A bicycle odometer (which measures distanc wsoy uz74*v d*e;vqx.e traveled) is attached near the wheel hu v47y;z.vw qsxoe**d ub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constantu(a3q/ma1 ))rse w asv angular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases unif(r 1amv3s) waqase/ )uormly, 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 body3z+ e. l umspixc2jyc;),b5u5y eb8go be described by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque thaht ,+tgg(*apa n 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 difficultb*)g8 kzc or9n1l8bv ky0,gpg to do a sit-up with your hands behind your head than when your arms are stretched out in front of you? A diagram may help you pg1v8rk,gz)c9gk* bn0y ol8b to answer this.

A 21-speed bicycle has seven sprockets at the rear whee w;e s-2;jlx so2-qtgll and three at the pedal cr let-q-; xowsl2g; 2jsanks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legs with fleb( ,wv9 jz w)t/xn 6 fw:hp*.tlh+fx0hhwyej)sh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain wh)xp6e . +b 0*:yfhwljh/v(, )zwh9nxfwtw hjty this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35)1yvzabdm rpt zyaui.-m86,) xv, ,m9d carry a long, narrow beam?

If the net force on a system is z:r1dkhdfca in.k0r- xv, + u8eero, is the net torque also zero? If the net torque on a system is zero, is the net force +vnkc, e0k-h8x .ri 1 dfra:udzero?

Two inclines have the same height but make f6/jl ipx;7jf g p ;j28sxnyf:different angles with the horizontal. The same steel ball is rolled down each incline. On which incline wilsji8/p ;pn lff7gjf 6 ;2y:xxjl the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolli9b pvn: z 0hc:w3+konnng (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the inclinwn 3kc0+h :bpvzn:on9e first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same masmok )16nom-9++qrv k td a-bars. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? W+ b qknvtro-a)+r a6-m19mdk ohich has the greater rotational KE?

We claim that momentum and angular momentu0dhe1*j yip-a )-i yd:futm,n 7j bvl*m are conserved. Yet most moving or rotating objects eventually slow down and stop. Explain. dl)n,1v bjd-u*:t0ym fephj-7ia* yi

If there were a great migration of people toward the Earth’s equator, how vz)vav(;z/ 1o/uk axmwould tvxz a/ ;m(k v)o/zu1avhis affect the length of the day?

Can the diver of Fig. 8–*v2 -oo 8y2)urylraka 29 do a somersault without having any initial rotation when she leaves the boaalao2yr* )k28u rvo y-rd?

The moment of inertia of a rotating solid disk about anv sy(t0m(+s;bn6u qxk axis through its center of mass isvnk( ;bs qsy +mx(u0t6 $\frac{1}{2}WR^2$ (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 2-kg mass in eat.c w808fgh(spql ; p0. lsgusch outstretched hand. If you sudden(.w 0s8f;u8gq0 sh.pc ps llgtly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. yl3 5mauq0ydwr (x+h:However, one is hollow and the other is solid. Describe an experiment to dete(+ hda5yxr qu m3l0wy:rmine which is which.

The angular velocity of a wheel rotating on a horizontal axle points wesq8m+*tjiahc 8t. 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 angularm cj+*t8haq 8i speed increasing or decreasing?

Suppose you are standing on the edge of a large fh/8g*7zvtxx tx-v8-j2j *e 5ud +vf9w rznpdoreely rotating turntable. What happenhxzz8jx2v w8 * -en-r5/vo+ x dtdtu p9jfg*v7s if you walk toward the center?

A shortstop may leap into the air to catch a ball and throw it quickly. As he u7unz1dtt bc7 *5wa1bthrows 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 directionwd 7a uubtb15 cz1n*t7 (Fig. 8–36). Explain.

