A bicycle odometer (which measures distance traveled j:y+-kw(woj2gvi7ka) is attached near the wheel hub and is designed for 27-inch wheels. What hkok (2:7ia+wjy jvg- wappens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular veygfn*v* ovd;0,fd 6a v:5 yrw j3+mqgeu s/+pklocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of eitqf0kvfdgs3*vym/++w ye,6vj da r5g; pu :*n oher component of linear acceleration change?

Could a nonrigid body be described by a single value of v6eiw (+il.5i qgi/cr7f xxg9 the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger forkxgk09mi+cysw6x* rv x*; xss37z(tpce? Explain.

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

Why is it more difficult to do a sit-up with your hands behind your head thaf6(d h*t.ewac*(bf . keo ol;jw/dh9bn when your arms arojk6aldew ( .hec;ow (*fh/.*btd 9bfe stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sproll7ks5a r8 b2ockets at the rear wheel and three 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 i l8br52k7alo ss it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on gw5 qyt085a gt5tzi5 bbeing able to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageo5585ti5 wgtzgtyq0 a bus.

Why do tightrope walkwhmpu 4e3y*xm3fk ( 8hk( z8*g3avwqu ers (Fig. 8–35) carry a long, narrow beam?

If the net force on a system is zero,g9g. 9wg abh7wtf 7-nb is the net torque also zero? If the net torque on a system is zero, is the net forcew.b7 9tng-a f ghwg9b7 zero?

Two inclines have the same height but make different angles with the horizontav y59mhdt :i l7fx+bw3d mkukx84cs **l. The same std myfktu 83 w*cx9kv7sl dh:x5i*bm4+ eel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an in-u lr;7c8-iej ve2rhn cline. 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 cv lj-rehiu-nre ;287energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. They star +wi lxlx/n-w(q lb,2a2n2bug t from rest at the top of an in 2nblb2/l2 gwxqw-xn(lua +,icline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most mov5nq)i/g(9r i;ty 0zj zhb:v peing or rotating objects / zie0y(jg5 tr;:hz) b qipv9neventually slow down and stop. Explain.

If there were a great migration of people toward the Earth’sj jj 4t-h7vr3y+h,v d m-w*beq equator, how would this affect the length of the d7 +*-4jr ,hvjywd qthj-v3em bay?

Can the diver of Fig. 8–29 do a somersault without having any initi;jan,: sb+ym:f dw*gual rotation when she leaves the boar: ,m+;aywud :jsnf gb*d?

The moment of inertia of a rotating solid disk about an axis through its center ica;qfm9j usyc3bvi7e; z(n v/, 43yzof masmu4ay3zc,bc ni93jzv;(;q7 is / veyfs 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 r(clk)qbd6ea. otating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay th k6)l bqa.e(dce same? Explain.

Two spheres look identical and have the same mass. Howevbiip v. y6k9u8er, one is hollow and the other is solid. y bki.vu8ip69 Describe an experiment to determine which is which.

The angular velocity z*/ a7ywr 3 ef5ywuzrk8k3rz4of a wheel rotating on a horizontal axle points west. In what direc*y 7 u3zrw/rfaz ek54y3 8krwztion 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 tsmh/x5 0x jv-kee7l/the edge of a large freely rotating turntable. What happens if you walk toward the celvj ht57m e/s-/ek xx0nter?

A shortstop may leap into the air to catch a ball and throw it quicklyyk1i:nz(sp y3 . As he throws the ball, the upper part of his body rotates. If you look quickly you will n:knpyi1y3 (zsotice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservatjx jlcic*r( /;ion of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second ;i/*jjr(lccx propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (a)2fm0:f q *ject;r.ivp 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. Calculat.,z8,f +h quakp;n qyce, using the information inside the Front Cover, the angular diameters (ina q,8;nyu czfpk h,+.q radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Ea u7 *ph.mw:bimrth. The beam diverges at a 7 mpi.wh:*mbun 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 r.(iohtw*fxl :otate at a rate of 6500 rpm. When the motor is turned off during operation, the bladwol .f: (i*txhes slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.n s(x:kb:yhw- 5 m to another child. If the ball makes 15.0 revolutions, what is its dw: (:xn sbhky-iameter?    m

