A bicycle odometer (which measures distf/-5cj 1notfi ance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle witj-fntoi /1c5f h 24-inch wheels?

Suppose a disk rotates at constant angular vec7ux tmq7sp-c0i: kl1 locity. 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 either component of linear acceleration chacpsc 7ktu -7 :ixml0q1nge?

Could a nonrigid body be described by a single value galf2t5, d /ipof the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger foc *endqvbjf y)1)+;legd skleih/* 99 rce? 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 q kt.+m+zt mb r*pw5e6to do a sit-up with your hands behind your head than when your arms are stretched out ir.5+mwm etk+qtbp*z6 n front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel andoz bn.gy/a.8s three at the pedal cranks. In which gear is it harder bngas. oz8y /.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 hek2 0ta4ydu 7i*daq8flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rota*qd 7ai y0 2e8th4akudtional dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walk+bc9bjsy t) .lers (Fig. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the net torqu:gqe*vc38o ime also zero? If the net torque on a system is zero, is the net force zero?*: iogcmve83q

Two inclines have the sam;dps on7) 0rsde height but make different angles with the horizontal. The same steel balo;dds0ps7 )nrl is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres si3,wok dyhd;/cj al+5z multaneously start rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total a kzy5ocw3+dld j; /h,kinetic energy at the bottom?

A sphere and a cylindeonv4o2i sl 2b3xpi l31r have the same radius and the same mass. They start from rest at the top of an incline. Which n4l2 vlp i1 io3x32bsoreaches 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 mom4gm)6erb4sy/e h eb(rentum are conserved. Yet most moving or rotating objects eventually slow down)4gy / er4rembhb (se6 and stop. Explain.

If there were a great migration of people toward the Earth’s e5aeg*3hm+ waiquator, how would this affect the li3awea+ 5*hmg ength of the day?

Can the diver of Fig.l9h asq c*zy98 8–29 do a somersault without having any initial rotation when she leaves the board?lq hza8s9y*9 c

The moment of inertia of a rotating solid disk about an a) (vng.pza2 ap bow1lo:(6x ccxis through its center of mass v1c)2 : wxcapo(6opnb .(aglzis $\frac{1}{2}WR^2$ (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 eac i,xa eh+(vvygzgvd6)+ , gv7ah outstretched hand. If you suddenly drox 7g v+vih,()v +ey6vgdg,zaa p the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have thehfjz4. 4gpj l, same mass. However, one is hollow and the ot,gfhz l4p4 j.jher is solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points west. e:m0j9pgvz(j4xp st 6In what direction is the linet(46jz9xmej s pg:pv0 ar 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 lar8z n/ty41 ze81 rptgxzge freely rotating turntable. What happens r8pntzt18y1xg/z e 4 zif you walk toward the center?

A shortstop may leap into t6x5m7b hrh y o54gf+5au f ybwy;6l5pohe air to catch a ball and throw it quickly. As he throws the ball, the u4 au5w olxg7b phy+56f5mob;5yhr f6ypper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the lawasf ee -c .rnwk 1*x8m)7ep8s/ etz/ti of conservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can ope x8wt7)ns-.eer*ct1pae k/ iz/s8femrate to keep the helicopter stable.

  Express the following angles in rad52bsad bfj3o) ians: (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 coincidentf/xqazd::f ; ce. Calculate, using the information inside the Front aqz:t d;:ff x/Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is direct826.j wgpr w0m cflkv0ed at the Moon, 380,000 km from Earth. The beam diverges at an ancpw 0r j2mgwfv.lk 086gle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rob 8y56dpm aw;6vpowg/ 2rss hn-hr0+htate 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 8hv by /06arspwog6 dr2m;ns-h+5 phwblades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the bajf ky1e7co,3h 9-a41y ie rwfvll makes 15.0 revolutio4 yfr jvk 19af -1eeci3wyho7,ns, what is its diameter?    m

  A bicycle with tires 68 cm in diameter t0 una-bv vqtnvg( mc7*: mi8pz74vqx i6,a 2llravels 8.0 km. How many revolutions do the wheels make(bv6: 7cv na viqnlx 4a2gp uv,ilm7m80*z q-t?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 25004 tpc+oner)b e)kv8dw 2mo7c xj/(ou7 rpm. Calculate its angular velocity in od7ckx+ mob)p/ewo82 u4ce )n (vj7rt$rad/s$ $\omega$ =    $rad/sec$
(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-roun:wye a gx:s-t2d makes one complete revolution in 4.0 s (Fig. 8–38). (a) What isw:y-sgte: xa2 the linear speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (j arf 7d*hz/nq4g1( dya) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a poin.(wnx 5*mkrj)ew0 ix8a a -rqut
(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 centrifuxn(e) mbg t+ei 3:9/t5g rnraxox.nq;ge rotate if a particle 7.0 cm from the axis of rotation is to experience an acceleration of 100,000 btn gex)a.;/e5o3grxt+ 9:i (rnnmqx $g’s$?    $rpm$

