A bicycle odometer (which measures distance traveled) is attached near the whee.a*)8fqumu0u h ;32z xcewbh2 u yx*rol hub 82 mcx;zrf h3w*ou 0eyhuqb. )ua*2xuand is 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 a point on the rim hav,qb.mlm2;6q9m 8 g kfxa)vhwte radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear accelm)q .vl6 xq,mm 2; akfhgb8w9teration change?

Could a nonrigid body be described by a single value ofwup b)t7)os 9xee*l+p the angular velocity $\omega$ Explain.

Can a small force ever exe+yjtcnt.o1k73/n jrvtk)s d1rt a greater torque 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 with youni)r 4j5y hqe8r hands behind your head than when your armq54i ej rh)8nys are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and qh1 2jy, uqk/gthree at the pedal cranks. In which gear is it harder to p/uk2, hy 1qqjgedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legs with 6 bmv n(e -;mgaomla1w82)soy flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass -a)yo1owem;6a m b2nlvs m8( gis advantageous.

Why do tightrope walkers (Fig. 8lz ji3evq/tn 1w* c7n-+;9mlzm 3pbbb–35) carry a long, narrow beam?

If the net force on a :mr k6ab :qzfvr+/d8i4 a: sqhsystem is zero, is the net torque also zero? If the net torque on a system is z abk4:q hrs:+:zf6vma r/iq8d ero, is the net force zero?

Two inclines have the same height5.nvqz6;zshgr t3d/p 9hrh)6 * ablv t but make different angles with the horizontal. The same steel ball6 t qrh9lrd;nv35tvhbzph z.*a/g6s) is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres siuq m-:tm4 tt:+qn wjci83kf3imultaneously start rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom ofct- 8t iw3: mk:+qmjit3qufn4 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 the same radius and the same mass. They sta5;w 6pyf (y)bskyxqqb z:v) 1nrt 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 energy at the bottom? Which hwyfb6)kxq bsv(;q15 n)yzy :pas the greater rotational KE?

We claim that momentum and angular momentumr t4dsd tz0d:ia6*w 9v are conserved. Yet most moving or rotating objects eventually slow down and stop. Explaidt: dwr4a 6*zitds9v0n.

If there were a great migration of peoplevy/uwih8l4 9kk bcz8k9fc b (jai148h toward the Earth’s equator, how would this affe1lb4z9h/ka k yfjbhi9v8u c4w(8c8k i ct the length of the day?

Can the diver of Fig. 8–29 do a som-k 4r5 od,nf8-kqobx qcns22zersault without having any initial rotation when 8zqo2roxk4q 2 cds--,f n5kbnshe leaves the board?

The moment of inertia of a rotating solid disk abou4 e,gw ggec:r( y-4dny4d qt5pt an axis through its center otqcr( dg44:g54 ye,p-gw y nedf 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 holdi)x-7k.atl 1mh qu o(s +ozwkf,ng a 2-kg mass in each outstretched hand. If you suwa7,ulh qz+1-stkk).o mofx( ddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same 2joz j d hal,2 xwgpnw,,m1/2rmass. However, one is hollow and the other is solid. Describe an ex,n2z,g/j1lwo 2 djrh aw,p mx2periment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle po)umgks4 b--huh*u kg9ints west. In what direction is the linear velocity of a point on the4u kg9-- shu )mbkg*uh top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of afzyc+,r oz (+g large freely rotating turntable. What happens if yocfzz +gyo(r, +u walk toward the center?

A shortstop may leap into the a1sy4iklt)fdb lni t,l-9n7 4wir 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 directio-slf nt4wn4 ,i dlkl)1i79ytbn (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discusnyuz.-ann76y-6c fhrs why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the n-uhyry6z f.ca6 -n n7helicopter stable.

  Express the following angles in radiansv03uw au*8zfq : (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 coincidence. C:g9hjor 8y*vl alculate, using the information inside the Frontjyr 8ogh :v9l* Cover, the angular 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 fr/crdwb (l0yia 5g3d15c9c rvuom Earth. The beam diverges at an angle5a 1db w vcc05 /9lrdyi3gru(c $\theta$ (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. W(w **+cj5 2) a xsd:teivtqyywhen the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as a:+)v*w j5x(2cetq t wys*ydithe blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to anotherxxr4: / 2fapyt child. If the ball makes 1x4/rx f:y2tap5.0 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolut )a/**;oq :j)/hq lhjggjvtu(hw y0naions do thehl oh :aqg(yht */0)/ujjwgn ;j*)avq wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate its angulq. . bx6 fiu,ij..tp* qwiijzy 2u6by;ar 6 y.yub.*z2i u.f;6j.bxtqjpq wiii, 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 /z,)ncp mdce2 one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from the cente/)n e mccpdz2,r?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its onb;e;6 jx7 -po sslxjt+j; -hod2sej9rbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a p-tnw0o kf9+a qoint
(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 aiok,c ynr v s-e ex3gv.3ja344 centrifuge rotate if a particle 7.0 cm from the axis of rotation is to experience an accelerj xo4i3 a. -,kes4ry 3vnegc3vation of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its cen nz r r-gtifw(ocrl7d1 *20ph9ter from 130 rpm to 2801 p zdrr nht*092co f-7(iglwr rpm in 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 radius5a c nze3s9fu5 $R_1$ 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 a9dc1: of3.mtn.tux3 b /g hr2lfztxh1 board the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start1 /xm.o3l u1r t bg:x2z 9fhf3c.dthnt 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 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 +zratt+dz ncojd2k:x8 t;; f)in 220 s. Through how many revolutions did it turknct f;xrz o2;zd+t +8dja):tn in this time?    $rev$

