A bicycle odometer (which measures distance traveled) is attachedic(-zn3m tk /j near the wheel hub and is designed for/m cnt(i j-kz3 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angularwe/q 9 hh82 ea.mgbm5f velocity. Does a point on the rim have radial 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/qhgb.mmw 5e h9 a8e2f the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the angular velo.h wkc(7 5pnd q.nlsu-o+vy 2ghdgf5 /city $\omega$ Explain.

Can a small force ever exert a greater torq/4x;vb-jyc6vlnum w5l - eg5khi8 +hlue 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 behind yc51qz 1ms3bflqg- ll6i).mn k our head than when your arms are stretched out in front of you? A diagram may help ib m l1qfms.)czgn 5-1kl3l6qyou to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and 7h5/nwc i *hkythree 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 hhk7y /i w5nh*carder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able u i6gykumide58)a4q2 to run fast have slender lower legs with flesh and u eqd2 ymi64)k uiag85muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narro9mm -umhy6c cnfe;l+9 w beam?

If the net force on a system is zero, rn-69ap w81rs*j unso is the net torque also zero? If the net torque aj-r9o6rp 1unns w*8 son a system is zero, is the net force zero?

Two inclines have the same height but make different ang;mh3t,1 -dygex ujsz9les with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottothzudg 91;ymjs3, x-em be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an incline. Onetkc nfac;2 t900m(k9tu dum8m 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 the bottt ;ca m9tkmdkm8(2 tfc9u0n0 uom?

A sphere and a cylinder have the same radius and the same mass. They start from 9 a+8 9rkjxnp iewkd8(b-y.cdrest at the top of an inclinen ar(dceby k 9 jiw.9d-p8+8kx. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum aoso; bb ;yp,s7n(q3qu iemj87nd angular momentum are conserved. Yet most moving or rotating objects eventually slow down and stop. E;j7sq;e,y(buiosn mbp8o73 q xplain.

If there were a great migration of people toward the Earth’s equator, h.9gt soh4ln)02 nhp gcow would this affect9n2so4ht p ) g0.gcnhl the length of the day?

Can the diver of Fig. 8–29 do a somersault without having any initial rotatio luof 7to,i(eb4yfy/k1( /thin when she leaves the boarie(7yb(/fofl i k4h/1o,tuyt d?

The moment of inertia of a rotating so/rj)ke4yf+q,s ph a )jlid disk about an axis through its center of mass is),4 fsj k+)p/ yeajrhq $\frac{1}{2}WR^2$ (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 o1pxk ,; edn;aeeqg;xi (i6 aj/n a rotating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Exn;6q; kg i(x/e1 ix edjapa,e;plain.

Two spheres look identical and have the same mass. However, one is hollow and th-706fs(6g nvjnb-npm* bp kk ce other is solvpnj nb-g7*s 6f pmc (nb-6k0kid. Describe an experiment to determine which is which.

The angular velocity of ))/qlq 8hafxehxe68 6w- z ebza 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 t hzlwq)6efqe88xb )-x/6 az ehop of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotatin,z9cq+mq8 jp ag turntable. What happens if you walkc+,aq 9mj qzp8 toward the center?

A shortstop may leap into the air to catch a ball and throw it quickly. As he t1tss yuzy:za;n) c;-fhrows the ball, the upper part of his body rotates. If you look qusuy ;zfna- 1t:y)sz; cickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of co0i z ts9 v*yqh(xsj:r;nservation of angular momentum, discuss why a helicopter must sh t90r:ysizv;x(j q *have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in r9vw.b u, /drnvadians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coi.y2) m+ph 3eocm npov+ncidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Eayo3e p2np+h m.cmo+v) rth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at irrq3*m;p;9i u: .p x uax0owlthe Moon, 380,000 km from Earth. The beam diverges at an anupila ;rq0 o pr:x3i*x ;.9uwmgle $\theta$ (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 th8bo8afc o; cd/e motor is turned off during operation, the blades slow to rest in 3.0 s. What is/bcocdfa; 88o the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 mxdmfnt // (n3ya q. postqm*8e4r i7h9 to another child. If the ball makes 15.0 revolutions, what is its di9o qt/nqt4p*7fin8ha(mm xsr/ e3dy.ameter?    m