  A 70-cm-diameter wheel ac( gn0)g9yncgycelerates uniformly about its center from 130 rpm to 280 rpm in 4n9cngg()yg y 0.0 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 radiusmis25 jas w/;4ayq, hs $R_1$ 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 Mpf8q v;ig 7h2xoon, 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 every minute during a 12-min time interval. The spacecraft can be thought of as v2i 8 xf7hpgq;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 22e6:6ugf: yodr0 s. Through how many revolutions did it turn in this tgdfe: 6ur yo6:ime?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calculate t:-740djllnnaqr mq3
(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+(k )tbpwi8r p jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final sp w+pr8kpb(t) ieed.
(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 unif1-zb 7 hxx2s ge0;vazhormly from 240 rpm to 360 rpm in 6.5 s. How far wil 1 7-hxs; b2xhezag0zvl a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned of advw7uh1) ,c -xyyl)rf when it is running at 850rev/min It turns 1500 revolutio-c)xwhl ,ay ryu 7dv)1ns before 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 cagtq/ q7yze0c4 ,li8rl69 jqjzkks,, xr reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.8s,yt9 lrg76kjz,q0 4c j8qliekxq,/z0 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 yds4.i9ui k xwd7 nn3*revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.nw*snyid7u49d .xk3i 80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on eaeoaz j c/5.bx4ch pedal when climbing a hill./.xezjo abc4 5 The pedals rotate 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 end of ;vm4f( b 6vf7( cphdwia door 74 cm wide. What is the magnitude of the torque i cvfd ;46hifpv(mb (7wf 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 the axle of the wheel shown in Fig. 8–39. *dbt0v lhu8ean+-f ;wAssume that a friction torque o-0t* andhwu 8;fv+le bf 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, av d.2ekn; ra4zyd8u 0dre attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude.vz e0a42u; n dy8dkdr and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening to a torque of 38zrv 9 -/ln8z1k0bhtl s lh8zv 10ktsz b /r-l9n$m \cdot N$ 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- ,ayp yhzmcl :n-sfnam0n1zj(c0 /b 4/kg sphere of radius 0.648 m when the axis of rotation is through its n04azajbc,y ms/:l c- 1zhnp0 mn yf/(center.    $kg \cdot m^2$

  Calculate the moment of inertia of ipezg3 )/gyqr.*xc h *-o ydi9a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignored (why?). ydx*ir-h *zi gg3)pq/.yco e9    $kg \cdot m^2$

