0u ueb2v:How 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 c/bxjpzj:* 2aat 850rev/min It turns 1500 revolutions before it come/2bz jxpac:*j s 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 se2.-duw qq gb7peed uniformly from 95km/h to 45km/h The tires h-72dqw ebug .qave 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 makx;qz *1 osfa4ojj5utl-2ggd :kw7qj +e 65 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h Theg 2qj 7-wk* z;f+j:lqodu1 j4og txa5s 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 w0:q b gk;jm k(x asfibet;j0::eight on each pedal when climbing a hill. The pedals rotate in a circle of radii;:jjs:0 gbfkxkt ae;0 b m(q:us 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 isa4k(w2;hc9 fx4ttkmr the magnitude of the torque if th2c hfxt4mawk rt 4(k9;e 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 wheegs 4wvp(q4 reyy7 m5m8l shown in Fig. 8–39. Assume that a frict4r( yg e4mwv5pq7my8sion 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 ends of a massless rod which pism)vvfslbek/zlk ) m//-cger1w2g4 , vots as shown in Fig. 8–40. Initially2wlev)- )r kf/e4/g/ , msgsmvl1 ckbz 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 x1ay/w, ah z69kxa so8engine require tightening to a torque of 38x9s186oawk/ yx,h aza $m \cdot N$ 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 .vn2g2 8b fy9wtude sg)q7w 2dof radius 0.648 m when the axis of rotation is through its cene . 9g2wu8)7n2fvbwg ddyt2s qter.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in5iaend3;fu0z ,0f odj diameter. The rim and tire hd0f5 0fi3jeuza o d,n;ave 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 thin, light rod is rotatena1ecyklr0 l9(7e5 ,vz9a f(y w)tnszhjqr-4d in a horizontal circle of radiz-yvl n0l 1(kzt4naja)rr 9eh s y q,e(c59fw7us 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 qxtmkt4c+58 0j kegqh 9 qmaazi156m0bowl on a potter’s wheel rotating at constant angular speed (F zt9txi54hak5q6kaqmm g8+q cem1j00ig. 8–42). 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 array8g 16p(kyqe (vhekk+ q 1afhs5 of point objects shown in Fig. 8–43 a pvk1khy(1 ehfga586eq s(k+ qbout
(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 who1o ejq1.iek2g se 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 cylindricald 9stluew i194 satellite 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 4.0ide l419w9uts 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 unu0 yg(c0yd iu9aih4 y0iform cylinder with a radius of 8.50 cm and a mass of 0.5800h 90ydiy g4c i0uyau( 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, accelerating iq*s(t f/ps p+)hv:q u9 nvs0fdt 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 lic*/:w,iitak 4futuo3 66nta 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 oppositwi*6cf, nu t4i6:ltokiu3/ ate 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 rotatinig7q6 nb98fg hg at 10,300 rpm is shut off and is eventually brought uniformly to rest by a frictional torque of 1.n7bg9fg86qi h 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 a6(k8 ;hehidbcjd r:n2 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 i ,doliy9m x+ 2v o-v 7v ;pw3;y5htvz.gkgydl(s thrown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is acceleroylwvv5dgz 2 ,7ly(;xm .dvvpgi9 koh+3-y;t ated 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 bla/gqgm 1yxjh 7e)8wop 0de can be considered a long thin rod, as shown in Fig. 8–46.h 7w/ pxjgmq)10eoy8 g
(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 tcjc2bywwkzq/ 9n io41 k8j0;g wo 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 +e+; n;zprhhqyxz*iek)xp 77 rest within four full turns (revolutions) and releases it axz e)e; hp q7rp ;yhn*i+7z+xkt 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 c,3 )2uq9wy czk eedgigj4s24a moment 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 oam:1hk7qjrs kt.g0lrx x 85*df 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..n md/6ju zchd h19,xz0 cm rolls without slipping down a lane at 3z9czn6xhjmd hd,u.1/.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of t6*fn0z9-vk3h, zix dqzl jr7t he Earth with respect to the Sun as the sum of two terms l7zvkqd -*z 6,t9firzn3j 0hx,
(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 ofmq8; su0+6zjd; xbvsdx9e .ka 1640 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.s;+ xa;vkdq8 z9 ds6je.0mxbu00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolli,mz mea0 68qjs without slipping down mej8mq za ,60ia 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 tip. It starts to fal s 90ujt n.n.am0k:vttl 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 oft:tk0.9nuj0ms .n tav energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ba2uc3tjs 6jc m5ll rotating on the end of a thin string in a circle of radius 1.10 m at an angular speed oct53cu jj6 ms2f 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 298d u4kun jyb+.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotating at 1500 rpmk+9 ub84unyd j?