A bicycle odometer (which measures distance travelei+ux8n(0 )uc, tnyvhgsk l,l 1d) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle withxu+ cg 1)v,un0 ( ,8ynslktihl 24-inch wheels?

Suppose a disk rotates at constant angula g,o*u2ngw0w wr velocity. 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 cogw0ung2o *,w wmponent of linear acceleration change?

Could a nonrigid body be described by a single value of the angular veljk 1f2 8n 4rrhyt5iny2q*m2moocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larg mx *n-7llzek(er force? Explain.

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

Why is it more difficult to do a sit-up with your hands behind your h h7l*na 9et7qgead than when your arms arg7l7hea9 * ntqe stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at th2ob*):l bq;ibk ;yfmnn4tv+ je 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 is it harder to pedal, a small front sprocket or a large front sprocket?fy qtnmj4)v k:l*;nbb+b2o;i Why?

Mammals that depend on being able to run fast have slender lower legs with flvhzpkbyp,4 /- esh and muscle concentrated high, close to the body (Fig.b4-z vp h,yk/p 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) ca.pcoe4(sse(h99pe 4b ukt3dzj / /e warry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? If thet.y34 q1i 0mori0 kai-zsf5dc net torque on m53 - y zi0i1toras0k4.cqdif a system is zero, is the net force zero?

Two inclines have the same height bu 9scpsir4 (l1ox, zzy,olb4,zt make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at thecpy bs,(z x,i zlz,or4os91l4 bottom be greater? Explain.

Two solid spheres simultcci h 4uo w4sale5(c*-aneously 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 kinetic e44w-*5eca uoisc(hc lnergy at the bottom?

A sphere and a cylinder have the8w t6i2 xg ,zyb0 87vjjz8eirj same radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energtybrzi728jej88 , 6jv0zw gxiy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yeyulancko 07;io5m a k5aa8-l(t most moving or rotating objects laauok5mi57-yol;acn0( k 8aeventually slow down and stop. Explain.

If there were a great migration of people toward the Earth’s eq3aw rrj5b:* du.4ectv flx-*arm2+ pquator, how would thic*u5j*2bpwatr q+4m. raf 3rx - vdl:es affect the length of the day?

Can the diver of Fig. 8–29 do a somersault without havinrlq +ik1+ gsu/g any initial rotation when s/ s+r ki1l+qguhe leaves the board?

The moment of inertia of a rotating sol.mekz h*a;7xi) 4eveh sz 2nr*id disk about an axis through its center of mass is x m2kev i*a4he7 z n;*).rszeh$\frac{1}{2}WR^2$ (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 m o*+dfzx5( xr6pm rn-wass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay zxpf w drmx6 *(-5ro+nthe same? Explain.

Two spheres look identical and have the same mass. However, one is hollow a -88,tee(daje 1 bngcdnd the other is solid. Descr,db18 e(-enga 8dtjce ibe an experiment to determine which is which.

The angular velocity of a wheel rota sfu,gd2zf0 -z3y o:u;l, 0ckpoawp0zting on 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, zl ; 0fo p:a3spgd0zou2kuzy,-0, fcw 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 thn -(b hm0 9i:3ve9)6 bdzelfnwv/m.fo 4zwb goe edge of a large freely rotating turntable. What happens if you walknmze)(wbh oo6v.4vnb 9/-f3iz9f: l0 dgmbew toward the center?

A shortstop may leap into thlp0,gjp-:- m 2sejrg xe air to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explaij g:x0 s- 2rpm-gljp,en.

On the basis of the law ofig9 h, rgylqds*30dac a*v 3d7 conservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more sd973hqairyggd*al, *3 cv d0ways the second propeller can operate to keep the helicopter stable.

  Express the followin ekwg+h ,v 3-i4yzm, rp.q3znng angles in radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Eart1djc r-j9 5 ycnb ;krxt/ept9ko 5xt94h because of an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Mot-/5t btpn4x j9x kr9de o y9j;1kccr5on, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 38;fc nt .gdyr2*m4tb m10,000 km from Earth. The beam diverges at an angledttgr1 m;mybn24* c .f $\theta$ (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 a+xfyt i6itn5ggq1 b95t a rate of 6500 rpm. When the motor is turned off during operation, the blades s g i1q+55tgt9ybf6nixlow 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 3.5 m to anoty;l7noe b bl+er4e)nn96 dp,kher child. If the ball makes 15.0 revolutions, what is its diame7dlbpn+ek6) lo;9eybn r, 4e nter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 2h mpmuo j5zwg05pj q;h4jxhx+og43 - km. How many revolutions do the h-5mh q4x4j;h0 +p3uzmj px5g 2goowjwheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculd)ev2 rz-b.p uate its angulardpv ezr u).b2- velocity 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 (Fignr.x +/ f2d61jmllnca. 8–38). (a) What is the linear speed of a child n ./dc lj2fr6xnmla1+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 Eart +iufv.,v8ooq h (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear spee k ufmj-(b* v7x+kb0gjd 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 ax mu0 e7pz/(qqw le;v7gis of 7z mwe ;pgq/0(uqvle7rotation is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel 4m j;ja1yov j-z:7z sraccelerates uniformly about its center from 130 rpm to 280 rpm in 4.jy;o1:-jajm 4r z7vsz0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radius *xbzkroxnn*wzz 5e,;f8z 3y- $R_1$ 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 theqx 1974 0qjf x 2ci2pcn,igvnh 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 a cylindvc9jiq,pq74 2 n2xh0fxn g 1icer 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. Thxp q;o.8s.fsdpi ):jnrough how many revolutions didf ;..pjdiqso: 8px)sn it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calculate t t)7nwr))aeb
(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 in a y;q83kxy w0c( (csngewhirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolc cq(yygn3sk8xew;0 ( utions 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 accelerat twa nt9*m/fj0es uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have tr/an w9*jtmtf0aveled in this time?    m

