A bicycle odometer (which measures distance tfpln.3 .di23lg;9yc ;reznf( sbyu7kraveled) is attached near the wheel hub y7pne r.fzu3b9. ;lgi(fkd 2;3ycnl s 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 constant angular velocity. Does a po:rju-p 2 gl.tm/tn. gu*5op usmx4: pxint on the rim have radial and/or tangential acceleratiuouppt-jn / g.mp5lgm u:2xx4.:ts r*on? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be descriu 9)x01hx iwilbed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exetfvk5 qho n./u3 w5ls,rt a greater torque 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 behinh6p3tbme pq3l) nme .4d your head than when your q e3)bppmte4.hl6 mn3arms are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and threevjda- u5 v h;tfn,93dw at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front5v,adhtd9uv3 wf ;n-j sprocket? Why?

Mammals that depend on being able to run fast have tv(*e 17hrp6zy gk7 caslender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribuev a*c7zr gk t67h1p(ytion of mass is advantageous.

Why do tightrope walkers (Fig. 8–3.8e c*vr 76s vggy4vhp5) carry a long, narrow beam?

If the net force on a system is zero, is the net torque alsov2vxn i7fd34m7zx -o. 8gg fxnkmzb57 zero? If the net torque on a system is zero, mzxg .5f7- 77vboxv 8dx nn24i3fkg zmis the net force zero?

Two inclines have the same height but make different angles with the horizontalu 7an/4tf npa/ms9iox-0 9e pk. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be us emx -k tanf4i9/9ao 07ppn/greater? Explain.

Two solid spheres simultaneously startuct) e0t2dm)w rolling (from rest) down an incline. One spheretu 2c)0wt)m de 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 bottom?

A sphere and a cylinder have the same radius and the/iugrc y00v1m(mv+ vn same mass. 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 gync gm0muv/riv 0+ v(1reater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conwxcjy;6 9z z*:39pygs/pa7(ttl ek s mserved. Yet most moving or rotating obe xmt(t969 lp* a /szkyyg3zwpcs;7: jjects eventually slow down and stop. Explain.

If there were a great migration of people toward the Earth’-f 6hic:wc t03.gd bzis equator, how would this affect z.cgwt03bi i:fhc-d6 the length of the day?

Can the diver of Fig. 8–29 do a somersault with9x2,w ,cklbw pout having any initial rotation when she leaves the boa ,w29 l,bxcwkprd?

The moment of inertia of a rotating solid disk about an axis throahr46 twr yp)j4.*z shugh its center of mas s.4 hthrpyjar6w )*4zs 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 a rotating stool holding a 2-kg mass in each outstre n 9nduxp1;(xw23bs agtched hand. If you suddenly drop the masses, will your angular velocity increase, g;(n dn 2xusbx 3a9pw1decrease, or stay the same? Explain.

Two spheres look identical and have the same ma8- y kgf9x*oyuss. However, one is hollow and the other is solid. Describe an experimen-xkuf gyo8y*9t to determine which is which.

The angular velocity of a wheel rotating on a h)0im 0k6h9fz,zpfw qr orizontal axle points west. In what direc,f0f9zwp z)ik6hq mr 0tion 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 angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rothc(q )xjitoi9+w ;td(ating turntable. What happens if you walk toward the cenojq c )(x9dt ihit+;(wter?

A shortstop may leap into the air to catch a ball and throw it quickpfmn04( atuc8ly. As he throws the ball, the upper part of his body rotate4atn fp(uc0 8ms. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular mome3zgrw5+f1ygub mf.xau38 +dqntum, discuss why a helicopter must have more than ozgy3. gf 3b uaqdw1x+ m8+uf5rne rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following ang-khevu,anu04c. /jean2jii : les in radians: (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 coincidence. Calculate, usicqd 5vbse1v5,bzc-jx p nc6:fzi)t1*ng the information inside1 )pz*b tf:vnx1 vz5c6-d,5s ebqcjci 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, 3kj(6ru6ol4s s9xmuuu . ,6es t80,000 km from Earth. The beam diverges at anxj9 u4s.6m urs 6uults(o6 ,ek 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 + sio t5:()afindtqb6ntwi5: at 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 angular acceleration as the b+(dttin n5) :aiof:bi w q5st6lades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the ball:zo+s e h6.q uz(4xobb makes 15.0 revolutions, what is ou+ob.6hz(4 s zbex:qits diameter?    m