On the basis of the law of cons uxc2-sv.au31pfgou :/niub1-. uf p yl,mhx* ervation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one fmy:-fo 1xp -ubx.hanicg , 3p2/l .usu1*vuuor more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radia 26h. klazjb)ans: (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 coinkf3 wa*c sod16-xs6wncidence. Calculate, using the information sa-ok df3w61wsc* x6ninside 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 k d ycxj0-np0cm, 8x2mbm from Earth. The beam diverges at an,m0cx 0-y2cd bmj8px n 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 axoy-cv k m+m;0 rate of 6500 rpm. When the motor is turned off during operation, the b;ym-mk+ xvo 0clades 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 d yn7 c97q7jqpl.ukas+7w;vka level floor 3.5 m to another child. If the ball makes 15.0 revolutionsk +9j7q7yu 7 nlqvwsc.a 7p;kd, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutions do vghd:qmtx,*5ffsbb ;x m (,s.thgv. :qx*ff dt m5bs(xs,b,hm ;e wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rota mvzzs *ki a;+wi;ryb106o.a3f;wntbtes at 2500 rpm. Calculate its angular v;bv0 w;yw.*nz kr6 b 3+a;zsiimta fo1elocity 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 c *ty mvj)qeon8el i/ir) +9q;s (Fig. 8–38). (a) What is the linear speed of a child seated qn+r9; emi je*tiy/ocqv))8l 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 En7 hvyma*yo u0ii .6wgn j4 yak.5w96aarth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear s9 jyv08wml.an(wus 6ipeed 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 5t)m wpjyk ex6n .x(v5the axis of(txkmvn5pjx6y we ). 5 rotation is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its centlan6g (b1 j:u3mv6ymn8 ntho 0er from 130 rpm to 280 rpm in 4.0 s. Determinvtl:3yuo(6g 8bj1nmm h 0nn6ae
(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 radiu6smqvb n+;8ox i7crw*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, astrondr 3 3c/8)xps9 zf o*9srxcogpauts 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 int/ zf*p s oxrxc)o3cp gd893sr9erval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm5u, avu;pr1(ra 8nvc b in 220 s. Through how many revolutions did it turn in1 au rr(v5ncbu v8p;,a this time?    $rev$

  An automobile engine slows down from 4500 rthjthd: )a op;oh*qp4 9g90ljmc;wl6 pm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed pg:kecgr4a6 9b3 idwp+)0fe 6- stzgv jets in a whirling “human centrifuget3cp6e4- 6s0dpv9bawi+: gz eg g fkr),” which takes 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter acj f4/2zczlwq rim +-mw24fd;vw5p8uc celerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a p;p uw4f2z2cwd / v+cmfl w84qm5z- rijoint on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned offe3 qia,:pfvk11tdu(h when it is running at 850rev/min It turns 1500 revolutions befo,1 p(d afkeh3t:iu 1qvre it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its spee. 1ih4mz7hvqhd uniformly from 95km/h to 45km/h The tires hzqhh.v4 71im 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 the car redpthjayn u o (956 dz coln(n9p7+mk41p5ck/sluces its speed uniformly from 95km/h to 45km/h The tir5cmopn hys1+ l pn(zj6pl4d u7(t cakk5o 99n/es 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 b:z2 cdo6 fpxw+as0;aucw e4-s ike puts all her weight on each pedal when climbing a hill. The pedals rotate in a circle of rad-6op;xccs w0dsa: efwz au+ 24ius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cm wide. What is tzs m0q 8x 8nx 0h6lyykn596(kbbtso3ehe magnitude ofx538l668ks z9yb( 0oy sh0knt mxbqn e the torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about tdid3xm4 l dj;95pbw 5bhe axle of the wheel shown in Fig. 8–39. Assume that a friction torque ofd345j;ml xdi pw95bbd 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, ai7y*sq2.tv9p0 t n/ yqucgbt/re 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. u pn/2tq9/ i tbvs qty.gyc*70Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine wsrs+b )wyq(i0eiz,r 49bf*ji5n db) require tightening to a torque of 38b y sqe9 b) wrdi5 *bz(+fn,4ws)jiir0 $m \cdot N$ 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 iner g 2o.c40xbd* ,pcedvctia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is through icdodc x,.*g2b0 ev4 cpts center.    $kg \cdot m^2$

