A bicycle odometer (which measures distance traveled) is attached;fqq0ejt ;3kkl3(ngp bhs-7b near the wheel hub and is designed for7;jq;03eb (thqbkgl 3fn s-k p 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular og:0 eo(; tlhnbicv5) b4i 9glvelocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocol )g l;cg ve n40bh:ti(i95obity 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 angulagb*7n m*uhvu+r velocity $\omega$ Explain.

Can a small force ever exert a greater torqueen6h0jn v: y(x 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 your l29c6 k 4drq7nil;rebhead than when your arms are stretched out in front of you? A diagram may help you to answer this.r6d7l4rck9bqil e;2 n

A 21-speed bicycle has seven sprockets at the rear wheel and three at t a0gtp.y1l.w whe pedal cranks. In which gear is it harder to pedal, a.l awpwgy.t 01 small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legs withy axhf4*k.5*6yc,ldus)wfe e flesh and muscle concentrated high, close to the body (Fig. 8–34). On lee fdx*5cs*6fy4uykah.,w) the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, n 2f kbzfk)fcq2 dd5e;9facxcto71/8k arrow beam?

If the net force on a system is zero, is the net torque alsblj3g ,k5hlhk/sp5 f:jx)1 qno zero? If the net torque on a system is zero, is the net lkx5:k)s, b5jn/ lhqgjf31 hpforce zero?

Two inclines have the same hei8-smf j.ioz y0ght but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball .zf-8j0 imoys at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) dwht-g/ ey107qo 6 d ynze31 3hokclkldl.p). jown 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 the bott3d6l q) kgz/.jykho01 h lw-enc.teo 317lpdy om?

A sphere and a cylinder have the same radius z4y9 vcghp+s;x 4oi 7land the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater spe+z lsoy 4x9p;h 74igcved at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conse uyjd4fan5 fnj y)071cdhc,a9rved. Yet most moving or rotating objects eventually sladn) nu7,j4fyychj15d9c 0a fow down and stop. Explain.

If there were a great migration of 7 7*0ernpe :freuek2tpeople toward the Earth’s equator, how would this affect t*r ntere27uee7fp k:0he length of the day?

Can the diver of Fig. 8–29d5l ;*p325vozhv ld e.u-v3tmvm in3f do a somersault without having any initial rotation when shevndm3l33v 25; hmfizev5pv .o u dl*-t leaves the board?

The moment of inertia of a )pe fynbv6p62 rotating solid disk about an axis through its center of v py6f62e bn)pmass 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 on,2/qh+pzudw 7o gx;pobd bm;3 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 thep7qb;u x,ho/ p+md o2gz3wdb; same? Explain.

Two spheres look identical and have thefbw k)jw c)-i 8gzdkv5y48nj- same mass. However, one is hollow and the other is si8k)8c-yd 5 -4)wzjkvgjbnf w olid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points west. In myt imb ); h:87-pktllwhat 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 th tl7k t)m: p;hib8myl-e top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating p mlhh 0;5+rrwturntable. What happensr5+ hrhp;wlm0 if you walk toward the center?

A shortstop may leap into the air to yknx 2: s-o4pvcatch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Expkn 2s4oxp:y-vlain.

On the basis of the law of conservation of angular momentum, discusc,h7bswj7xt )ffbq 1fw)6q lj +2xqu9 s why a helicopter must have more than one rotor 7tx b xc)ufsqj+q9)61,qlb7 w2jfw fh (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (a) mm 4 cegt 4v)b*fif(+h30 $^{\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 coincidence. Calculate, using 0;+ mpop hkbz* 3ts nb684jygxthe informati+t*46;m ksb83pbjz x0gp n ohyon 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 Mkx 7s yf(zkra03rrw28 oon, 380,000 km from Earth. The beam diverges aa3zwfr k2r(rx s7 ky08t an angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at +r -hl 10x0c(8c:gizhfl ylqc a rate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angulag-r:0l+hyz0f( lxq8i1 c cclh r acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a levv ng1 85qce1if(pr:sm el floor 3.5 m to another child. If the ball makes 15.0 revolutions, 1c:5ie8qfg np1s(vrm what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutik4h lq d-t2l9 u5t/brfons do the wheels makeb5t-/f u42t q l9kdhlr?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate its ni- h)gs)gxi9angular velocitn- i)ix)g h9sgy 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-r4kgx7 4s rnuh:h 5bh7-a7fuvoound makes one complete revolution in 4.0 s (Fig. 8–38). (a) What4nkhu vso74-7h x: r7hfagb5u is the linear 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 Ep3qut4g c4oy7r/g jp( arth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a poin nuu fbyc/-.0jt
(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 rotyo3r*: ar7gdd oh; a, v.qvl+eate if a particle 7.0 cm from the axis of rotatih.+vlrqg3a,7r od:yo e*v da;on is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel acceleratiwv- y )650ywcolq2p:orh ) rzes uniformly about its center from 130 rpm to 280 rpm in 4.yz)l5 w0-riow:) p2vc6q ohyr0 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 radiuttp p*cmbsq/ivef om1 ,/1 d)+s $R_1$ 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 Apoltq3t*j:* x s1ikecq c3lo 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 spacecraft can be thought of as a cylindeq:cqi3j txt 1k*3c*se r 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 accelerawfg fgb i 66t:g6fc.v7tes uniformly from rest to 15,000 rpm in 220 s. Through how many revolutions did it turn in this time:f7 6 66gbgtwgicf .vf?    $rev$