  A bicycle with tires 68 cm in diameter dhv:s :n*0n tpb (.qqutravels 8.0 km. How many revolutions do the wheels bsh .p q:udq0n v*n(:tmake?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate its:+6uofl.f;mfwuc rb mg .*s -z angular velogu z sl;*.w+r f.mc6f:fm-b oucity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.0 s ((co *) h;t ur3o z:mndg/,tivnFig. 8–38). (a) What is the linear;3( i:zd*uomnht,t o) v gn/rc 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 velocihl/k nye7*mj gkl8 9*v/fe1m +n,c2nfqz0ylfty 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 speed o/odj) 2uif+bw f 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 rotateg12eqfn 5 ip,l if a particle 7.0 cm from the axis of rotation is to experience an acceleratione, 1qgnl f2i5p of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from g 6cpkfab;,e:,) oxfz130 rpm to 280,aopkc exg;:)ff6 bz, rpm in 4.0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radius abzek7uf0t5b0yshzc(p 8 75s$R_1$ 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 ab fs0o1e9d9 n36ts :b hugiz4vcoard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenlyf0ous1ngzi9 he463b:tc v9s d . 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 cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm in ckofn7 c9hcf tbhhg3g. 0,*2 p220 s. Through how many revolutions did it turn inn gokgf7*chcb ,0ct32pfh9. h this time?    $rev$

