A bicycle odometer (which meas::k0 pql)tn crures distance traveled) is attached near the wheel hub and is de)t cr:pl:0n qksigned for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at coyoce8r-xi,ev6) 1y c4 eq* ve ng,*rgynstant angular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point yrxr 8og-,e** cg cqv6ie1ey,4)neyv have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body7 cx(/5 fc gckigz-ga* be described by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque tha aism;-3j+akrn a larger force? Explain.

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

Why is it more difficult to do a sit-up with your5m v0 +(pu od5dprg-tm hands behind your head than when your arms are stretched out in front of you? A diagram may help you to ans -pu5m (d rg0t5pomdv+wer this.

A 21-speed bicycle has seven sbseb*v;-tc m dxl7;ad+;d,d lprockets at the rear wheel and three at the pedal cranks. In which gearl+ce,;-* dbs7xlb dtm;d v d;a 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? Why?

Mammals that depend on being abgyv s4 wzhv. rem,;-i8le to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fg;i4 ,z.s hw 8vemyv-rig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow/,;jam s*sgkr zehj0* beam?

If the net force on a sydlz(,9pk yth(stem is zero, is the net torque also zero? If the net torque on a system is zero, is thdh ((ykpl9zt ,e net force zero?

Two inclines have the : x/j: k5qz x4px.uem8rdski8 same height but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball atqjmr8ixk/4pxx 8e. z:sk d:5u the bottom be greater? Explain.

Two solid spheres simultt-;zvcm d)fy8bcy:jg.wv 8(paneously start rolling (from rest) down an incline. One sphere has twice the radius and twice t:8vf ;.y zwvcmj-(pt cyg )d8bhe mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have th *qs xi;oo.f 44dm9ujke 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 kims;qi4kxo*4.of j u9dnetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum.6qkvk5pb(k fpt :oe3 are conserved. Yet most moving or rotating objects eventually slow down and stop. E.kqpetof (v5bp: 6k3kxplain.

If there were a great migration of people toward the Earth’s equyf-uhw 0ww3melh65q-ator, how would this affect the length o-5 mhly w-630 whwequff the day?

Can the diver of Fig. 8–29 do a somersault without having any inn wlz rfn+7r-9itial rotation when she leaves the bol9r z -+nfwr7nard?

The moment of inertia of a rotating solid dl ,iqh4j-bd8b9jk ::r 7ryvuskn;o, k isk about an axis through its center of r9nvkr:,4 yb, dkj8k7soliu-:b j;qhmass 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 rotatg5 pgid i9tt77ing stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Exg5ii9t7p 7 dgtplain.

Two spheres look identical and have the same xnalqi3 sz ,*qf7c-)tmass. However, one is hollow and the other is solid. Describe an experiment to determine which is whici q),xczsql3a n-*7fth.

The angular velocity of a wheel3jqaz;b,q s g( rotating 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 ea,jgzq q bs;a3(st, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turntable. What mp+g 4i1aww:u hap iaww14:p gmu+pens if you walk toward the center?

A shortstop may leap into the air to catch a ball and throw it2knfqgj ;k0i ey8z5t1 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)z 8e;0j1 fkgytniqk25 . Explain.

On the basis of the law of conservation of angular momentua78bol2hp;f km, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keo h27lapk;f8 bep the helicopter stable.

  Express the followin6o6 bo y f.ngp nuk.7a2b(hs ay5c1w1jg 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 Earth because of an amazing coinci:xnuf z bnst h2kwrn6 s186egx;4,i(f dence. Calculate, using the information inside the Front Cover, the angularei;tssxn 8frw nn:k,6 6 hx142(fg ubz diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. Tn0vwwla* 7uqbi pn6d)/ /-twtj3er; 4a k/fjfhe beam diverges at anwaf6;7utnvwnjk/fdiqt4 r 0bj- a*/3 pl/ w)e angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate b8v;cd2v v j(3d8nkir at a rate of 6500 rpm. When the motor is turned off 3bvikvn2; v(dr8djc8 during operation, 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 2eb5bpyehcb -3 ab5w8eoe;6 clevel floor 3.5 m to another child. If the ball makes 15.0 revolutions, what is its diamewa8e ;2 ebpbyb3ce5- 5cheo b6ter?    m

  A bicycle with tires 68 cm lp/x5yoovj yxmscf30t ;c9e4k,s p ,5in diameter travels 8.0 km. How many revolutions do txkpcp3,9 sy t5leom y0xo/f5 c,s; vj4he wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate its a7(2hy(rdio qrv 7yn)ck hz)8 gngularr )r(zv)(qnic78ho 2ydyk 7 hg 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 4xcc 5p7bcq-w7w e;zu 9.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from thw ; 95ubcxpw-czq7 7cee center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) * b81.ty6jvqfq5imd ain its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed oz( vlv ut:*wo(f a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0 cm from the axist7s hda, y7k6d2 wtn:j5dxp8xs tp/ +g of rotation is to experi7d n,: tw5j+sdxt2h pdtx/ys6 8 akg7pence an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its centerf vvxv+g0 ew;t6( lfw, from 130 rpm to 280 rpm ;vx fv t+0f6,(glwevwin 4.0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radiusi/ zp.cjlb5 k u,k+fj8)vbh; d $R_1$ 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 themsel gidu-ax*s -7xves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they i7uxxdg- as*-accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,00stw;f)e//a-jfc p w * fe)hn,r0 rpm in 220 s. Through how many revolutions did it turn *prnf;ffwe)c se / a)hw/jt-,in this time?    $rev$