  A bicycle with tires 68 c1 in:3 )orw1z2suc gmrm in diameter travels 8.0 km. How many revolutions do the whee2znscwrg3i 1u )m:o1rls make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Ca)aj;r ay)rc lum6kb j 6/r-1ollculate its angular velocit r6)-jurlk 6a;cmlb ayr)o/ 1jy 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-roundj* jq jysa5te,u+ku o(x23jo25yo+ af makes one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2u*+ja o xyjaoyk+ o2us j5e,f3q(t j25 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angula;o sk1fs06 bg(xc bzu5kpmu5) r velocity of the Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear sprd89;sbf( e mweed of 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 top uua:)f s+e+mx1z3whe axis of rz +p)1aewu:3sof+ mxuotation is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from e h4v-a9mo( yq130 rpm to 280 rpm ineav9oh m (4qy- 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 a33tj(gd7 b(.1ulq l5c au*drzox m4y $R_1$ 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, astronaut -.pndjov v/v z(cw+s,s aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecrajs-nvvwd+(oc .p/z, vft 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 uniformlyq5ne*s6.s l oavrpp/6 from rest to 15,000 rpm in 220 s. Through how manysep 56 rlqa 6p*./vons revolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calcfnmd moyz0.(qpr 5s *99;dgygulate
(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 flh+1t ov vq 4n,rx(ju2jying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions befor, 1v v+q ujhjr(ox2n4te 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 accelehd.va pf 70(jp7ku ;qel2 n gupay)15erates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge oqjkh572ae.y 7pda(vuuefpnlp 0;) 1gf the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 1500 re: (g:comors.q/i yzi5kk2 rw5 volutions before it comesmirk5:.rysk:/o 2 wz5 q(gi co 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 revolutl ;v; i8w.hpgie :ybyr3av+ b:ions as the car reduces its speed uniformly from 95km/h to 45km/h Thee:w;igvv.l y ;+b3ph:b airy8 tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as rmtr; t7yqsp9j, d t4,c9 g(g*luhz6f ;mhit/the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.80 mtr*g (m ;,tufpr6hqim9hdyc, 7;t49tgj z/ ls.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each pedal when cc7emqybo 0v3h l gh+v)td0*: dlimbing a hill. The pedals rotate iy) *7el vh0gm b3q0chdtv+:odn 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 a84 5+9ky l 5mwnnrcrccm wide. What is the magnitude of the torque if the for9 c a5k5cnm rn+wryl48ce 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 Figdf pulua84 :doj 4e29h. 8–39. Assume tha d u8l f4h2e9p:jdoau4t a friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, ks :vv5 jb .dhnp/9t-mare attached to the ends of a massless rod which pivots 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 nens. kd-jpv: mtb/h5v 9t torque on this system.

  The bolts on the cylinder head of an enr(pvb9mrp l -53fm , 5iy;rlp*/yuftwgine require tightening to a torque of 38w;-5 py 5f/9impm (vltyprlfu b3rr*, $m \cdot N$ 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 mq2p eq8e9 mi7o when the axis of rotation is through its cen7qo9mq8ie p2 eter.    $kg \cdot m^2$

  Calculate the moment of inertia - ko5kh 9(ejrjd,s bkb2b1 m/jof a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The kj-kmbkbos5b/9reh,djj1( 2 mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram balvznltuz8 (e/5:f/y)nv v c/hpl on the end of a thin, light rod is rotated in a horizontal circle of radius 1.2 m. C 5v //8v(vcphzzn )y:ufltn /ealculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating at constant angular sp,0c c(/ 6mzcum6 esomyeem/ s0c c66o,mmc e(uzyd (Fig. 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 of2 k8mt 3 bmuhb;no,xqzjdn;4z,ij( 8d inertia of the array of point objects shown in Fig. 8–43 ku3(b d; 2mnz, h dxqzb8m4j8tnj;,oiabout
(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 atomwt1 uow;vi,)s s whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the cjw 6p :rh k:e*npfec3.s 24ylaorrect rate, engineers fire four tangential roe6 :sp* :nh2.l 3ce rwfpaj4kyckets 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 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 cylindengt7zg .- +azqr with a radius of 8.50 cm and a mass of 0.580 kg+ zta7gn .qgz-. 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, acc 8kp4hxwvmnxz2( b(l7 elerating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a smarrn3yu/3oi3 k (l/v-bx xxtt9ll 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 -x/r t/ xv oy3i3lx3krb(t u9nthe acceleration, neglecting frictional 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-ij6x*6d txe,h17ktef cof0mg+o 9ma eventually brought uniformly to rest by a frictional torque of 1.2x+09,* igf6to o6mkxaj7 1hdft e-mce $m \cdot N$ 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-gf4 gwi5ucm: six i6+j(w1 2 tji9icl1kg 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 -xwk 3 d.r uyzgb0nirw3es+2p+l bx* ;solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45+3kbi2-pb 3 gw. w0+xe *zdursy;lnx r. 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 cvxd9 ylju1cybr4b4)npw-jb mzmk+1* 6 lpn,;an be considered a long thin rod, as shown in Fig. 8–4 4 rmvyb-j1*p6,clnp9 )dbj;xuk zmwl+nby 416.
(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 consistsj) w.b2(5jsufbki 4gi 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 ;-:/1xwuhxu iiuiob6 from rest within four full turns (revolutions) and rel;xiui 1o/ xb huw-iu:6eases 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 moment of lbgo9;tuls )8 oa+7 bww,eh7minertia 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 to6g ;(sm.kvhx 5b+ fshvrque 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 withje;x 5:z.dw 1p;b zjpuout slipping down a lane at 3. :.zu wzp5jd1xe;;b jp3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of th ynbyn;et i-6f+-f9lde Earth with respect to the Sun as the sum of two te tny-fb+leyn; 9 if-d6rms,
(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 2hyg5lt t ;p1ia 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 8.00 s? Assume it is a s5 1iyhplg2 tt;olid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls without s ctdt7lfs-v;zyea+v, m6gb1 8lipping down a 30eg7 t816,cdzsam fyvl ;bt-v +.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. It5y cku/v0f/va;v qalm7*fc8h starts to fall 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 /7vkf8/* au5a c0 vychvlmq;fof energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on d8b):dq d+wc bthe end of a thin string in a circle odcbb)d dw 8q:+f radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg unifki+yy 1q1 kd5corm cylindrical grinding wheel of 1yy c+k5k1dqi 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, han)kt.a tj -s41lfl s/rids at his side, on a platform that is rotating at a rate of 1.3rev/s If he raises jrss)l1i 4.fl ta t-k/his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce her mh,, xg9( gabyyuu2:uooment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she make9hguy xy,a:o2(b uu,g s 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial rhfh a; bk :frfiux 9rze/co5f 774+of3j* zv1eate of 1.0 rev every 2.0 s to af5crzf: 1hfef3kv/ ;7ruoxjeb7f hi+4 9oza* final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis throul fbfw hrilt76ym 9) :zt3tw:)gh 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, appttt3fm9b)) h fry liw:7l6z:w roximately 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 b0mm 4p*rk.wyk) hd0 nof 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 angularwmg,)p 2ycc yt(a0uav8.i i0 b momentum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$