  An automobile engine slows downs8lsvs o7 / hr.bxm2k0 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 str2z- 34xf ;oua;jp 4uc lsnmn)besses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before)fo2ax z;n spcm-u3j4 ;l nbu4 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 fromcw8j)+hu4l as;e v,bl 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in this time? ,ju cv l+b)e4l;wha8s    m

  A cooling fan is turned off when it is running at 850rev/min It turns 150r.suzqfeu:7,3. v 8qkn(f zvj0 revolutions before it comes tqq :e,k 8fv3 vzj7uz(.f s.nuro 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 itsi3ibpt2.. ce y )on6qp speed uniformly from 95km/2bq y i3n.e6pc)o tp.ih 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 65 revolutionsh- 9dr)uepf6l8xil 9c as the car reduces its speed uniformly from 95km/h6)9xf -9 8diculpe rhl 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

  A 55-kg person riding a bike puts all her weight on each pedal when climbing a hwhfbw t/t7-wf k)/rx4ill. The pedals rotate in a circle of radius 17 cm.7b) w/ 4ftt-r kw/fxhw
(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 5 bm0 st aic-fbb)x3nnp+d;a-door 74 cm wide. What is the magnitude of the torque if the force is exn 0);ast dp-fb3+n bi mcb5x-aerted
(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 axlzoo(f kqp1/wt3bvfm farn0 dz87h 2:k3 4t1hhe of the wheel shown in Fig. 8–39. Assume that a moh3 f fzzp1hdtqk04ohk73 wbv8nar(f/t:1 2 friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of m,/wn0 fumhj/-;rzb zo ass m, are attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal pos -0zj;bh/,wuorz/ n fmition 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 n*8 v 6o+cscyga torque of 38 coys+vc8 gn *6$m \cdot N$ 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/e*c5g uih0zu a 10.8-kg sphere of radius 0.648 m when the axis of rotation is ie uhu zg05*/cthrough its center.    $kg \cdot m^2$