  An automobile engine slows 7)x8*vo .atu 3y pfllbdown from 4500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed jets in a whirl*nzj( w ,m/nf+u( fctcing “human centrifuge,” which takes 1.0 min to cf znw tc/(mf*,(n +ujturn 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 accelesh wy0*7. qyklgnkc4jwic28+rates uniformly from 240 rpm to 360 rpm in 6.5 s. How far nc0k4qykih7 2wl g+c js y.*w8will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is runn6 4, p,uw3 lhtqeu+2c7ytgkh6ckadf c x43yg 8ing at 850rev/min It turns 1500 revolu6f ldcua7,,y3u6x38pgce 44c hgt wqk2hy tk+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 revo (kos9hcc7l:mcz.-z s lutions as the car reduces its speed uniformly from 9c9 kzcs7.sl- : cmz(ho5km/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 6sg49q rm -lyumyiy5 h :5woz()5 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.y9sz u ho m:wy-y 5rl5mi(4gq)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. ge gfgs,,vn8 climbing a hill. The pedals rotate in a circle of rngg8, ,g. esfvadius 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 cm wide. What isk: .uhmxs0z1 a the magnitude of the torque if the forcea0m u.h zs1:kx 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 x:2zjsx c3yj oxa n9: q,r.qp,of the wheel shown in Fig. 8–39. Assume that a frictioxsp2,zx9.3j:xrj,oa qc:nqy n torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m dq)f6zn9 m+i +xz1jvjabx)6n, are 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+m q+69na)b x zj )jzx1ifdv6n the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require mgnc c+j5lw.m +l ;67bbjto w.tightening to a torque of 38 .b+l nmbt+5;wjc.7lc 6mgojw $m \cdot N$ 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 x/1hc0.grv ltfdll, x mgf-m 3n8c4p,of inertia of a 10.8-kg sphere of radius 0.648 m when the axis of mr -tggf.dc ,fnmc/8xv 4l,l3x1h0pl rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle r uk6ib39xb e/ fl(o4/rg(ysb)9ghl m 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 ignoegbbryih3f( k/s9ub ml g/49xo l)(r 6red (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in a horizttk85;naz n b-t2+let; nz*uxontal circle of radius 1.2 m. Calculatltu z-;z25*n n kttt;x8 a+nbee
(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 0hk 8 7 8owqsexu3trs:-vdw/m lf at1+n4lyj( a potter’s wheel rotating at constant angular speed (Fig. 8–42). wtnjd3l1xrsaeq w o7yk/4u8v8 mt( f-sl:h+ 0The 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 of point objects shown in Figna6+wl gu3ex3. 8–43 awge33 l6+unxabout
(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 total m46g qzg0 9g o+tq6g9 5x /arsxaetwf7eass 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 cod91nsf2 57q xdnlo7rarrect rate, engineers fire four tangential rockets as shown in xdo7qda5rs 9 21nf7lnFig. 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 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 f.ey991a:,ufycs ujp with a radius of 8.50 cm and a mass of 0.580 k99aecpj: .fu,u fy 1syg. 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 swi1gy gji1bl9hbr+((x dngs 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 small hs /4x)8 oq*u pleam8asand-driven merry-go-round and is able to accelerate it from rest to ao axs ql4)a*pems88/u 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 2wzc 8:y7k fwvlfd( *jshut off and is eventually brought uniformly to res* jv7 c l2d8:zw(fkyfwt by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 azuu02*gpmyb a c. :cn4ccelerates 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 th-2gts ymook3* e action of the forearm, which *k yoo sm2t-g3rotates about 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 tha3 + lci22dh2qrzkg k3in rod, as shown in Fig. 8–46i2 hakzkl32d 3gq+cr2.
(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 cm5krb 0br:- 3ls2xcyaonsists of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest wlbj .v e5ltzg; o-lv )g528gbeithin four full turns (revolutions) and releases it at a speedv52;85jeotzlg)le gv bl .g- b 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 inertia oqp3r/qf (uwsfn:c i .73t-ovuf $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine deve:xv/u;ao )p fflops 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 +n dt(+xp pi,kb o/rly6b.(xo without slipping down a lannkobxti+ xy+. , d(b 6(olp/pre at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to(e ,w: 3 q(2 ns:blaqrqxnigg5 the Sun as the sum of twox(5i qgnr(:3 aw2,qln g sbeq: 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 k)ha uaq y; 51kl eq8,ndtq45yw/fhik7g and a radius of 7.50 m. How much net work is requirekq5wh dfnie5 l4 71aq8 yqu)h/ ka,yt;d to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts froj0f4; e ztb7qom rest and rolls without sl;oe0 7b4qf tjzipping down a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is balanced vertically on v (vcc9fk5y t;8eyzg( its tip. It starts to fall and its lower end does not(e9cy ;( gv5 fykctv8z slip. What will 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 theww jk4 ma4495syk jo.u end of a thin string in a circle of radius 1.10 m at a 4ousjkwkwa45y 4jm.9n angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylind7kojxymy ciq*z9d 56 (rical grinding wheel oj (7qz c95yd*kx6o yimf 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, handijv,i 1nli**hvmq8x1 s at his side, on a platform that is rotating at a rate of 1.3rev/s If he 1ii *nlj1vm qvh8*xi,raises his arms to a horizontal position, 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) can redwipx-le7it7 f e8 r:dw8w(:xos mjk6(uce 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 hermdw (xkofj7 xl6wie:ri- wst 8ep8(:7 angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial rate orjc9 6tjl* o x;nufuow12 i i1ph0c8)uf 1.0 rev every 2.0 s to a;np*0ih 1jl)uxu1u w6ofo t9i 8r c2cj 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 vertrjh)s)i4l * u.j uf)gtical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg anr ))tj )j *4giuuflhs.d diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinning atuj8x1b w11uafs 6 5aor 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 tzv f1v7z1(rqtq 4t3 4q6-5vaznzqj jEarth
(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 eleolw, o 68v73p gzc1of moment of inertia I is dropped onto an identical disk rotating at angularleooc 3lg p6,w71 8vze speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.6emxw6z0,4etey / if:; rcfn o4 $rev/s$ 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 oljny0h q c3t/mrs:b4+ f a rotating merry-go-round platform/: q sbmtlcj3yh n+r40 of 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 r+m/gttr )f 5zyotating freely with an angular velocity of 0.8+tt5 )yf/ zrgm $rad/s$ 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 collapse/+m ypf/a9wep khqygzc5 af;2n- y/ul1* k;fqs into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away n9py/1l+ewkqmfa;- /ugpaz * y hfkf 2;cq 5/yno 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 excebr0ma kg7: vb)a9gdm:ss 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 n :wbsgm1am5 8 (evnx4$ 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, initiy(w63x stos0z f ,7hopally at rest, that can rotate freely without friction. The moment of inertia of the person plus thxpyw ohfss(zo03 t7 6,e platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diametz -ru )l8axlbca)lhenw2 5x +6er merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia l)l+2r -865z nuaex lhwc)axbof 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 opp f0,sh, 3yrnwop+lz*-o* oground with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, w0 * ohp-oln*fzp ,oy3r+pos,holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side alwaysxyr,1 k wbie5hi(ke9hoo 8)i 5 faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Eartr 5i ( h)i9yeokbik1oh5,8ewxh.)    $\times10^{ -6}$

