A bicycle odometer (which meay .al.lbq mm(8pcptu4:03jza3 ; f5d9fxcgh msures distance traveled) is attached near the wheel hub ac u.l qlhzyaj 43fp9bdt5c8p m0gx (m;.:fm3and 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 p:/ +vs./cb(g nhpmj mqoint on the rim have radg/nm svhp+bm. c:/j(q ial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the angular v x +g(u3l rh7nz/cbsa7elocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larg:83ds;o w9j54-h nfwepr usgqer force? Explain.

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

Why is it more difficult to do a sit-up with your hands behind your head than whk+ffml;oyb r+)35v shwhzf,+b fx6*ten your arms are stretched out in front of you? A diagram may hel3*x+vof6 ,f szrh)5k yfh+ wf;bb+tm lp you to answer this.

A 21-speed bicycle has seven sprockets 9epvw+v bp jpb42/ut-at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket ou/ b -jptb 24vvwe9pp+r 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 exx(i.3it; y4sdfw z8to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain whyix48d t(i ;3.fey xszw this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, ,n eecl2 7*3lnh.nkbsnarrow beam?

If the net force on a system is zero, is the net torque also zero? If the ner a(rk0uapa .,t toa,(0upr. rka arque on a system is zero, is the net force zero?

Two inclines have the same height but make different angles wit.+6yu vmphz6tpz .qb2 h the horizontal. The same steel ball zb+p2v6yhm6 tqu.p .zis rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (dgf / 1o ov zl3ni )akky+hc7+,dy0hj;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 tk ga+f,j v+7zhn1)cy hyk3iodl /do;0otal kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. Theym-bo) n+u m0vprp4s i( start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at thmv+( posibpm0r-)4n ue bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum 2 wv:o2e-dcsf and angular momentum are conserved. Yet most moving or rotating objects eventually slow down and stop. Expld2ev-:fs2 ocw ain.

If there were a great migration of people toward the Earth’s equator, how ozzqn 53tboqs2ph82l ( tcz10would this affect tpn2 lz h(o3s1oz q0tctq2z85 bhe length of the day?

Can the diver of Fig. 8–29 do a somersault witd j-+-eizh d4thout having any initial rotation when she leaves eijd 4+ dzh-t-the board?

The moment of inertia of a rot-w pmln*lf9buwp5+ .f ue;c/( v ryjw(ating solid disk about an axis through its center of mass iur+w .f wfm95 ln pe-clpw (jby*v;u(/s $\frac{1}{2}WR^2$ (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 3sobqbtif6k b0/e(5j 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angula5ib e3q so6(b/tj0kf br velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hos w;bmftkyu6 -u/:u;allow and the other is solid. Describe an experiment to debu:/a- kf6;uumst;yw termine which is which.

The angular velocity of a wheel rotating on a horizontrp/qh64)z3/n h za ii ;:njmth91r vgesp)c )cal 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 angularnrac h4)npz/s) ;1cvpi h:r9g3qhmjt)6i e/ z speed increasing or decreasing?

Suppose you are standing on the edge of a la,+- b7cpyu8lv8uq *cjc l+d5sb3nm xirge freely rotating turntable. What happens if you walk towayvi+ 7qbl l*8s xc-,jdb8c3 cup5n+umrd the center?

A shortstop may leap into the air to catch a ball and throw it quickl; nxkopw0n/r)(vx c3slup ; i)y. As he throws the ball, the upper part of his body rotates. If you look quickly0));onc /3kr pw i;vnlx(usxp 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 eyo/b 6 qw-es5angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more waysqyeb6w/ o5 -es the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (a) 30/mtl-4s-hc (*a lumav o 1yqa:2r 4mib $^{\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 becausetmt-t4xip2 ),phj3nnno (8dz of an amazing coincidence. Calculate, using the infonn4pn3x ,d28 j htio t)pmtz-(rmation 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 Earth. wtr.yajsir 9+ah0;s /2twtjl+ v:uv1 The beam diverges at an angle jjt;h9l/+ t+ 1rv2:taarv uswi0 sy.w$\theta$ (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 ratxv7o-,2u i jpiribg5m9 7-owve of 6500 rpm. When the motor is turned off during operation, the,bo-x ii9g7v i2-orv p 75ujwm blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another c-pbcx:uz0/i jiu4ry ,hild. If the ball makes 15.0 revolutions, what is its4i,c /iu:r -xupyz j0b diameter?    m

  A bicycle with tires 68 cm in f0 haarz+1xh i/x9gk )diameter travels 8.0 km. How many revolutions do the /ihhzf)9+ kg x0xra 1awheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Cbo/ f:wny4pvd v.ouz12 q3 6lsalculate its angular velocils6vu : o p 1z2wv.4noq3fybd/ty 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. 4s. /yfci:cc p8–38). (a) What is the linear speed of a child se /4.c:cfsyipc ated 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) in its orbit -xz rd071or tr;coz 2earound the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed oxi2xvn0u( y/ kf 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 :p*art-xy o yxxf4.u :rotate if a particle 7.0 cm from the axis of rotation is to experience an acceleration of 1 a pot.x- fr4*:yyxu:x00,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uni1vz ,5x-zlwra0r m2 uiformly about its center from 130 rpm to 280 0z-u1za r,wli r2m5xvrpm 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 radiubkq h-km76 bt+ wwg2coj *g4ipsb7/-ds $R_1$ 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 spacg-ssi8 *br+zsr9*tz necraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotaz**g nrst8- i+b9s rzstion 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 z6zsi-3 wl:kh ;s ifv.rest to 15,000 rpm in 220 s. Through how many revolutions did it turn in i3w:6kss f i;v -hzlz.this time?    $rev$