  A bicycle with tires 68 cm in diamftuuuh 3i)+h5eter travels 8.0 km. How many revolutions do th3+ uth 5fhuui)e wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calcul/ipf 7m1os,(/jpg ma swmh68cate its angular vswsmao /m(87g/1 6m,hjpf ipcelocity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.0 s (Fig. 8–38)x+ jsz1x r. c-r/b(f irijq)t+. (a) What is the ltq /.1+r )(sjr xjii+f-cxzbrinear 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 Earth (a) in its orbit around the s yh0/x x1h*kyg p2r0wSun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear sws,r y 3uu9gx;q ej(3lpeed 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 the 8vx7)inzcn+l axis of roizl)nc+ 7xnv8 tation is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly(wm ps;c+fhc41a4 y ig about its center from 130 rpm to 280 rpm in 4.0 s. Detycwgf 4( s1pcia+h;4mermine
(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 avzz 7*lv05 s t6cib2r+ vg5/zr juy)x$R_1$ 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 8 7,d+l)b:or lwdscts2*q e0elvzn* sMoon, astronauts 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 everbsvqesn*odc +, esw78 )2dt0rl*ll:z y 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 rpmz hvt01jgmg 61im6(* 1vdsb fx in 220 s. Through how many revolutions did it turn in this time?z6mggx mf(t01d1vv6ih*bs1j    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm nw5 1kcwz ).hcin 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 stfb lpcy3fb0o/4jx5v) resses of flying highspeed jets in a whirling “human centrifuge,” which tak/of3xlpb4yfj )vc50b es 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 accelerates uniformly from 240 rpm to 3m7.o 58t,txy5dbc6b go0o 6 hchggs;r60 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in this 0ty58gx 65t7orhd .gomsbh;cb, o6gc time?    m

  A cooling fan is turned off when it is running at 850rev/min It turn /d1qj0has/dynr5 *lt s 1500 revolutions before it comes to a sto/qa/dj05* tlnyd hs1rp.
(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 aso7 .(bo( f upue7r.kmv the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter ue7 .pv (kom(fu7rb.o 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 reduces its speed unifor yaxnm3z ;w5iok1p0p2mly from 95km/h to 45w5y mkz1aoin2 p;30pxkm/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 bik y ff0jee qxgbf7501/ae puts all her weight on each pedal when climbing a hill. The pedals rota5fg0xfqe a107byjf/ete 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 thefciv ze m4.gx54nq,8 jy0h7mu end of a door 74 cm wide. What is the magnitude of the torque if th,qxm44 y708cfhv u emn. 5gijze force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown i mrd2, z14zo7 ir7vfo*y+cn nen Fig. 8–39. Assume that a friction torque o dci*o7f4nze7 m o2ryv,z1+rnf 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to th b6 d3uh ehdpi**-ve8ye ends of a massless rod which pivots as *y6hi-8 3ueehbd pv*dshown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engi(ucd z+7r6c baw:0g5e;sjby une require tightening to a torque of 38gsyj:czw7+ucrb ub;a 560(d e $m \cdot N$ 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 i-:dbjin/t ,jfp*))iyho hr d3nertia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is through its center3 )pidoi)bjydr/ ,-h*h: tnjf.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cmej9 -qo /osofvf9zp8. in diameter. The rim and tire have a combined ma.qf s9vpoo 9zo ej/8f-ss of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the ejp p1:y*y0xcu kwc*/xnd of a thin, light rod is rotated in a horizontal wpxy 0yk*xc:/*cpj u1circle 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 d6wp3 uee 39qsbowl on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her s9d6eep w3 q3uhands 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. 8–4yesui:1w;-fe* g x vl-3 as*xug evyfl --1we i;:bout
(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 conrghwsfc1t-3f +45h* 7fy f vbnsists of two 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 cylf-uq fvq5 o40sindrical satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellfsv- q4q5of u0ite 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 unu,v 6 h96 2 bfcghz-bqgrfum/)iform cylinder with a radius of 8.50 cm and a mass of 0.580 kg. C c )9 bhfvfz2m,h/gubq6rg6- ualculate
(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 rest to cu pl(.uc(rs89-q qkf9iw umm1yt+/b 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 ant7 k0a90hspogr*2hd *3mw nopd is able to accelerate it from p*7 0o3d*w0gt2sh 9r knaomp hrest 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 shut off and is evef78w -o5opbu ontually brought unifor wo75ofbpuo8- mly to rest by a frictional 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 acceb eox*)hi+8yt+qt;rk5gsml3 lerates 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/b cw 1wr o.gu/4ev*y/8 xi7r1fppgil is thrown solely by the action of the forearm, which rotates ab 7 .e8 wyp/1c1*r f wlxiigvb/4pgo/urout the 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 considered a long e(-b2v a4q 9bqlf*,o*l3t yy0es pqcethin rod, as shown in Fig. 8–46e4qle-pysq*v(0e2lqt yb ab* 39 ,cfo.
(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 consin;( yo3k+eddtsts of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest within fou/ 5 bw:a 66owrkm*h2ude6;rm2f:. h skij tndwr full turns (revolutir/ nrtk6o :.km b6dj 2f:i *wwmh5u;2ad6wsehons) 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 inertiad)rec yng 3 1oy) uci3e++2wkj 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 developsd5 lo30a7k gx6s;0r:fvqcfas 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 an7wvx b,biffd94)obx ; d radius 9.0 cm rolls without slipping down a lane at 3.3 xi9;obxdvf) fw b 47b,$m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Suncx b.cglg61m67byq 2 e as the sum of two ter 7clxe b26ymq. 1g6bgcms,
(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 of 7.50 m. How munin;g 5k4d0o8q x. rr7e0 taozch 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 s0noqrtxk0g oe 7ad8nz 5i4. r;olid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts fromkaoqun0i,sbc 1m ; /w) rest and rolls without sl 1;,i)mnq owaucb0sk /ipping down a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is balanced verticall liyc0jlg c+,.o+qtkbq0k6i,xoqq0( y 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 it hits the ground? [Hint: Uoli+ q+0j(qcik c,q t 60kxqy go0l,b.se conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end of a thin sqkmm;i3ovi. mxw:q0xwixz7 lc01 .g6tring in a circle of ra w71.v wc0z3mmmxx6q.x i;klgii q0o:dius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical gr2a74+se + jeotb gkm/zinding wheel of radiute e+7b4mgs a2kz +/jos 18 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is ro)ft/bclr 4 ;ok8njzq+i z7j*utating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speedu7jl z zn ir;)f+48ko t*q/cbj 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 onc50m:e y0z cgwe shown in Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular spc0gwc5y0:e zm eed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate frkiv6pr :in,.mwx/ f*l.l i/m oom an initial rate of 1.0 rev every 2.0 s to a finalmi6mxn ,.wkoi ifp v:ll */./r rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis through its ck).uxw03 oeddenter at a frequency of 1.5rev/s The whee0d). d wouex3kl 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 frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of ak5*zbewje;wqh () ya7 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 momentt)en3kr 1 8qpoum 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 dro *g4 ,i/:/ ghpeal noh hw.4wq7,fcditpped onto an identical disk rotatingdal q n7*.h,//i4op4w:iefgw,hh tcg at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2p2snb/ysov m9g+; o* ue2pu;n;al;i i.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 stc-ho7 toyc2, jands at the center of a rotating merry-go-round platform of radius 3.0 m -ct2oy,c 7o hjand 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 rotatinls lfa 9y6)b:og freely with an angular velocity of 0.8 f6a)9y:olb ls$rad/s$ 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 x,15.dhvpxkpd ;2i xcits md .ixpxkch5v1 , pxd;2ass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to 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 12 epjo2 ;u8 81ukli2a dr(nq:zx0 $km/h$ 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 mwp-z jzaib/95q y5 z)$ 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 atm5j .(9zhy *5h lzpqoijbo,-t rest, that can rotate freely without friction. The moment of iotm zh9j oizjb,ly h*p (5-q5.nertia 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 merry-go-round tjft7z1bmtyl30z f+ ta u8n 8e;urntable that is mounted on frictionless bearings and t 08 83jeftmnyz1bzatu l;7f+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 rouh -o8gs8fob u0l.l 7xlls on the ground with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does l87oh 0x.u-ugoslf b8the spool’s center of mass move?

