A bicycle odometer (which measures dzlfk*2x7 3aak istance traveled) is attached near the wheel hub and isfk3k*x2za7 al 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 f+(.dowmyqcwa:5uht j:2*bh 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 matuw:y+2m j* oc.whfa( hb:5dq gnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the angular veloht20ko78 y*zvouglw, city $\omega$ Explain.

Can a small force ever exert a greater torq7;edqhxcg /if */ 24d; l1sr;tugjfrnue 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:m wc- x+z.gb brp5x(v3fp jq/ with your hands behind your head than when yov:wczg+xj3xrb - (.p fp5mq/bur arms are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seve2wknp6 .;- 7 kpozmqbon sprockets at the rear wheel and three at the pedal cranks. In which gear isk op6o2nbp;qm-z7 w.k it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend onfn; bc 1/0ztlb being 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 advantageouln 1/z;c0fbb ts.

Why do tightrope walker khhk/ a*9dg*bs (Fig. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the net torque i o82fjm:hf (em mc(:xalso zero? If the net torque on a system is zero, is the net force ziohmf mx( :fe2j:m(8cero?

Two inclines have the same height but make different angles with the ho 7)mp fyp5auu0/o/9tnna+xw rrizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the btnx+w/) na a7ru0u9p/pom f5yottom be greater? Explain.

Two solid spheres simultanexbdwyoz:uo/f *8zg 1 *ously 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? Wugwdx : *yo8oz fb/*1zhich has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. They s-e 9n uypcud5gs(ve-80hc;hq 0)mvfqtart from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greatc85ue9nf0-;mv( c es)hp-y qqh 0ugv der total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet moqgu/elk(c y-: st moving or rotating objects eventually slow down- y/c:e klqu(g and stop. Explain.

If there were a great migration of people toward the Earth’s equator, how woul1pra/n4 +ez f2egkgo 6d this affect the length41go k+ z2fe nape/6rg of the day?

Can the diver of Fig. :si)8,b 1l:n rs4mmd/6oyw6 nkzfm zj8–29 do a somersault without having any initial rotation when she leavess/1z, dwno::mbnz6jmkr ys 4f6m li8) the board?

The moment of inertia of a rotating solid disk aboew-) ar+:im c1kzj c*yut an axis through its center a*rji-ek ymcz1c : +w)of mass 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 mas-k /zed /hrrb.t (x8bfze/p8 rs in each outstretched hand. If br e.8 /pt/eb/z-d(kxzfhr8ryou suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. a tx) :ntpg1i8However, one is hollow and the other is solid. Describe an experiment to determine which isa1tixp t)n g:8 which.

The angular velocity of a wheel rotating tos9fe9dkb*(8) fea ion a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential la* fs)k8ti9b o(dfee9inear 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 rotating turntable. Wbhsvwm92+ad3 fv z9e7w/ jn8khat hfd 72m9v j+zv nhbs9a k8ew3/wappens if you walk toward the center?

A shortstop may leap into the air to catch a ball and x9/c0(g u/ 0btphsmxjthrow it quickly. As he throws the ball, the upper pab/sm9gp( cxt x00h uj/rt 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 law of conservation of angular mompek). 9gt: jgoxy04irentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keeprg .9:4io0egpyk jxt) the helicopter stable.

  Express the following angles in radians: (a) 7om;-an n ox6mr 0*1 udmksc1n30 $^{\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 abc5la5g0*x dk mazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the S*g5ba x0 5clkdun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth.gl dhp1 bx0+ *3ef*itn The beam diverges at an n3bhlxg p d1 et+i*f*0angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. Wi7/)p dezrm*khen the motor is turned off during operat e)z r7kd/*impion, the blades 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 lem./ix7t vj +6uwvam1avel floor 3.5 m to another child. If the ball makes 15.0 revolutions, what isia.mw7x uvm1+ j6av/t its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How z-uzh n /ivf4u4dde4l ,; 6skb2lgmt9many revolutions do the wheels mak4ldhe 6ufl;dui vbg/n-4, t9z4zmk2 se?    $rev$