  An automobile engine slows down from 4500 rpm to i*pl8nrfucw5o0e vru+ bfu (5 zpu7m ic+65b5 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 1* it1yhry6zzthe stresses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 compleyz 6zh*1yr1i tte 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 from 240 rpm to 360 rpm in 6.5 s8z qb2lbp5wzmto95 b). How far will a point on the edge of the wheel have traveled in this 2z9p w5b)btqob z l8m5time?    m

  A cooling fan is turned off when it is rungsev :0-rrvr+ifsb9l)dd ;-x ning at 850rev/min It turns 1500 revolutions b9xd)d:rs-esv0gf -; rilv +r before 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 maj, k lt: z+ irphf.qw(+8qb+ibke 65 revolutions as the car reduces its speed uniformly from +tf,:(ii zbk.qj w h+q8b+prl 95km/h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65g)eaqm*q d-3 zo/b1xm revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.80 m.*-ozdeb3 /qq mx m)ga1
(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 onu.f*ki)t jrwnf-8 p/c each pedal when climbing a hill. T fn8/w ftkj ir)u*p-.che 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 wi9 jv6e in,q5okigtbo00 t7 g44ofb+wide. What is the magnitude of the04bfg5bg 0 iqjn6iw ti9o, 4+votoke7 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 wheeld/g 6j3 co enl:u4u h3z )x:b+zt8bnor shown in Fig. 8–39. Assume that a frictionbdl: 34t 6x:n) gobnzeh3 uo+zc /rju8 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 p2(1p e7swni65v5lkx,fjmjh k 5d 2js tiv25ktepkjl jw 5xj6fi( mn,72s 5 shvd1ots as shown 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 require tightening to a toodqr ::4eo/ch 8 xq3tsrque of 38 :c8t xs qordqe/:o3h4$m \cdot N$ 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 inert-3 gmhr4. avsi6 qwm vs-j,kg:ia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is through irm gia -jq4v -skh3s:w,.6vgmts center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameteop(5ybfgu lma607 k(nr. The rim ab ya(gk7m0 5 ln6(fopund tire 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 t)wiyq 1i:.cw.lwe 0w a thin, light rod is rotated in a horizontal circle of radius 1.w)yi01: i .twelqww.c2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotatingfa/ cq89p2yex at constant angular speed (Fig. 8–42). The friction force between her hands and tp /a9eyf2 q8cxhe 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 nqs.82 4yc ybug6t5 0ktt.va qarray of point objects shown in Fig. 8–43 asqqygt4.ykb cvt nt.8au50 26bout
(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 ox/:k6djs .gsj;df*fzpvcz q9:w i9bv9ygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at thea3nu)o(3)dh)g7pn 1c pdipth 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 3h u n3t7d)op(1hgppn c))dai 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 cm and a mass -;79mh pl -iddag9vpd of 0.5-9 -7am; i9hldpgpd dv80 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;wt*jt gci(s, t df-p) a bat, 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 tangezr +)bw rz.dvt91 lu5vd90l sta2 q8gtntially 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 acceleration, neglecting fricti8agb d 0+z1vsrl .z)9l92rvtqudtw5tonal torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is fx687a+o h)ayll k+r leventually brought uniformly to rest by a ll7) xa6rfal8 o h+y+kfrictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelm9 ) k;v3h dgbm+cxfylae3*n*erates 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 s9mh1ljqkj8y7/ fib wx sm26)py 09ed uolely by the action of the forearm, which rotates about the elbow joint under the action of the triceps musc01jleyj9 k7i8phw9 )fuq 2x 6y/sdbmmle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade ca scpw (b;dih/j pcikg3rm;792n be considered a long thin rod, as shown in Fig. 8–46.7/ms 9rpc;wpg(h cd2 kbj;i3i
(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 masses, 8m0z-s55d y sffl*t.b8rvkr3 hs ch n9$m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest within four full tu fb gdvu2hy 2yzk-,.;g,zu;dyrns (revolutions) and releases it ; hkd,u;y2ugbgd z,2yvf-.yz 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 inertia of oj2,0 p /sr/7asvhmu /(nkh kur(z9c v$3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops 7l j+ otk0.4b9n 1u mqur-igqrxywbhma*,0 8p 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 a 6cmc up-*aiebe(3:bnsqn 36xnd radius 9.0 cm rolls without slipping down a lane ape qb ms:a (x3niceub-c6n*63 t 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic enewfcq 4gw 2xornq)v8x. 9h :3fargy of the Earth with respect to the Sun as the sum of two termscq:ofxghx)waw4 n.239 vq8 rf,