  The Moon orbits the Earth such l i*/i2i z)fazrp65 :mcgpn1dthat the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting zrc/ga 2imn5 i1*6) zpdl:p fithe Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate, swjj)st,94lzesj -o 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 cylimhk9i3e+-0 p lj8v khfnder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at const9l+ k- ki8p0vh3jfehmant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 kg cpowi l r:946kxp*n9uand diameter 0.075 m ioc n:6*w9p l9krxp4u, 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 is the angular speed of thwn5b *zjd5kiazm- t;-/g/rcf e 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 mhcz1k93k kpm8;-s gdwith 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,g-cmzhp sm3k;kdk981 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-pollution automobile is for it to use energy stored inilaxzx *;kjt2 :e4rpi8b05 vh a heavy rotating flywheel. Suppose such a car 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 without0kri2 vajx 4pb xz;elith58:* 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 illustrat me4t/2u lphb9es 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 on a horizontal surfac4 v 2depvi2hg-e 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 frev3kn; q.jeq c7ely (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 .j3e;n7q cq kvrod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on.y-t e.yi d8jq 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 ha8ji y .e.dtyq-s height h, where h

  A bicyclist traveling with -h( 6s) q-vu6 pe-mtsuv5qx wji0wx *vspeed v=4.2m/s on a flat road is making a turn with a radius The forces acting on the c 6hu xe*-mv)pqs 0qu-v6vjwt5iw x(s-yclist 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 bqyc5)g6l(x4 ;b s*g kho6 hddlength 1.5 m and begins whirling the (6kh 5q 4*ybhx6ocsg d)lbgd; rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as t3z gdiw/ls9e-.5 c jfjhin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held t3ld.5sgwc 9jjfei-/z ightly against her body.   

  You are designing a clutch assd(cubsa z4 79eembly which consists of two cylindrical plates, of m97zeb(sd u4c aass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 traci9;r basd.cz* k of Fig. 8–58. What is the minimum value of th .ib9as; dzc*re vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but wh ouarnr,k6um,0u5 x 46oft5do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutir06pre12(xu8 n1b7 ,h; rsct vyj.w8qjxd wx xons as the car reduces its speed uniformly from 90km/h to 60km wcwdjx8j6 r yu,(0 .she72rrv1 ;pbqtxxx 81n/h The tires have a diameter of 0.90 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