  An automobile engine slows dowc*b n7f ,agugjsh. ab1f-lh; /n 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 flsijz76.;c0itn d2;t xi atn- wying highspeed jets in a whirling “human centrifuge,” which c -x;ins26ja tz7wdt0.int;itakes 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates unb.1x1b2gjhe1; xg e fhw ij4m+iformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheehbgx 1 4jwmf+1i.j1 e b2;gxhel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/mi93m +jx+wa(k0twv ne r(t0 jean It turns 1500 revolutions beforjnet vw +mr3t9(a wxje (0ka0+e 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 k:rggnu 2zx4-its speed uniformly from 95km/h rx-g2gzun:k4to 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 as the car reduces itsl ovarkrh* 7j0j6*2rp speed uniformly from 95km/h to 45km/h Theaprk vo jrh0*ljr62*7 tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each pedal when climy ues2pgd(h35 bing a hill. The pedals rotate inpse3g5ud2 h( y 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 cm wide. What is t 4ak2r+;jku egjbe5+che magnituda + kber;+ke4cu2 j5jge of the torque 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 about5t+btd6w1 pz n the axle of the wheel shown in Fig. 8–39. Assume tzw+p5d nt61b that a friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends,q(rz8o.mb d w 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 d8o (wzm br.,qthe magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine requiregpx:kg fdeyw9n-6u97po 4/ yc tightening to a torque of 38 y n:u fpk9 od4/6g pgx7wyce9-$m \cdot N$ 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 whx ;a;hxdx0n l2 ppw.m*en the axis of rotation is through itl2p.xadxxp;w0n ;hm* s center.    $kg \cdot m^2$

  Calculate the moment 15*ci8 ygve- v t0wznunt js+,of inertia of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of th1-nt sug , +0*ynezvc5v t8ijwe hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball7.pnqe /ft:ny cx)4,a fxbj 8q on the end of a thin, light rod is rotated in a horizontal circle ,fx a)j y tb.4qen:c/xfp7n8q 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 a1w* r5 wnnw:ld potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and:n5 1lwnwd w*r 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 sq,;znjpt/pr o9)t:gajcs82of point objects shown in Fig. 8–43 about o/9;rn gc sqjpjs 8ztt:pa,2)
(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 iezpe 0c/cb.m6/ t 5utof two oxygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite sping(zz,;lru7btgd gi3x r5 .a 7hning 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 thebrhz3 g z,.l gri77 5udgta;(x 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 w+l1 gp,3q /2madirc .rb n5vdvith a radius of 8.50 cm and a mass of 0.580 a/,2 bpq nmi3rd5vvdr+g1c. l kg. Calculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, a/ fei1k,e dh0juf3w)m ccelerating 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 fi *: 0o5s twcgkdxab*o:2bh/hand-driven merry-go-round and is able 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 and has a mass of 760 kg, and two children (each with at2csgobwd:o 5 /*axf :i*hb0k 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 sg)s;0ys 4cqy and is eventually brought uniformly toqsy0y)s sg;c 4 rest by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelei/7 jo w./bp(iazddt.rates 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 throwh:ego5;s)9dne fas i7 n solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 7s;g)9e :de ih5 sonaf8–45. 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 thin rod,/fe3rxio *8ssfi )a(r as shown in Fig. 8–46. sx *s) fio38raf/eir(
(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 tw2 t:.gq9d203)yd vud hxk fiwco 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 e9;4)yi(/ol ui:aeo kgqev 4hrest within four full turns (revolutions):lu4yghieo9vei a ) 4;oek(q/ and releases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor h:b li7qpup pggu(2. 1tas 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 develops a torquy,-(zw zzb dp2e 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 sli ovr1psnn+1-u pping down a lane atpnn1sovr+1 -u 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic ed9k 0(l0fybn o /lp,gynergy of the Earth with respect to the Sun as the sum of two terl ,pl(d 0 gb9ky0y/nofms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius of 7.50 m. How much net worm+nfhv3l r/p + lx8l8lk is required to accelerate it from restlf8l xv+/3 +hplnm8l r 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 mc.a1gx,mxrh zf,- t( v9p9jieass 1.80 kg starts from rest and rolls without slipping down mtjzx9a9r ghcxf,(,i1p . -ev 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 fall hbp*+ (/k( x.ijiw+ rdyeyk-aand its lower end does not slip. What will be the speed of the upper end of the a k ir(bx-ky+*hpy .e(j+wid/ pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-ka;g9 dzun3rp7 g ball rotating on the end of a thin string in a circle of radius 1 ; 39udpna7grz.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a-6egwa;gek bt/y 5 c9s 2.8-kg uniform cylindrical grinding wheel o5;kye-c ggetbs 9/wa6f radius 18 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a plat7f.s n6dse(d.o xm ghcca*u7+ form that is rotating at a rate of 1.3rev/s If hea 6o( *gcsf cu.dd+.hm s77nex 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 ivyt/x 6,md/l in Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the str/6 m,dyv xit/laight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her sp38z,ubg w2dw bin rotation rate from an initial rate of 1.0 rev every 2.0 s to a final rate of 3 8,gbdw wzbu 32$rev/s$ 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 at3 f/bs gp0j:yxis through 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 b/gs: jp0ft3 y8.0 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 spinnw*e8iw.6 r q9uzy jpg,ing 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 of4eu*6qij t:h ct/g,11gg t fjq the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$