  A cooling fan is turned off when it is runnglaxy71p:irc e d0+3ling at 850rev/min It turns 1500 revolutioncge x3l:0p+7ldia r1ys 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 eh/yy;mc7a u6tn9 ) tecar reduces its speed uniformly from 95kmemt tauye9h/ 6cn7;)y /h to 45km/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

  The tires of a car make 65 revolutions as7gsrp +m -;.yg9orj )zrs p+xg the car reduces its speed uniformly from 95km/h to 45km/h The tires havyps.g9r s+r jorg);7x m p+g-ze 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 bike puts all her weight on each pedal when.iynnzwza n: ef0p5pcf( n;f*w;6kc 9 climbing a hill. The pedals rotate in a circle of radiziz9f c pcny w kn nfe;6(a5fw*n;:0.pus 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 ther pr (0t/ax3 chwa0*ez- bjss:3r jx8v end of a door 74 cm wide. What is the magnitude of the torq/xe0zts*sajcjr8: h pv x b303wrar(-ue 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 f9eub +uy( 4( :6nag rlnmsow676prpvthe axle of the wheel shown in Fig. 8–39. Assume that a friction torque of 649 u a +vn:mrre6pny(( gubflwp o6s70.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a massless r dkh3;i8nqj*qu* t:gwod which pivots as shown in Fig. 8–408wuqgji n3q** ;:th kd. 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 engine require tightening to a torque oo.uxql.seh 04hro*nk+u 8 k)h+vi /zjf 38x. surhiz)hknqv/8*jlu e oo+4 . +0kh $m \cdot N$ 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 ra .p*ecxeu n4f9dius 0.648 m when the ap. fuec 4*9xenxis of rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. The rim--z aam5:jmkb and tire have a combined mass of 1.25 kg. The mass of the hub can be i:bma -za-m5k jgnored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotatgyxa p1 20nm+wed in a horizontal 0xwa npm1y2 +gcircle 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 potigs*:s0,wl* zhhrqh b 0n4x.nter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the clay hxn w0ns,i*s r*h.4qzbgh:0l 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 xo13ca+,o su / tzxh/rthe array of point objects shown in Fig. 8–43ozx/1t/3arxuhs ,+c o about
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of t3hf3rdxof2g,e dmv6nz +7o.a wo 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 cylindrical satellite spinning at the correct rad9 vj-0 ad5.jcdf8dt 2wlnlb j )c6nf.te, engineers fire four tangential rockets as shown in Fig. 8–44. If the 9 dn8 jaf n-5wtdj ..cj6bld)vdfl c02satellite 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 cylindz+ e:umx3ziyb p3;i0ver with a radius of 8.50 cm and a mass of 0.580 kg. Calcum+3;pz:e u bviiz0y3xlate
(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 r1 3lp zj9cgnf0,r ps 4f6smml2est 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 a95pwjz+m5ub*vwx 0ge ble to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m an5*w wvmjepx 59ug0z+ bd 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 bz220,j zeamt grought uniformly to rest by a frictional t gj0 aztme2,2zorque 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 acceleratew(z fw:x+v3lb2apqr 2s 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 .p8vhhme .em:, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest tom h :.8.hepemv 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 can0 ejy6q1+bgvt-ssk*w z :f4gy be considered a long thin rod, as shown in Fig. 8–46ve w6y1 yfjkbss-q+ 4g:g*0zt.
(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 co5 pj.d b (cajq.r2cn2znsists 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 hia 39my(d7a,gjr;c l dot26kzammer from rest within four full turns (revolutions) and rel3,( ;jd i7zmal29gdr c o6yktaeases 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 mom/roe;cn1bue7g0t0ut 4 , ltdoent 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 deve 0hpq2vf j(*fyhu)*3 *vijd rolops a torque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls without slipping down a lft-0o lotj-7qo;ks e5 1vlzg) ane at 3.31t-sev o-o lj 0t)g7lk; qoz5f $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the sum ofe.dn5h *z.- xsawjnf1 tw* z-e.f d5jxwnhn a1s.o 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 (0/9d h9wb jmjqrx5uo of 1640 kg and a radius of 7.50 m. How much net workqm (w0dbru/j 9ho9x5j is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.q0-yk;mo4fx y80 kg starts from rest and rolls without slipping down k-o 4;m yxy0qfa 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 4 kff)5y/8q:w o )o bafyqtdm*to fall and its lower end does not slip. What will be the speed of the upper end of the pka*)48)q of y ywtmbdo/:5ff qole 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 n tb dxba+n5owrrp93s89v-u us.)n *ayw1j -yon the end of a thin string in a circle of radius 1.10 m at rt)yb - rs*od9wxu-na .n1y5 j+uns3a9pwbv 8an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angulan e.3 -hzc(ov;q8wrcx r momentum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotating at 1500 rpmrcon(8zveh ;.3x-w qc?    $kg \cdot m^2$
(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 platforz*d)c 5l+bdc b7 v0e)zg u5pnlm that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, d*b eu +c0nzc bd)llz5gpv7)5 Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) ca4gqn)9 ,d l zuq;7jrxgn 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 rotation4;9zju7) qxgnldq,rg s in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initia bd oc*9 f(y.s9pk5w zownw1)p6,wrmz l rate of 1.0 rev every1(f,bz9 s * zwkr6cmw)nw pw5ypo.od9 2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis through it utf8m8ny58h zs center at a frequency of 1.5rev/s The wheel can be consi8 y5nzh utmf88dered 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 molxo ;nyau oru o7;5t(;mentum 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 l r.89pwb6doo rcoa 24./yayathe Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$

