A bicycle odometer (which measures 6(-qr)mvwrm9 cxl0 cs distance traveled) is attached near the wheel hubrc9 0lrms cx-6v( qmw) and 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 rkq e0a)w+z7e zim 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 cas+e7)ezw 0zaq kes would the magnitude of either component of linear acceleration change?

Could a nonrigid body be desci3jeho 3n;j( iribed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exerj 0- hd*u.b 8ucq*j6iee ukzi/9),vfe piz8omt 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 your hands behinty i)(yb9 3w8* tw* zjcum8vfdd your head than when your arms are strdu*ji(88 yf 3tvycw * 9)wbztmetched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at tvgw m:e.u9zbw+ld m0 3he rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small re+dgb0mw3vlz.wm9 e:u ar 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 lowef:1w f bqji6n6r legs with flesh and muscle concentrated high, close to fw ji6:6b qfn1the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry zk-bpnf7 jq 9c8uvcq8i a il/g 2)kv8+a long, narrow beam?

If the net force on a system is zero, is the;uvy)fq r,sn9 net torque also zero? If the net torque on a sfnyv )9qru;s,ystem is zero, is the net force zero?

Two inclines have the same height but makbh/h e,)64ztlzg p8tp e different angles with the horizontal. The same steel ball is rolled down each incline. On which inczglez th)p6/84tb h,p line will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously 9t5kpj+h rab2jkme 6 nm80a)gstart rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches+t 5n6kmpjk9b rm8ga 2 0)ajeh 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 the same radius and the same mass. They startrir53) 1su )h cpkli6h 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 energk)l3 15rp6ih hsc iru)y at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conservex2:j e g7kg3vqd. Yet most moving or rotx7e2vq: g3kgj ating objects eventually slow down and stop. Explain.

If there were a great migration of people t0uv 5kxsg7 n6howard the Earth’s equator, how would g76xns5uh0vk this affect the length of the day?

Can the diver of Fig. 8–29 do a somersault without having any initiahi/l9qn kjo0x )5vo s6l rotation when she leaves the board?5 v 6iqxoosk9jn/)0lh

The moment of inertia of a rotating solid disk aboutbw/uit:ztlz+y)yba.5v k23e an axis through its center ofat5:b /vk uwiby.y+)tz3lze2 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 a skf0jz.;j 5kstool holding a 2-kg mass in each outstretch. ;kjkas0 z5fjed hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look idetlsxvq8zx -)4 ntical and have the same mass. However, one is hollow and the ots- qx)tvl4xz8 her is solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a hori, 7eq.,zzvdx8jp/cuji ua8 rg (z6uy4 zontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the6zuzv i8pe 8jjduu,ga, xz74(q./ cyr angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freelskw2o 33z.3exa8rqjjp ;cr5x)n6lbcy rotating turntable. What happens if you wall3z8xo k2 jc35)ns.er j r3waqcbxp6;k toward the center?

A shortstop may leap into the air to catch a ball and throw it qus7k +)6y iuy f0v)rwshickly. As he throws the ball, the upper part of his body rotates. If you look )u +ysf)hksywirv607quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservt15u5(siimr e lxt5)r ation of angular momentum, discuss why a helicopter musttlr1x isr( 5i te)u55m have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angleb.dzr ao,t;xa -a7j 6r (lk*79czdck es 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 Exltzshnk1 dt.u6d-ow qf4* 6l-p*wn/lp-5 idarth because of an amazing coincidence. Calculate, using the information inside the Front Cover, the angular dia * pl.k-5qu-noiwh4/*6dstn fl ltd1z x-pd6wmeters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon,zyhbv o4*3l+o 380,000 km from Earth. The beam diverges at an angle*h4 lov 3zboy+ $\theta$ (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 ox paz23l+ p7dal6yc )rpm. When the motor is turned off during operation, the blades slow 7c po dzpxl+y23 a6la)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 th+b5qjm* ra)h8 eidr v71 /jdwo another child. If the ball makes 15.0 revolutions, what isr+b/jrh7i m *ej8h )wv1 dd5qa its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutions do t zay0lgu94xnu.mq : l)k rb19wa9lb q*he wheels mab: k*yb0lurx 4.9wq1 l9lma9zunaq)g ke?    $rev$