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his sidexa7h 6ya fbd2*, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48,dxha fy 7*a62b 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 9h.w2uc 3anfyxv pt09 moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2ty9 f .uxnw 2ph0a9v3c.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 rot58-pm kp:dqms ation rate from an initial rate of 1.0 rev every 2.0 s to a8mdmpk: ps5 -q 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 rotatin0lnxwg,e73ughhg 8 0av,a;dmg around 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 fre03;v7eha,w0 uga gx 8,md lhngquency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skatqw 1qm rs-1k5cer 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;kc7s3 6we n0bu iduh, 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 drojnj/me a ;6nz2cmcf:g6ic4 boeu3* .opped onto an identical disk rotating at angule/; :3o bc26unzc6 na c*e4 mjgofijm.ar speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.x-q)lt3h)z0b ;;kq.ouglky g 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 atpjz hdo3bv* g; tjv8yd4(:0ls the center of a rotating merry-go-round platform of radius 3.0 m and moment o(dj v08g loh;vy*pzbj:s3d 4tf 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 merryf5b-dz(nx;:io 8fc-( w jqprz -go-round is rotating freely with an angular velocity of 0.8fjzfnp; 5w(-oq c zi8r(bx:d- $rad/s$ 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 cl-x/ *pllq5p htwr5h iy9ok,4ollapses into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass rh,l-kliq5 /*4xpopwh 5t l9y 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 obp)ov+ .x4q rlzd, j4tf 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 w*tlo+ , yslc8 .)dqyh dcvq(+$ 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 wd-,3qyub vj1txds(3jo;esp ) ithout friction. The moment of ine3do,xj-qy s;3tu dvbe)(spj 1rtia 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 at3 r s/uo db-svk52t-diameter merry-go-round turntable that is mountedtrs-5ka3- 2 tvod sbu/ 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 ropeuxcq k6k5t84/cxp6u m rolls on the ground with the end of the rope lying on the top edge of the spool. A person grabs the exm64uc8kk u5qx cpt/6nd 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 g;wkr w. 0yktu /+2+ t h1xezqopjud.t1mbp54the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular mh/11 tw;xq0gu.m zk yokpew. pr 54bt+ d+tju2omentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates fr2hee, 3yew80o umk+ 1;evu nafm)(gbtom 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 shape of e.gntjrp5 j.,mutd *p.cn5twbb ;k8* a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculabptw,gtj ckj 5 e*. nurnb. d*p;5t.8mte the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass55n )r .5 rjsjx)mjx.en u;rqk 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter .ejx);5uj n jkqx5nrs)rm . r50.010 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 nw/tguy,3owv2a1jj;4 3ptua the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as greh*34-upprpj /vq,kqn )i6sbsat, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all t 3 vsp4ukbqqp-i s,*nj 6r)/hpimes, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it tm3o fhi*s+o)lc41 h:rgk ruv +o use energy stored in a heavy rotating flywheel. Su+vo3rl mu 1:*) fchi+khg4ors ppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without 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)xamn5d zs i+qtt,:v)9( spass 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 horizou./ct7gk; wm k5hz:3c qke s h ;/l/sji0nu8tontal 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 (mhwx2r 0-t6p:im ,* 6 ynsjscyi.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of releaseps6*,myn x0s6y -2:ihw mtcrj, 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 tg -9 d.l fa,l+ *q1ks*zk unev:dsyi5is standing vertically on the floor, and we want to exert a horizontal for-k.zy k :suaf,nslde+ *i1d9vlqt5*gce 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 v=4.2m/s on a flat road is may ax6x 1:/yk8hbjn 43pomx vb*ks.;ki king a turn with a radius The forces acting oj4xb/*k 8 y i 3pnkavok.hxy1:b6;s mxn 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 1.5 m and begi4jqjocj+ oi.xk1m7g, ns whirling the rock in a nearly horizontal c ,xkgc.4ji+om 71q ojjircle 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 torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her armx-j72voo 9f/ gxtoq) ls as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater o-x )l2o 7fqvxto 9jg/with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindrical plates, l5oae39+vgw0 d a:cr ;ulaw o+of l+5u +03oaewardog9a w:v;lcmass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 oy6 ;wtal xyt1 a2;sb1along the looped rough track of Fig. 8–58. What is the minimum value of the vertical het y2la1o;w6xtab1 ys ;ight 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 not assume hel q)sx3y z1n*5sl1z$r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed unifor14j:u ialm0 wc 5ke5p:1ciurwmly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the anguji1uc4wec:mr k5u p1wil a0:5 lar 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