(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dropped onld:61u rmdu2*hkw g5.ir4nal te8v4mto an identical disk rotatinedlvrmu61 liuk a 4ght284 d.w:nr*5 mg at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns aoy9d0 n*5 ddkrt 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 rotatinn2eqo2 qttw.2il9w 8ug merry-go-round platform of radius 3.0 m 9t2 .quwol82nt wq i2eand 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+dr 0-4vk 1snf a7ebo/)y fr0 mtuf:yu rotating freely with an angular velocity of 0.8 rymf7 r-bfnv/ed )afuy:k0014us +ot $rad/s$ 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 h9n7.k:cxvhaw4 5;vd aqy nbl0 alf its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming thevacwda: ;9xv4 y lqhn n0b5.7k 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 ee 5bcr7mjiy81 xcess 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 2n qn h9w; nh*:-yxdb8va;wd w$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, that can rotate fr2bynu)cm0 h -v3uxtgnit5e) r8kp6 9n eely without friction. The moment of inertia of the person plus the platform 0tv3m-n6bpy hcni 2k) gu 9etr) u58xnis $I_P$ 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 merrglxbi. w:4 gnb-3 9qtqy-go-round turntable that is mounted on frictionless bearings anb g3 .4t:9liq-q gxwnbd 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 the end of the rope lying o1o6 vrkxbu9 in;knl (fjy ;x0)n 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 0j r;61lvxuiyk; b xf9onn )k(far does the spool’s center of mass move?

  The Moon orbits the Earth such that thr9 pmnios;bbz*v- ( 1xe 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, treas*pzbm;- (1n9borx i vt the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at hf3ckgoy6y e n9cq;4(,hbcq8a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a un .:d5enwlhtcm2t;h z2iform 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 torque delivered by the motor.5t mh; hw 2tz2cn:led.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, esai6nm47pqj1s dk2( ti 37g2*ijuqr kach 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 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 re t2g* uaim sjr7 24(idi n7p61kkqqs3jleased from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how nw mqo8i-+6q3nk(-hh*xr. ylr pmi d.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 d4kc(.0ju+ryl9v x wghhk*h. i7/y s iour Sun, 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 coy(h ih04 *.s+ xlu79yghvw/.crk djik llapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use energy stor02l(v22*kg1 f gio)hzwlr:owbaqw y- ed 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 izl-h(w*k:v2yq lb12 org0 awg)f2 ow m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates mcxj1 a3pwty:de78*xan $H_2O$ 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 qh r3.-27w*rv zq vau l,s3egtat speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we ignore frictioncpt:o-spw z - m) rr.i wd341(oub8kqs) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assu81 otp4urcd(sko bwz)3w.rp -mi-:q sme 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 vv+mn+vf1+)wfi ( w d/1xxwdyx ertically 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. 8–55). wnfiy)+ + /mx vv1xxd1(wwf+d The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a turn wsh6 jfs2ff k01kq 8m(vith a radius The forces acting on the cyclist and cycle are the n 0 (v82kffksfjhs q6m1ormal 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 slin+gwp 491d7 6fc ;mwl3 dcuxpnng of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torqpnucf1w w3l6dnd4p; m g97+xcue 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 thin rods, makoipk4/xr0 d:h ing reasonable estimates for the dimensions. Then calculate the h/ 0xiok4:pdrratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindrical p mea9d iw/h/9plates, of m9 ph/d9eai/wm ass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 track ralr8c (: gff3of Fig. 8–58. What 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? Assurf8r(l3a :cg fme $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, b6ik;a a qdu2..+49jgwi-y 2aaimdyeev+o 1 knut do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uniforwlhl* qczag eh;;3,vnocoo. p9ihd)5,/4hfs mly from 90km/h to 60nazo,;sc ohio c *le4dlpgh9h .q fw/,3;h)5vkm/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