  (a) A grinding wheel 0.35 m in4vmv5b t:c8o n diameter rotates at 2500 rpm. Calculate its angular velocity8 tbon5 cvmv:4 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 4.0k* n-r+4qa 4b18bknm- nfmpp kx2:keb/9lwow s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from the cente-92feb4kkx8l anmnmp roqw: / +1nkbk- pw4*br?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the 7sk5us4 uz1.eahes3 h Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a pot*-z+ lf7zudh(j c/-rk .wg xnacfm3 (int
(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 par;r(t k1c+fbtwticle 7.0 cm from the axis of rotation is to experience an accelerationbrcwt (kf+1; t of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel/lm*vd j/cea : accelerates uniformly about its center from 130 rpm to 280 re//l ja*c mvd:pm 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 radius *je+ r)kyq::-x pbt2k4l3 4ycz gcr-bc zt 0pz$R_1$ 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, astrona8,buq 6r-g 5 k xskskfv3b,0xjuts aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they a6kb8gf jrs ux-k5vbx0,,3 ksq ccelerated 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 in 220 s. Throudnayh 6vq9l 6)gh how my69 lqdvhn 6a)any revolutions did it turn in this time?    $rev$

  An automobile engine slows down fro;8:uhaz jjr :vm 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 highspek3o6fe31z() tq dktcx wy p;v3ed jets in a whirling “human centrifuge,” which takes 1.0 mq3wkpvcf xt)y (oz3 ke;6td1 3in 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 chpfu .hfb5th4s6 g1 0uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a poing c 6f5104b.hhpuhtsf t on the edge of the wheel have traveled in this time?    m

  A cooling fan is turn(-* yvrlpss;:7 df, ylj9qkbled off when it is running at 850rev/min It turns 1500 revolutions before v9 yflqj *lls7 ,pyk(r- ;b:sdit comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its speeu+,nokll fhx;x 3(sxt+d qs*j.gx ( f8d uniformly from 95km/h to 45km/h The tires have a diameld3;f( u.ns tlh+f,qx jx x(gks+x*8oter 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 its speed upa:n s1n5u9oi 1d-imcniformly from 95km/h to 45km/hi 59u 1mdcapon-n :is1 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