  A small 650-gram ball on the end of a2k0 c*mogo)w w thin, light rod is rotated in a horizontal circl )cog wk*o2m0we 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 rotating at constant angul/zq3jq18o i)ds o1ara-u2y1w1 dhl g +6 xnhnaar speed (F3wadq1iy81h zo/ 6)hn2l-g1u+1 nrs d aqx aojig. 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 of0k+q tqt7polso )0a5i point objects shown in Fig. 8–43 qpqs)7oko5tl+ 0t 0iaabout
(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 totor:1d9bwn su d e)y9shh66;c jal 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 cylik(ra :su 9jedhh0 n*l5ndrical satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satelli*0 hl5u9 sdhjk(a:n erte has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a rad v)(wwyey h1h.mj2use5k z:;xius of 8.50 cm and a mass of 0.5805ske v1 jwhzmyxw:(;.u2hye) kg. 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 swinggdq-x aa,2 m4awr/ g9ps a bat, accelerating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a s0:gcoiwza-v h r/7 nu4mall hand-driven merry-go-round and is able to accelerate it from rcaig/ohw nu:z4 7vr0-est 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 eventually br fyb*e,meh:k.(1 .unmcq7s bt ought uniformly to rest by a frictional torque ( tu.,bmy*fe hb1ce: q7skmn.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 accelerate (ndap1 gds+v yc 11sh/tjx19bs a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the action of the foreaibo*eph1l;o2 1 czy czn-*p:crm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from res*b 1h;1o*zc l-2p :pconie yczt 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 thi n n +ufx4hv,3+t 9qu/ljzwx/cg jdn,75tm/uon rod, as shown in Fig. 8–46gdn+/h+nzq37w4 tj/ tm9 luj, nuufx x/,o5cv .
(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 two massekw;2,; ikrd.tdm5q py s, $m_1$ 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 acceleratex;aqp (ern a-3 t j3lwk8)za;as the hammer from rest within four full turns (revolutions) and releases it 3zr(ea akjaxpnt;a38lq;)w-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 inskx9 4sw6tbx-1 pf2 glertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torqu1zsiq c94fj6 ie 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 ra 2n:f;n5h jqfodius 9.0 cm rolls without slipping down a lane at 3.qjnf :2on;h5 f3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Ea v5gm .1z-nw:oxs i2girth with respect to the Sun as the sum of tn1w. i- szo:ig52vgmxwo terms,
(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 much net wovbu,d sv en84gv- j wzu55j13erk is required to accelerate it from rest to a rotation rate of 1.00 revolutioue8ds-njz, b1 g5w4 ve35vu jvn 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 witt vdwo/zyx96b 7l w(umn(4n 3ehout slipping down a 30.04ylvenu/ 6 z( d wbm9(ton73xw $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is balanced vertically on its tip. It starts to fall and ir- 5 yyf72e3 aq;qm-z(d odeouts lower end does not slip. What will be the speed of the uppe 3rd2uy-qa5of( d7mz;e-oy qer end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.21b.;zp)p zls3)w mrytt 01 wl*y0-kg ball rotating on the end of a thin string l pbtm3l y;0 1wz )py)z.wsr*tin a circle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum a uei)-o 8-byqof a 2.8-kg uniform cylindrical grinding wheel ofayqiu -e)-b8o 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 his side, on a platform tm **wn)7 elcpkf-ox1 wi(x ,ushat is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decr )u-psn *eoc7*mf1xk(iw,x wleases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one showzl0k9zc vyk,zg* k b/:n in Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing lkk 0 :cg9v,b/z* ykzzfrom 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 speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation ra. x p5(lvhq2 ;ry6vdo8mhz( inte from an initial rate of 1.0 rev every 2.0 s to a final rate of 3 6(5 dhh8(;2ivx oyvzq .prlmn$rev/s$ 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 cu, 1 r4462llzdei rnra vertical axis through its center at a frequenz lnrdr644c1e2l,rui cy 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 frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momu mwvw 7 iq3h;gh5g+;tentum 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 momentum of the Earc/:q)uh 3q.psuzg0kt,gyym*r - kvlx7 .d t*wth
(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 o)e)hmq-vwsip 7(e r+kf moment of inertia I is dropped onto an identical di)qkm7 r (ve h+s-epwi)sk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turnsrjcs2 y(aep)g,( 9m zy at 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-pb,vdase x.us x( 07f0v8d/v ugo-round platform of radius 3.e, v 0v8pux7sd.(df v buxa/0s0 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 7saj*77bv q xzdt7w gcu.k*-d4l*eal an angular velocity of 0.8 v7-eaj.74*dg *b dclazu7kx s w*7qlt$rad/s$ 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 abo1g+ms4fvp2wjz0v je*y (c1 ht wpl(t*ut half its mass in the process, and winding up with a radius 1.0% of its e+jf1tp1 *pt(wzvsm2 ewlycj *4 v(hg0xisting 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 wnhy7lvj( ts)2)/ynrcdi7 ml5 inds 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 pqc)e*/urqon2 nik /1b9qh.r$ 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 cankapdi:(, 8gbiv tnf yf0uni--36es/i rotate freely without friction. The moment of inertia of the person plus the plt(u8:spvni e,nifikad b0iy/ -g f63 -atform 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- asje6+ mi*ht0m diameter merry-go-round turntable that is mmi+hjt* sa06eounted on frictionless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with thepa.*-iscf bc/ e75gep end of the rope lying on the top edge of the 7i 5f/.agb-*peepcsc 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 s d21wssq1m/* *wlnrgaide always faces the Earth. Determine the ratio of the Moon’s spin angular momentumds/1swa*l2*rwgn q 1 m (about 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 rest at a rate oar b 48(w7pnxg(hufj5 f 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 sha:mm+ 5hp6edja b h0t1zip *l1dpe of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the tor*l pjtea60d+h15z:m1hi b dp mque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrx j70 w/sbdl.bical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg0bbsd 7jwx l/. 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, hoyg+m/fwh;jy 3 w is the angular speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our S f woa2edwf,wtq03 o*7un, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, w32fodw7wt, qaf o*w 0ehat 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 co.b ruzr;2x xx;+ 5vmis 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 cylindrical flywheel of diameter 1.50 m and ma5.xzu;+c xmxvror ;b 2ss 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 illustrat(dmu5(wmccy m 5 m-l+mes 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 e uur:1,+tjc sa 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 masr;pq 9q,giwdjj95q j ,s M and length L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tipiq j;r qj9 d,jgp9qw,5 of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on t2ahl+kyv4y:rvl- )39uc w b)gz r3nrbhe floor, and we want to exert a horizonk )arubly vbh3g3r:n2+vcl z -49)yrwtal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.43;mtfwhxisi* 0k j4, kje7a o2m/s on a flat road is making a turn with a rfjs wmjix0e a7,ti4 h*kk3o4;adius The forces acting 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 anbnn33v cnf+4cmxo08 fd begins whirling the rock in a nearly horizontal vxoc+33n8b0mn cffn 4circle 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 soliv3sgfuv60 sox a0q91gd cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretchedg6xsvqo0fag10 s9u3v arms, and with arms held tightly against her body.   

  You are designing a c:ilb g +dh5fw(lutch assembly which consists of two cylindrical plates, of mass : hldwb5f+ig($M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 rol25u, xgxpsd. buq0oy3/g i m2eow41mrls along the looped rough track of Fig. 8–58. Wh p2 5rq2/bg.u3ox0xso,uw4mge y d1miat is the minimum value of the 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 do not assupt0f i/ 2r y8wi( :mnnnkyu.y+me $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uniformly hg(hrb6 7w0jvfrom 90km/h to 60km/h The tires have a drhg07vb6 wh(jiameter 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