  Calculate the moment9 a8tzo7o5gv x of inertia of a bicycle wheel 66.7 cm in diameter. The rim and tire hav x5o9zo78gvt ae 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 rotateden(q( l7o bvf( -ssea, in a horizontal circle of radiusvf(eeba -ss7o l(, nq( 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on affi67m.w 7 q+f4hwumx potter’s wheel rotating at constant angular speedfimw uf+.74h q7mfw6x (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. 8–43d:a8ij uvv8+c aij 8v: avc+du8bout
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen atoms whose total ma)6m/ thoaupti0d5b) w ss 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 correct ratev5ysue3ev bny dfj31:7 yv)z,, engineers fire four tangential rockets as show,vfzvud3 :ys3nyy7eb1evj5)n 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 q3z bd tik0t-6r /p-nouniform cylinder with a radius of 8.50 cm and a mass of 0.580 kg. Calculat/z nr-tqok03b -pit6d e
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerating it from r:w6ozj)jjp/ex0 p ym:est 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 small hand-driven merry-go-round and q*7/rn7s.x xe mj 9vx)vhc4fxis 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, neglecti9q /cnxxev rvhx).mx s7j74 *fng frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating iuq0opmc* t1g(2jjb-m/ sf c: at 10,300 rpm is shut off and is eventually brought uniformly to rest by a fb0j(g21 u*qsmjof i :/pt-mccrictional 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 accelerates a 3.67 wrw7 zrb3kl2-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 d+ff;y kywc35/s i,ujof the forearm, which rotates about the elbow joint under th5,yc3yj di+u fk/w;s fe 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 considered a lonq +gsvo 6 7inm1)fe1qpg thin rod, as shown in Fig. 8–41 v7no6) iq+qgp 1smfe6.
(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 o,l;694qjas npq pqi1 if 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 from rest within four full turnbi: ag0(w9ulv ln0o n ;u+wpy5s (revolutions) and releases it at a speed of 28 alw5 pg(n0lo:uw 0n+ib;9 vuy$m/s$ 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 mo bje;z m;op2se q5a2i2ment of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine deved)o b m7gxz+q:lops a torque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls without sl1xp w ,6dqzvc 7k8hw-z58l kjpipping down a lane at 5kw 8-ldqz8j 76p,c1vhxkpw z 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with resph/t leo mi.gtwi2; m5o*9to7hect to the Sun as the sum of two too9omml* 5 ./g;tw it hih2et7erms,
(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 zbo(q2*:8wy-xii e etof 1640 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to-8w2zy *xo(i ebteiq: 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 kg starts from rest and rolls wit*knu t2 -,fprynr/o3 -( ozohvhout slipping dzr,n k(f-/ovp3t* -hnuro2oy own 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 balankc e/zcxaq .l(emyr6ozah; ; d581ne5ced vertically on its tip. It starts to fall and its lower end does not slip. Whay8.c omrd z6a1/ ;xe;5ahlcz(e ke5nqt 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 ar( 2xzx (rolb( 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an angular speed of 10.42xxo(r zrb((l $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grindf8 :n9qb:e3b+s 3vzuck sp5 tfing wheel uz8nt q:e c3p fb+5:3k9s bsvfof 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 h+ :t-rzao: va,7rajpr-ot 2qbis 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 ro, rp tbj+-:7zra:avo2a qrot-tation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shy ddp9vb+n57wk5kgeut:z p*n)1j) c aown 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 edptd1yg5:b*p+wu7ncjk95))n vzk a tuck position. If she makes 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 rate of wq mntm62s n-21.0 rev every 2.0 s to a fin26n2mtwmqs - nal 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 rotatin3 cl*1pd*7db/ peafo qg around a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the freq f*ep1* db7d/a3oqclp uency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinning at 3 t u vk.)4z9e;lu ;x9anxv.qyr.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 mom*c-h g) ojb)pi/rf fmf7n1v9 aentum 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 ofwz6ltj3xxw* ffjh1i-uyv:4)i4 w5px moment of inertia I is dropped onto an identical disk rotating x1vfww:wij6pyx fjz)4*h iu4 t-x53l at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns ati-6 s qya:d/f ma(d:+4jfhctt 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 stands a,:tbz1 k/ zqibt the center of a rotating merry-go-round platform of radi:bzz,q t k1bi/us 3.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angular velocity o,q 9c6g 3gif(b7 esntef 0.8cb (eig3efq n7sg96t , $rad/s$ 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 al zttw6b;/ 7g5wc 6pl8syhks7 white dwarf, losing about half its mass in the process, and wt z5 clwbtslpk 7g766s/wy;8 hinding 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 the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in excess of 120 fd s ,i,whae1q.v8if1 $km/h$ 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 d (k/vy6 bsuop.yr*v 2$ 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, initial8h k 5 rvhbz,dfm)yn-.ly at rest, that can rotate freely without friction. The moment of inertia8d,z. -f rkhh) vymnb5 of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diameter merryil2)ncs2 (qatc8 h)md -go-round turntable that is mounted on frictionless )nc 8h2a2stm c qli(d)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 thed./k 8fp8m sv1 .ppamf ground with the end of the rope lying on the top kp. dpf amp /vsf.81m8edge 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 the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces the Earth. Detnya1exj**x ( x ks7g)xermine the ratio of the Moon’s spin angular momentum (aexxx1gn7sx*(a *)kjy bout its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from re; h be7ner6:w,wfq:post at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of rah+tl6y7 e x*asdius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by* he s7tylax6+ the motor.    $m \cdot N$