(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dropped onto an ide*t3q*ra)tz t 3ntni6 pntical disk rotating at angular spetp 3 *a3)rnzq *t6ttined $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4 36hhc;mzta 655 mcryl $rev/s$ 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 merry-fa8f5t vw b 3ola,kbrx- 68pz7k2n/awgo-round platform of radius 3 kxo,-26f 7 8wr/pbv8a5a3bznftwlk a.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 witq ex* n;8ar a:5lehpywuo:3-h h an angular velocity of 0.uxoa e*3p ; rnh:-:qaeyl wh858 $rad/s$ 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 wh h:5g00 l xstnjwi:1b;bs)pag ite 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 caxnt)1:5i:pggl0habs sw b0 ;j rries 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 sc43-ylezt- wm j,ga*winds 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 .,cfpuk -r*fd $ 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 platformh( wzox 83cvfu0wk7 k9, initially at rest, that can rotate freely without friction. The moment of inertia of the person plus the platform is w3fk(c9o87 wxukhvz0 $I_P$ 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 off(pl1afc tgwt/ kkm;o6inif 5u-n0 +1n2v vr/ a 6.5-m diameter merry-go-round turntable that-gk 1t(fp knf1v 6witi r;+a0cn2fon v/5u/lm is mounted 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 rl7w)p h 59zcdjope lying on the top edge of the spool. A person grabs thhc 7dlzp5jw9)e end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth suc*gqvr485hgsbt i1a o (h 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 sqi514t(grh* gab8o vmomentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of /esqe2hlq hm :hdkh2su v7 q0xu:i)p7(/ka 5e1 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 cylind ;k)f9j chdt :ul7lmqfz*zr; 9er of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. 9)djt9r:h lc;l;m* fzk7ufqz Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.0 wv7.uqw 8b)ut50 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the l)t7bq v8uwuw.inear 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, hoipi3 z71(lb wtw is the angular speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0igppvl7+/w6g 3m/cnb times as great, 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 rv i 3ppm/cg7b/wng6 l+otation 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-pol vpz/+ x)df8md.6 quwn-zd.idz,x- ymlution 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 zz -ydiv uw/+p,)xqd.n dz8f.m- xd6m 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 illustrates ao6 ec2xibz- /2s drine/-/psl7x zoi4 n $H_2O$ 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 horizontal surface at spewhryoo, + 74qped v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we igtjkbc.7e(oxz h28u5b- 4zteunore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontalkj buezb.uot 5 (h2e-t847z xcly 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 vertically on the floor, and welam d.0fy w1yg;x 87vm want to exert a horizontal force F at its axle so that it will climb a step against whiawlg.1vy0;md 8m f7x ych it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling wfw-axcthx1 zu 75h (b;ith speed v=4.2m/s on a flat road is making a turn with a radius The forces actitc- wx bxz7 h(hfa1;u5ng on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5 m and begins qmhm: +ybc+;fwhirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the t :m+ byc+hqf;morque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thin rods, makid 9lzuw s3d6a;ng reasonable estimates for the dimensions. Then calculate the ratio of the angula wld6dau3 ;s9zr speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindri ixue252jymk-:i/dd gcal plates, of me5k -gjyixd/:2ud2 im ass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 track of Fig. yzg9tw7v j2,v-zawcdfh ff- cfv8.51 8–58. What is the minimum value of the vertical height h that thed-,7zgtc8y5 -a fzwvvw21ffvcf. h j 9 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 assuivt.4i 76 dgp/wmtkg(s(v dh8me $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car ;scko. ygb *i8,efm-vreduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (ayi8;og c m ebvf.k,*-s) 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