(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 an7e( g( h pwge,bl+t 3elcwnj73d a radius of 7.50 m. How much net work is required to accele ,jwh(nt+w 77pleecl bg3g3(e rate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 khtv* ro, 4j9psg starts from rest and rolls without slippi tohr9*jvs ,4png down a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is bailfsz4/o.,1v4(nl pva vq x 8elanced vertically on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of th axpn z/l.v f4iqse,(vv481lo e pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular moment --2 qdytfi3mvo i,5qrum of a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an angular spei, i3fmt -rdqv2yqo-5ed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindriz0cm 9+a jff-jcal grinding wheel fzj 9f-m c0j+aof radius 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 rate of p -e fig.3s)-4+sqrvpy gjmw81.3rev/s If he raises his arms to a horizontal position, Fig. pw-)sqr gm84y +3e .s-gfipjv 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 3pe;s4krsf6i in Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed (k; pe 34ir6ssf$rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial r .pbuj9g7: caoate of 1.0 rev every 2.0 s to a fb7:ujoa cg9p. inal 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 vertical axiuxk0xi:+rk:qpe :h -i s through its center at a frequency of 1.5rev/s The wheel can be qhp: e-rkk:u0:i i+x xconsidered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angularhnf7v 6l/t 7s2gduavvbjzr)13 02fl o momentum of a figure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular mo 0l7fipjo2 3vh+ 0lxokmentum 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 cylindricj)43be5lgd (l d um8upal disk of moment of inertia I is dropped onto an identical disk rotating dle54mu8(3 lju pgbd)at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns 6 -fcxelm:sw6 at 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg standstq gb:/p 1ga*-(g*po5u ny j5wwita c, at the center of a rotating merry-go-round platform of r( yja*, 1qgngb: -gc aw/tw5iopp5u*t adius 3.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry2kfee8+nuof faf4qry7v/, *d -go-round is rotating freely with an angular velocity of 0.8fq24y , ue7/ade vrnf+ o8ff*k $rad/s$ 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 white dwarf, losing arz) -1doeybkh (zsa;h0u4 7 yrbout half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the1hkh y0r-e)yzsza; urdb4( o7 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 exnbdv/rr p)9( vxvk)/.bbg ch2 cess of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass 25u ehw:cc)lrt;;)zgm mde*a$ 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 cnro.7h n871c3uk*i2ot;re sq jrt;mjan rotate freely without friction. The moment of inertia of the person plus the pla ;s*ik7on hqo 7m28u.c;ntje3 rtr1rjtform 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 persob30xq.g4 a djncz c0t5ug8y :gn stands at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a momjaqtx3 :50c0 dgcbu4zyngg. 8ent 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 grshu mo / izl*2e)br1k7ound with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rollek*b /)7uzh moslr1i2s 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 seq 01*x7rfvhs ) oysrz h6v4v(uch that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting te 71zvy(x64v)r*rov hs0h fq she Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate 3ve. nf4gz(djof 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 cylinder of radius 0.20 m acquir m3 ufe( x p ,mapt7k2r2hxbl++l7f)ttes a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calcufpxrt(a,+ m)f+lxe b 2tkpl th37m7 2ulate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cyji;6g;1ifne: zu.zrkst s0a 9lindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 i6a:.g1; irz ste zjsfnu;09km. 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 wh+)gt;r sxso0o eel ($\omega_R$) 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 rj:ap+3 lp6s-v0ifqx /*ozlwgreat, 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 pl3 zjx f :iv +wpo-a0ls/qr*6its 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-polk/q6x jl nh3k*xd 0oq.lution automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical3d*0 /qkh koj6.lxnqx 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 illustrates andc)9 hizb2 lm, $H_2O$ 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 p8h.d. h*i mmhu1- suwon 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 xm ki.z48so k;L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, deteroikmk8.x;zs 4mine (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 d -d- daa5-x qwlzwu0j2nd3i/ standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The stepx/znjwd -5wd-a ddq -lia 23u0 has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road isn lt:o g9vt x+ch jh*d5+5cw9n making a turn with a radius The forces acting on the cyclist and cycle are the normal force 9 n+gvn+lcx 5d9oj5 *wt:t chh$\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ 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 aix:7f 6m9cu)2h pqe pt 6rd0qi sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. Whadxc67phmie0tq :q) fi 2rpu69t 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 arms as thin rods, makire ys w2*m6.fwng reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms reyw.sw 6f2*m held tightly against her body.   

  You are designing a clutch assembly which cond;ys uuk2d66x ey al7t:5)dnjsists of two cylindrical plates, of mass 2k ud5:ly t)nyd;6axuj6 d 7es$M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped rough tra5k6xi: v)jkcuc2-ch od5u.sn/ + s alqck of Fig. 8–58. What is the minimum value of the vertical height h that the marble must dro/cikuv jokdsl2.snc uc-x: 55a6+ hq)p 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 :cajcppt 0 jvan41++i not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the msvvrq0zh1s a1)0 ar+ car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular ac1rs0hv) +1avrm 0szaqceleration 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