  (a) A yo-yo is made qs)dq1kmzzc9 sbswcr x9 ,c4j .3 1n9sof two solid cylindrical disks, each of mass 0.050 kg and diameter 0.kz j1 zbc,q dsw9.sq4sxr)3s9m1c nc9075 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 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 ioisyatgtpc)s) b9p, )ps,kdh) t8 6 t,s 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 8.0 times as great, were )diqs) i 98h c:ri8nzyrotating at a speed of 1.0 revolution every 12 days. If it were to underg9 ic8i) d8 izhnsyr:q)o 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 lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-vngur x+agv20 +ik, 37tlk;ilpollution 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 cylindritligk,+u+rxg27;v3nl v0ia kcal 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 an 4qr+e+n vsc)p: qmfk2 $H_2O$ 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 (hooe+hdp3 v kf //5q/bl85ftsblmp) is rolling on 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+m; fq8,hr88 ev(pabf-tlpd: dxvaq w:c du .) 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 of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, ac.d bpa;l, w8r:q)tuafd ehv8:mqf (d8+vpx-s shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the floor, and wpge*ve 0e l69mp10e9p nbc2 :c qqgn0se want to exert a horizontal force F at its axle so that it will climb a step against which it rep nm2p960lvs pe:g c9 eg*n1c qq0be0ests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with spe)g-jwm1c ewz+ ed v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclm zw-ejcg+)1wist 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 2)te61d1ruh5whg ajb-kg rock into a 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. What is the wd5ga6e2h j)ut1hrb 1 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 cyhl zv9r4pjmjz.2i7 6 vlinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinni4hm9v jz .lvz 62irp7jng skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of l( ciej *)q17o,lpoww two cylindrical plates, of m1,7(*w pcoql )lew iojass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 thw:1ix2n rvgn + ib.w2bq. ii:ee looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble muswng 2ii v2b1wb: :qin.ex+ri.t 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 dok; epsoy/ s; hq3(s3yh not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed 2drd9v; 7t wlauniformly from 90km/h to 60km/h The tires have a diameter of 0.2 ;td7l9drwv a90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s