  Calculate the moment o l6b)zkvwgl9 n27vhz,x. qr6r f inertia of a bicycle wheel 66.7 cm in diameter. The rim and tire hav lk6,qgrv .lnhzv b6w7r 2)z9xe 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 rotated in a horizonbq bl vmcww)xf61rl4 7l*6 5nutal circle of radius 1.2 m. Calculw1ulmb q)wrl567n4 fbl * 6vxcate
(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 o) kbid7oa- vgtwc zra.3n)2 e*n a potter’s wheel rotating at constant angular spee )- *arvoazn7ce2bg.k3)dwti d (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 of inertia of the array of point objects shown in Fig. 7 m tl hv,vf+l 17*ontboa:z0l8–43 an *vo ola+h:bfvl, mtt10l7 z7bout
(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 2t/a5h4bjpjp oxygen 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 atbv(q-w yxh8rk,d) 7xe the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady forc ,x7rdx)yvq 8eh w(kb-e 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 witaae27v71)r00wm olch dzj ua9h a radius of 8.50 cm and a mass of 0.580 kg. Calc wzl0o27he7aj 0a)1dac mvr9uulate
(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 swiz,xwa *rh0k8jd ka*d 7ngs 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 tangentia3x bleuzscl .),6zz *zk5 nooam;l(5clly 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 fricozzla)u oz ec,n3;5b llxm .c5 zks(6*tional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rota*hci fnf6+u (uting at 10,300 rpm is shut off and is eventually brought uniformly to rest by a frictional torque of 1.2 iuc (unh6ff*+ $m \cdot N$ 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 acceler v 0ac7ag3,fvaates a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the action of the forearm, - c(vv dl768trl8 /0cvalfe ewwhich rotates about the v(flv-6c/ d wetve a78rl80cl elbow joint under the action of the triceps muscle, 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 can be consideredhl ek2 (n394a6qy(udeysd5 y qj.sq *a a long thin rod, as shown in Fig. 8–4* qe qs (d.s(h3uqaya52nklye 9j yd646.
(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 c)l j2*, d:d(w-ho kc ibi3knwlonsists 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 hammeh+ h;c abx mu b47d7x0dx+9zeir from rest within four full turns (revolutioa4b dm;h7deh0xu97xib + x+zcns) and releases 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 inerti*(frf*c; usfk jk9n diw; s93na 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 of 280 ,s8+ce z /pqts$m \cdot N$ 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 with8ftz93dn*yqfly5eb6g sx(- at4dsi p3m*d .pout slipping down a laddezpg* f.9tslp d83(-yf6yimb x*s3nq4a t 5ne at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Ear) 264t1;5age 5bzupcoogg sktth with respect to the Sun as the sum of two termscpoa56 s2b t)uogeg;z4gkt 15,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius o duao n:6n mv0.q3xjf.f 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 0v dja nn..3ufmo6:x qsolid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls without1x,fy8 ov 9 z7 mbwhoit(4x6qd slipping dowx7 ,wt( 1dyz b9hxv86fim4oo qn a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is balanced vertically on its tip. It starts to fall ahx7fg.o 9;h7 kzmyq:sfg sbh: xii-h +14-wdx nd its low dhhi7;bf m+-z1s iy:xo7gh-.w9q h4g x:fk xser end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end of a thit4i5 agg5l- k57yd9xo y5rbh ;:ofzxmn string in a circle of radius 1.10 m rag;m:g7tkyz9foybxld5 55ho 4 x -i5 at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uyrbsz;cqec2/( (ee a(vub7 )hniform cylindrical grinding wheel of radius hb2 z v;(er c)abe7q e(scy(/u18 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 hp0 ulv+s92kub is side, 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, the speed of rotation decrease9ksl+ u0vp2bus 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 re8ehel nt2/z(xduce 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/8(2x henz tel is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increhj8dl0e jgi25nw0 mh(ase her spin rotation rate from an initial rate of 1.0 rev every 2.h(jm 5n208j0l dgihw e0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vvo6r e1 /8*r-r yfqo ln28kdxlertical 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 t8vony ldefl*/x6r o2r-qkr 81 he rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinnicq6.gr ln;p6l ng 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 lfq gk-o(9kk: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 in6 woifh,y7,ie ertia I is dropped onto an identical6w ,e7iiyf, oh disk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.j v 4gr wl+j*/a vx.:nt6hc;js4 $rev/s$ 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 rothv(yoaz e+f 0u;u 0f2oating merry-go-round platform of radius 3.0 m and moment of inertia 9 fu0 o 0(;favz+eou2yh20 $kg \cdot m^2$ 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 merrylr5n45 xn(ktw6c y. ti-go-round is rotating freely with an angular velocity of 0.8 nc( w5y5n ik tlr.x46t$rad/s$ 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 about half its )5lzsw8 e2 dh n0ru;yzsa: ok*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 ratu :r 0dz8h k;aos)szy5el n2*we 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 dqi3b a 77 zdj65txixe8*p(mtwinds 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 *mz)t jfx1y8h)u:b c c)mugs 8$ 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, init -nl2.+1o vssiu9lam10gp bvoially at rest, that can rotate freely without friction. The moment of inertia of ml l +v 1pn1vo2ua-i9bso.sg0 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 ax4 1n;stkpf9*lwi3j 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and jn*1api9 l wx4fk ;st3has 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 t p5;7dmdfrc4 xhe 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 rolls behind the person without slipping. What length of rope unwinds from the spool? How far does t ;cd4xp7 dr5mfhe spool’s center of mass move?

  The Moon orbits the Earth such that the s*bh/nkpx u+8kjg9hkk3o 4qw; ame 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 thejkqpw*kkugknhhx/b; 8+ o349 Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at9m.hn83/2z gnl6r c :qrpkrj g 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 a uniform cylinder of radius 0.20 m acquirh)h:vz1 sy7 *gljnvi*es a rotational rate of from rest over a 6.0-s interval at constant angul:i7s vhg)*nh jv1zyl *ar acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical diskszuz zz+ld8csroui hmi7m*.al5447a ,, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass rdm i hza ,sz+57 a4lzmuzo7u4*8lci .0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of the/ qurwx*qm35k q0g2fv 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 v 8tzjuikc8u;4mu 75zwith mass 8.0 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 rotation speed be? Assume that the star is a uniform sphere at all times, and that the loscv tj4ziumk78u 8;zu5t mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution a8:kky kwm: g,outomobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kmk:,g woyk:8k g, 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 illustrates asyxt g zpr,:*3n $H_2O$ 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 on0lu1+v (/a*tb wxoozc 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 massgl9)9bodor -h M and length 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, determine (a) the angular acceleration of the rod, and (b) the linear acceleration o9 9 hd-lb)orogf 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 g6uemk.pmc6 -e:rw 4hvertically 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 step hhmp.w4g m6ruck:e -e6 as height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a turn with syk t, sizu:1hrn,,7 wa radius The forces acting on the cyclist and cycle are t :kst uz,7irny,1h w,she 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 begi7sj4 g iyse a3*.rqc2t7brvg ;a:f j(tns whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where rqsijvgr7t es. 3y:*bfc 7a jt(;g4a2does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as txybzm h jkfdw(12758hhin rods, making reasonable estimawmfh2h x(yb j8k 157dztes for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which 4lkuq5e;i6-t cej.v xconsists of two cylindrical plates, of mcx4.jvke-q6e 5t;ilu 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 r* u nrn,: 3adf*xxo8hfough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the hnfuf ,h*o*r3d xx8:an ighest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not :j 2,gpnin+zvassume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed u rgwbapn;s 0;9wi 8en0niformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular accelera0;9n wwp8re0 abi gs;ntion 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