  (a) A grinding wheel 0.35 m in d1)x dbqvl 9 gcdv.o5ux--. fsviameter rotates at 2500 rpm. Calculate its angular veo-vgfvxs 9 ) 5dc.ul.bq1vxd-locity 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 revolut 8 vhn/p:szr+yion in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from the cer hspz8+y:nv/ nter?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (hz j q9/b;j4ga1gun3la) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed ofms,v3 re :*vzu 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 particl6w)m.(oinnsa3h,(hf zh hnu 3e 7.0 cm from the axis of rotation is to experience an acceleration h ao fhnni3m(h3nzsh)6u(.w,of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rppnrf6 g 10d+zxm to 280 rpm in 4.0 s. Degf 6 +drx0nz1ptermine
(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 ,d6m le: 55i hsc7f72o5zhq;etx n qss$R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard the Apollo spacecraft put themselv5qzw ):) la5zegtgkr+ es 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. re5t+zq kzg):l 5 w)agThe 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 220 s.jn, /cif ;vr:q Through how many revolutionsqif cn,jv; :r/ did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calcul-yx0+ ub 4l:qh9x obp) r4jonnotl9n( ate
(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 jets w. (qyfl:zk ; inv.fxp3 fo9*ain a whirling “human centrifuge,ak nwv.x;39 fipq:f*yl .f(oz” which takes 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter acc6k2 ,s mx1abr2qp23 v v6msy ek41i;z(nq idwselerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the(s6 dz;v32ysk1, i 41s6kqn ea xm bqwripv2m2 wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 1500 yi..t x ) ;sxpp0eeh*mrevolup yhpesimx .;.t* 0xe)tions 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 car reduces its speedz-ajbgas*.m/ z x7 b*k uniformly from 95km/h to 45km/h T*aabg xz s z-kj.7*m/bhe tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the caf 41dvdfx-8(ats5wb w*ktc9*n hh) o ur reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.80 t(1vck da)n4 *f8d*hh-x ows 5t9wbufm.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each p0i ;c)mmk 0fvledal when climbing a hili ;cl0mf)vkm0l. 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 a door 74 l48*vsv ax lb7hmz +9 q s,f7zj/u+t4y3zsamkcm wide. What is the magnitude of the tor q7azu48jtv mlh+z/* sbf ayx 9 +34,sz7lmskvque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel5nh -0+dm: mpc* , /uwuhhmiqv shown in Fig. 8–39. Assume that a friction torque of 0./ 0hu hnp*+ qv:-hui,m5cwdmm4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, aretb :*l y*d(xmf attached to the ends of a massless rod which pivots as shown in Fig. 8–: ytx*(*dbml f40. 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 eod o 2zs)cm2kj:x.6kzngine require tightening to a torque of 38 kok. jz26mo s)dz2cx: $m \cdot N$ 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 sphere of radius 0.648 m whe 6pw nzz7 s2amvlzr+,.n the axis of rotation ivz.swz7 +m l2pan,rz6s through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 667di1y2.ul nx t.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hx7niy1dl. t 2uub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod iw2zdxg4g .venze.c(i b/ 8b*ws rotated in a horizontal circle of radius 1.2 m. . gde.*bizg(vz/ n4we wbx82cCalculate
(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 bowzeegs (j5z-m*l on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction forcee5-gzme( sz *j 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 avbs 4y i*y(5dov9ihk9 rray of point objects shown in Fig. 8–43 abou*v5dv4i(ib9yysk9 o ht
(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 twofxobn8l*g ynl4j . d8/di, ms6 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, uniform0p0+ cinujhi 8 cylindrical satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each roccni+8p0h0uji ket 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 cyavd6*.2t c: oc.dqb hbe.lil6linder with a radius of 8.50 cm and a mass of 0.580 kg.lb. he* b a2cvl6iqcdd .:6o.t 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, accelerating it froctw.xymzj22 :wqf 3:)y2dhww m 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-go-round and is able o-/vn q04lr 9(tmxxgr to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume o0rv xx( /9l -g4nmqrtthe 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 ib. c25aui 9tmys shut off and is eventually brought uniformly to rest by abu myt2.59ia c 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–nir *dp5am r*34mujg 245 accelerates a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the action of the forearm,3bs4su0+lk cc which rotates aboutslc c4 30bu+ks 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 thi-ac1w r7 c9hfvn rod, as shown in Fig. 8–4a r1vcf79h-wc6.
(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 enhy5s w5h6j/ *dolp) 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;gji)v3vvtxosi3 : dh)x1m- u within four full turns (revolutions) and rele:3u1jxs iig-vtx;hvv ) )d o3mases 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 ha+n/m:yai6i s qt(r wrchnd( n+xtz5/xi8 no(*s a moment of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine w cur4( 1fat52g8xkz/ewtzt6sl s33 j develops 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 andc/t68w f8zl 9dv-f l1ong0t js radius 9.0 cm rolls without slipping down a lane at 3. 8jnltcf0d 19lzow/ 6 -fsg8tv3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth wit(w wrma u) bsxn-w**m(h respect to the Sun as the sum of two termw*-(wws*mar (n umxb )s,
(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 .0+knui/gw rjf5 lmr7 much net work is required to accelera +rng7umjr0k l 5fw/i.te 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 stayhv6z)y da1)hrs97d jrts from rest and rolls without slipping downd j h76sr) y1hd9z)yav a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is balanced vertically on its tip. It starts to falo7s rs: :b/fodl and its lower end does not slip. What wi /rsfb:7odso: ll be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on tt.xg 2;qx0 kiepp ,s1xd/r0fkhe end of a thin string in a circle of ;x/ kp. si1 d,0xfpgt ek2qxr0radius 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 grinding wheer6y cs)s9 x*chhl x)v 5qrx:*ia(va y2l of radius vcla* qr2xs59hyax xsi:6hv))*(ycr18 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 rotating89y6os1v3 d5p vm aytw at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the 18w svm95ypda o6t y3vspeed 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 F 6 d;fga3ahgg9ig. 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 angugf39gg a 6;hadlar speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase ho a g29 lw)-mtms-1lj,tcvv7jer spin rotation rate from an initial rate of 1.0 rev every 2.0 s to a final rate of 3 j-1) tvo9-llm2vwsmt ,gc7ja $rev/s$ 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 thkihjea-+ gw83:l 0yp frough its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?e-:kp l a8yh30g+jwif    $rev/s$