  An automobile engine slows downlq*c ;e,9+mv y+bhur. ym,uys from 4500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed je+ou 2bsgem1n :ts in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching :m2une g+sob1its 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 unifor;bp4r:wy 0mzc ;3zhrx (ls1xxmly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheex03w mhrzzb:1 c4y;pr lx (xs;l have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 15gw *nj. i(i6 hus 10t(kov 6thwh4jj2z00 revolutions before nv(tho .skz(1wj46i h*htj0w6gu2 ji 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 6 u*o cxutv8 :o9 +po8ovxnc(mkcw;c725 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.8v8oocowknmtx c(p9v c2u; +7 o 8x:*uc0 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 reduces its speed unifo vd)7pe p8,0g .hox*x,wrzlg trmly from 95km/h to 45km/h The tires have a diametervx,ph). ewdgor xg0l ,8z7 *tp 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 weigh+yzb42.yyr5fpf8cq)w;/x b n)iocx 4 ep .jwrt on each pedal when climbing a hill. The pedals rotc.rw2w)bpb5 e ;j.+4zy 4yqfc)o nf/xrx yp8iate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the e27w xl2yp xsjeklf 9;0nd of a door 74 cm wide. What is the magnitude of the torque if the force i 9y0j7 xwl2k;psle2xfs 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 0lxy 6 qkz*2t ,lsqh2cn;f/ clfl5,azthe axle of the wheel shown in Fig. 8–39. Assume that a friction torqu 25 qalhq6l *t zncxc,0 lfk2/s;zl,fye of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are u4.bbnmw5 ;0sfk(qt.c,u bm eattached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rodq5t.e,k (;f wc.b4mmubn0bs u is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening to a torque o0v*edux r,0 suf 38 *use, 0d ruvx0$m \cdot N$ 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 spheren, /t2rcv3o0wv 9 wrgx of radius 0.648 m when the axis of rotation is through its center. xvnt0g9r2o3cwv, /w r   $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diam /m( if/f .mekt0o/5gm niltr9eter. The rim and tire have a combined mass of 1.25 kg. The mass n/ mf9r 5 ilo.mm/e0fk/gi(ttof 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 horiz lat4*6 gh/wfzontal/zg wthlf46* a circle of radius 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 a potter’s wheel r l; otl/sqe8.cotating at constant angular speed (Fig. 8–42). The friction force between her hands and the clay is 1.5 N e;sc q8t.ol/l 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.k(p;i rv h4i5t 8–43tir 5; 4(khpiv about
(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 oxz iw7. is+unw 7,qkwzvg4sel (*d-i3l/0dlal ygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the correct rarwp yusb c 8n.g j)1m/a:iid5;te, 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 steadyb;5 :cds pgi1mayn u 8/i.j)wr force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with amau7- rtz.i(unv *-4 /tzcjf.hr4po h radius of 8.50 cm and a mass of 0.580 kg.*..t4imjh--vf74 aprz(huo /rctn u z 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,b ok)/t- ynk3z 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 tangentially on a small hand-driven merry-.8u n4rhfi5 dm yjp+p9go-round and is able to accelerate it from ry5 m+89firh pun.p 4djest 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 frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shutl.7ezs52 *v us;6o(z gzt5 5lz60aiu whdrkx q off and is eventually brought uniformly to rest by a fthxl5w 0 *s(usk5zvz6z2d;.z5l q giu 7oaer6rictional 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 3.6-kg balll, nva;d6t1y9 w -rndg 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, wh ouf71ag5q,*qfc*9xbt+s m nu ich rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest too a bmtcf1sf*5qq*7+x9gu,n u 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 long thin rod, as shown l8 d)z6lr b7lmin Fig. 8–46.6bl)l7zd lm r8
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consists of t:p8j viar .4 rn9xaor*k :(mvpwo 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 /oay 2ja9qm3w.er2) owv*tyerest within four full turns (revolutions) and releases it at a spe ty9 v)2ew/.aoj wem3yo2*aqred 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 ini/ov.e.ox wt + ;oi5veertia 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 devek 46km dt u;9tv* vo5xsorm8:llops 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 slipping down a 7.i1vyk+h *3ookf. i-fh hokh lane at 3.3 h kh *f v.i31hkyh-+o7ok.foi$m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic enr+ uf;vk4f+h2s v 2gzaergy of the Earth with respect to the Sun as the sum of two terms, a2+2rkfuv vsf+;h4zg
(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 164069ja2d.cgymu kg and a radius of 7.50 m. How much net work iscj9 6dym.g2 au required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest andf8mn0sez, , + /sbdqgi rolls without slipping down a 30. fm/0 ,e gb+8sizqs,dn0 $^{\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 vertically2pix9pbpt6m* , yq0ov on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it9*mv2pbq0p,y6 xpo it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotatingv9b2z,y,dmh onxe7rr jag. +. on the end of a thin string in a circle of razmg+ yo dare v,9.,bhxn 7r.2jdius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular 7 la g/mdvb(:blgo.ry9q3ct) momentum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 7bv3ty:.clo /qab) (d mrlgg9 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rot bo4 c-t/dahe kba(e6z-z(0p6pm tsy 1ating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. a /a01ty(msphp4z ck be6-t6- eo bd(z8–48, 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 7afwuf)35xe91jkzxqor :s1 c reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed9usrx ckx:3f)7o ja1z w5f1qe ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotaspaubem )/ -;stion rate from an initial rate of 1.0 rev ev ;saeu/ mb)s-pery 2.0 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 aroylepb. p-dh(*po qvx3qc -s;x2riu)-und 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-prsvch)lxi.q y d2xpbu q3 *ep(;o- --kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentux* d,njxc tiz5m4 2ivsfe 4f4*m of a figure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular m a)b+;s t8x1;irctg0ur 7ipgiomentum 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 idsrb aq: 8v;7 fo4;9s *xovrrdqfuv7d0 entical disk rotating at a7*0do b ov7qqr vx v;uf8;s4d:fs9rarngular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at6oxzq l47zh z* 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 at the center of a rotating merry-go-round platfo:,7qcb q t;iosrm of radius 3.0 m and moment of inertits;qi:ob7cq ,a 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 wittst nf52. l0hyh an angular velocity of 0.8 5hn. 2ftst 0ly$rad/s$ 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 inx 2k*6bfm) a, xuqb.t/v.iudj to a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, v ., t)xxbd jukm/.62qa*bufiwhat 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 involw8bgso ov-1n*w f8 u)lve 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 5wx )emr+j:ec p8cy -elnv/6y$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, that can rotate fr5n5 7uj14sckb 6eozk meely without fro6c14skk7ujzemb 5 5niction. The moment of inertia 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 staj engd( csgfgi0:a :1-4 eer a,z+az4ands at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and hac,fa:gge1:az4 j4 ie0+ged n(-z saras 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 rop3s5*nq9c /yh7 oamagb e lying on the top edge of the spool. A person grabs the end of the rope an nay75chbgm qs*ao /93d 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 r or07cxpm/ in/z qr)+the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orb+i0 zx/q/ )rmrp7nrcoital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 1 i8zx j 8a3er;gm/$s^2$ 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 cylinajo 1uuym.b .)8qp4/ddk btz ;der of radius 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 the mo18;.db4uo pt).z/ym kdaubj q tor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0. -5u 2kt37 elljb4ryec050 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 linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released froluej-y tbe3 75cl 2k4rm rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of the reari pl 4bcry4g1inz.efjh 1vg- c9:.8pw 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 as0lx g-tz,wi:vt/x6 :y sky6jh 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 i0j:6kihx-6/vsy xy: gz,l twtn 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-pollution automobile pj37ir lr/rv-is for it to use energy stored in a heavy rotating flywheel. Suppose such a cr li-7r3pv/jrar has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustratesgi5+ua mw/ jopx; )/ss an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on1pp 2 9z*zijks a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we ignore -j z ;lcl:fc.xdec 8h8friction) about a hinge attached hjce : d.fclc 8xz8-l;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, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has rn453f yiu nmps.5vyo(f5x xr mq bq-(9adius R. It is standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it will cyfbuqs4(9po 55 y-5f nvmr3.( mxiqx nlimb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a tudkg5 (8pj zw3q 8dj0knrn with a radius The forces acting g 80 kpdj3 8qjz5ndwk(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 sqo)kf p 76l(n (cpul7and 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 torque required to achieve this featn lukp 6 s)qc(l7of7p(, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms aszrl tw i7f(5:+ ulb)fv thin rods, making reasonf+:z lbtr)v li5wf7u (able estimates 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 )xfv:99a-n: g p h)zl*qqi fur22xetcassembly which consists of two cylindrical plates, of mn2rqgif 9xpc z9t*e-u:qah v))xl :2fass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 tracscrk 5 cu4-m846obcsftzk: d3k of Fig. 8–58. What is the minimum value of the vertical height h that the marble mustk mscz- 5rtcbd osu43k6f8 :c4 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 no:k/ djbrv.sy: t assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed hwbma w,e :)0f98g uwh p-k5couniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tire?-9mp5 g ku e):waw, 0hfoc8bhw $\approx$    $rad/s^2$(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