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

A nonrotating cylindrical disk of moment of inertia I is dcc:yui4 -dv npy6j.bb f0 (vvlk**nc-ropped onto an identical disk rotc-vv0:c6ybbki(.* vunj* dy-f c4n plating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at *ers+ajgb-*1 u xupf;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 r/ud/z 3s hrl-cotating merry-go-round platform of radius 3.0 m and momdcl hs/z/u3 -rent 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 velocitlt. b p: ,b 0zoofj7z)54jjd8zncqm1uy of 0.84j1bt,c 8z j)lomnb:. p0oz5 f udj7zq $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, losing about half its jr4+n nd*a12sx-lg7u lv14id t x5ntimass in the procesi5*s4inn -x+ 2xnll4vj7tdu 1t1a gd rs, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve +i2 uvasq 03iao9*fzjwinds in excess of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass 9ocpk (v*.7hv0o aahfrf xs0-$ 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 rot1 t3ogelz3zwe/o61op ate freely without friction. The moment of inertia of the person plus the platfztelg3e3ow 6 1poz/1 oorm 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 ofuj)v3lyfxng7 .6vhah-31pol a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia ofh 6 j7pya lfnlv)voh3u-x3.g1 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 wimj u8d 4oo ft8*fe96j.zp*ckfth the end of the rope lying on the top edge of j meo4u6df*9tzffc*pjk8o.8the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side alwaysu b*)ga281u;gs7ko0 npjel1ct :nnms faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (2t* kgp s bn0 m)sua;l1 e18j7unc:ongIn the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at qqz+unn z(4e-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 radius 0sg-9v j 8j;jjg.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torqu;ggj-v8j sj9 je delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two so3u:d1b xtxylvwq ,4a1lid cylindrical disks, 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 0.010 m. Use conservation of energy to calculate the lxtu134x l,q:a yvw1bdinear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of the rear wheeyj xenk;tq ;mdwgds5.5 b13*u l ($\omega_R$) 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 makj rv-33zra,nw9 dd s1ss 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries of wndrs1 39v-,kd jz3raf no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use energy stored ;tos.ghzopqx ek 2wz15w+ q;-in a heavy ;x5t w2o.ehzg-k1s zo;w+qpqrotating 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 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 illustrai2lluux2f1b+ 22ov dqtes 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 horizolf -ez8 2evgl0ntal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we ignore frictio85a w,co xjk6ln) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontallj ,kx8w 5a6olcy 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 standing vertically on the floor, and we +6bfu12 bydahzg gxff2); td :want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has he+z tdbb f6f2:g1hx)fg2 ;ydau ight h, where h

  A bicyclist traveling with :6f-my 6t errvspeed v=4.2m/s on a flat road is making a turn with a radiusfv66:rerymt - 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 ro+ vvpb8u 2-msr,u aphgda3)j(ck into a sling of length 1.5 m and begins whirling the rock in a nearly hourhv- 3p +2)(agbpjd8u ,mvasrizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as x4hu0bgly /2+cry ,i tgmz7e: thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with lgy2x7rm 0, ceub+z4yith /:goutstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindrical platk j * v r+a4bg9sde;evxgo*s6:es, of6x9 v+*bss doger4g ea :j*kv; 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 rollnz 5akhabzq (/a2gu9wfgvx03w)i5l 8 aaj68 as along the looped 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 h aag5gafwl()zkqaza 3vn9u8 ab0xhi6j28/5w ighest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, bu t)q7;l2ldn2tksbo1kt do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the caxbzu3x gxg4 nd6lh- 2-r reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each ti2lgxb xxd-h u -6zg43nre? $\approx$    $rad/s^2$(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