  A cyclist accelerates from;my-vki-e lm4 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 shape of a uniform cylinder of n17o3 jx;ns6mr- vftj 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 mo t6m -s3rojnxnv1jf;7 tor.    $m \cdot N$

  (a) A yo-yo is made oo ,hzx. nj2l (zfmr4x1f two solid cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) tr.x2no l,xhfz41zm (j hin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how0 o-za fmrrp 44d-t8p8cder) v 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 Sun, but with mass 8.0 times as great, wereh q,x w5yy9ot/a12rkh rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neua rxy1th9oy 5qk,h/2 wtron 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 for0o;0rblng8; mr-gan r it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylind; 0nbrgo0a-r;m ngl8r rical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates an o-xbfqq s3 eeji7:9m 5$H_2O$ 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=3.3 2hmxi 0 w)h; li0gku3a$m/s$ 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.g8mxu-q :zrt+ opm6h3 e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held hooxm-8r3 qmh gt6u z+:prizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standbvmp gxcw , )sa2.4ib zbkl+.6ing 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. 8–55). Th,+sk gbw..6 bvxl c pi4bm2za)e step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a fl4o t8/-8 5qoixz+z df usbbjx2at road is making a turn with a radius qbj fu 2zx8bdt o xsi8o+5z4-/ 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 0.5/ 2e hhch+wpp ( 9jimg+a6pj-z0-kg rock into a sling 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 ajpcha - gh96+w2m +i peph/zj(fter 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thin bz3 ky71r79 lxqamff*rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a7r 1mkx*q79af 3l yzbf spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindrical pl-djp50pc v) ;a/bl9o ;vwqdqr ates, ofbp/9add;v qcplj ow )v;r 0-5q mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 loh;uz5 ,/.bupn wto.4s(h z ctyoped rough track of 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 .tnuo yzpc( . 4/wthb ,szh5;upoint of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but d j1ht;si+ +h+;n+,njt/zsl kwjxy a 5jo not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its spe)-g xnvr/pyhthlus*w s04b a) q+2un: ed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular a)ph4*at+rlvw- q 0un)y2h gx:u/bssncceleration 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