  (a) What is the angular a9pn3d09qgx z momentum 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 Earth.xtbx.8l/w ax
(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 drp kn.pw8 j(x9xopped onto an identical disk rotati k(9x p8pn.xjwng at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at tt zbyix,b1;,ao.;lpr /qm*c2.4 $rev/s$ 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 t:cnr qv1qjvrgo4+ /y 4he center of a rotating merry-go-round platform ofqyv +q v4rro4jn g/1c: radius 3.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angular velocity of )ee/ n+rj3xcq0.83)jcxnr eq e+/ $rad/s$ 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 qg8jy-gd i2nem+n;:/h *lfj.n j fwn 8about 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, what would the Sun’s new rotation rate be?(rof ;geinjhqwm8n-j2/dy n:*.8g+f lnjund 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 excessvq /)qn y*xz-c 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 ro;dnok+nm)tea dm2tv*:xr 0 1d; 1vj $ 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 plr 84 z)dsifo,0djdk q.atform, initially at rest, that can rotate freely without fr.doj4 d 8zsi,dfr)k0qiction. 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 personr/ fk3*fjcdb xu)*/iu stands at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a mokjf/)u3x * rifcd bu/*ment 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 wi)j g36t v2qtshgt wd)w+w-) up6hzey jyh,d48 th the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walk)t) g t,jj +6 6w-3hh2wwgzu8edsdpqvhyy)t 4s 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 suc0*jeu v) 6s-yr;vvgybh that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to -*brv)v6vyyje0s g ;uits orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates fromkaz yv .p3i beee-be23ld/-zh3)e lu, rest at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the fa9zxno4y)p hga39rzyykj 0n.o7 *d+ shape 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 thej ) 4.zhorgya+fd7n3x9 0n*pya9okz y torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disksy jp3 .d2ata2i, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 3iyj ap2t2d.a 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 speej) bgs-* 6pxm 6g3 zttf8uw.yad of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, -ljo1h(6x k qb/8af lhbut 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 threexf6-(h hb 1 o/q8ljkal-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 +)khlg2ot 7d0wu86 e8qojmx qs:hqi f* 8yy*va low-pollution automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needi0qyog 7wx o*l+sy2hh8fv 8 mjq8di6:eqk tu)*ng 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 illustraf/zq cu5g sr5*o6hp)c tes 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 surface at speed v=3.z. sd:taw l(7u3 $m/s$ 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 ignore0 )f(0)gce fw5rbr51fezy w te 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 5z(fc e bw0e5t0)f1)ewgy rfrof 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 +m(h/5cv),ke zr+a5obsz mls vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55omb5sz csr)/l+aez,vmhk5 +( ). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road isbs15 aq48bq4*dhbe 3uh h8r cg making a turn with a radius 4dbcq 4hb1 rgq8b3shh5a8ue* 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 and m3g5wiu*cy) h(i-3, fw fsn iobegins whirling the rock in a nearly horizontal circlg )3ycnisu*i( w5f,wm h-3fioe 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 asj7ris2ge 7jyi c1 r,bc 1uh:6o thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater wit61h , cbs7r:2g7iey 1jjocriuh outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly whictvg8h/3f w*-u sv ;w uvhps5b:h consists of two cylindrical plates, of mass: pwu/v*;t8 whsgv s-v 53hbfu $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 ;b( ukak21-gxqt6q8+i-q-fmbfz j 1itq mk)b along the looped rough track of Fig. 8–58. What is the minimum value of the 1f-qti1k jq)-ifz qkua m+mgkbb2 8-;6(txbq 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 bnc4hld / xf0c.:d;wfnot assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as thwt+/ik::abuzz9 hw (oe car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What wab(tau /:9wzhk+ io:wz s 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