  A 55-kg person riding a bike puts all her weight on each peh vq*qrfg9 53mdal when climbing a hill. The pedals rotate in hmq q3f*g59rv 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 55yg l5at4o : e9vylw-;e N on the end of a door 74 cm wide. What is the magnitude of the torque if the forc-l elg y :ta9oy5w4e;ve 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 torqq;25 0ssnz: ;dap1xqi6igr tbue about the axle of the wheel shown in Fig. 8–39. Assume that a fq d16 ; sbn;tsz0xig2 :ri5paqriction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass a.hjvnj+.d 8)anh ytm5e: :0pek- cclm, are attached to the ends of a massless rod which pivot.nykeejdv: 8ch ):cmnp.taj+ a5 l- h0s as shown in Fig. 8–40. 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 q1djqbs2p . 4rto a torque of 38qrsd.b14 j qp2 $m \cdot N$ 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 sphere4/liwekp *b8t*nsyi) 6 kvgo2 of radius 0.648 m when the axis of rotation is through its 82*ke/wy4psig ov k) *ti6lnbcenter.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in l -daft4q d*s.f0u 8bidiameter. The rim and tire have a combined mass of 1.25 kg. The ma4- si80 dfd*talbqf u.ss of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a4 kqa luqo22 *3tzdyv, thin, light rod is rotated in a horizontal circle of radius 3 qt4o*ayldz2,qu2kv 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating at constant k-sxh ewjw2+6uc/f;bs:gu g)angular speed (Fig. 8–42). The friction force between her hands and the cu2k h6+s)xgej u;/wfw: c- bsglay 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 9go kpyl;ykozo ye n0p :661+gpoint objects shown in Fig. 8–43 ap;oo:6ezy 1 kyyon+kpg0 l6g 9bout
(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 consmi):ke/*ms-v33v3i xoh0 o; guvj kc6c*ma u pists of 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, uniformq0a(m9:,.qa m roq ahdvej7e7 cylindrical satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44jm7erm qae( o,a .vdq07 :h9aq. 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 with- jd2-lalcm 7n a radius of 8.50 cm and a mass of 0.580 kg. Caalj -7d lmc-2nlculate
(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, accelerao rk5xr trz2,w9e3zk2 ting 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 gf(0 *xk4fu z xhdp.;hhand-driven merry-go-round and is able to accelerate it from rest to a zpx; uf(hgd*x4 .khf0 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 10,300j;i2ixx8xhx047c d1uk7pw r n rpm is shut off and is eventually brought uniformly to rest bnwxhuxrc71j4d2 x7i;0 i8k pxy 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 accelerqd 5hy;wvm( 9v +fze+wates 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 forearmw(ynqt -(:3 qucykk. v, which rotates about the elbow joint under n ( q(vwy-y.ck3:quktthe 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 8ha5heeg*wh + 01c lym)*o lkwconsidered a long thin rod, as shown in Fig. 8–46ywel *m*ka5h1+ lhoch 80g)we.
(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 tuzai/a)5r i)3 ix(h1lhrp/w5gq m +htwo 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 within four full turn bvfv6fi5:/vo sficj(7z q 3)ba94: esqdel+rs (revolutionss+v4j iq76zsi(3 ) /fbavqvffr ebdc 9:5:elo) 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 has a moment of in,b+ fh6rlj d+eertia 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 tv ( tu; n6iik2msu)6hmorque 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 ajpss cd7wtn .0(c-5nb0ya .z j lane yc 0zj .7wcjpnast5ns0d-. (b at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energyga5a y jxdk3;24 7lyt5itzu,c of the Earth with respect to the Sun as the sum of two terc a5jt ga 34 idkyuly,27xt5z;ms,
(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 164r/ jz;5o cy uhzzl0o80v r:jz80 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate ofoz j8ror ;y0u0j:8hlz/zc5z v 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mas .bo:3yv:xh;g kb wsm6s 1.80 kg starts from rest and rolls without slipping down a 3xso ;kbmb::6 yhvw.g30.0 $^{\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 t vvfz1*m7 )zqhnfq(p6 o fall and its lower end does not slip. What will be the speed off6pz)qz1m7h v*v q(fn 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 the end of a thin fjmq5 6 aih*tw5r94tn string in a circle of radius 1.10 m at an angular speed of afm 5t nrqt6h*j5iw4 910.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grinding whe urwf9+ l(bb9x6 qcnf7el of radius l (bunfc 7xb+w9rf96q18 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotating at a raty)e f 1ar+h8,m x 2utinri6q(lrj qc07e of 1.3rev/s If he raises his armr)i2fejru x 6al8n+7hq r(cqy1,im 0ts 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 reduce her momeri9 -b q q4iwbo.t9og*nt of inertia by a factor of about 3.5 when changing from the straight position to the tu q*o4-q .9wt ib9rgoibck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial rate of 1.oco asp)r,k5dxr1q 8rw 9d9i 50 rev every 2.0 s to a final rasr1o8 cdka5pq,dowr9 r5 xi 9)te 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 its center at aqvz;(tzc tw5w7t,v sy,q6(mr qh21 sn frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. Wtqz t 1s,rqsw2 wyn7v mht,(5;(zcv6qhat is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figtt*;lmm4)1w hayw6gvbdd ) 3gure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of the Eartqi9.fo55qwh a h
(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 :se; )rw tvo6k8l.f1 e)h.ujjntas9 s I is dropped onto an identical disk rotatint1: .9eej o8 aw)fjul s6rhssv;.t) kng at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4m67ibf 2ton c-g:xs +sn, kwyqj (y13e $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotating merry-go-round platfoccgr5mgk ( p6z .z 26i*dezjo(rm of radius 3.0 m anzzep.6 k5j6gorci (c( *g z2dmd 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 rotawnsrw6q46j z)ting freely with an angular velocity of 0.8 zs6 wr 4q)nj6w$rad/s$ 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 dwa-x:d 6 ns0d uiri0obf(z in;.grf, 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 no angular momentum, wha;i r(.n006ibnu sdfgoi- :x dzt 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 igw7i/i y :wfrg9y/u5xl n(v+f n 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 en :p;j6 )yebqpg-e9 ,r:hqen$ 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 whl4w 9n(+rhkve1n+n freely without frnn+e1rk v+w l4w9hh(niction. The moment of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m deb7nvbxv 69f; iameter merry-go-round turntable that is mounted on frictionless bearings and hvb ;6bnevx7 f9as 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 tht*o)mw cg +7zve ground 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, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. c*+gw7 otzvm)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 always faces the Earth. Determu(auw tiub 8(9ine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentuauwui(8t9 ub (m. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at r dda0lz5wzzpd uiivf,8c 1 r7 q92l::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 0.20jccs6ain 9h x n,:qgf/n55*pp m acquires a rotational rate of from rest over a 6/ 5 hngi59nspnfjc,*qx:cpa 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050d:7kmtl pw 2)s:kqca ;u 4:dql 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 pcw)u;sdatk q: 7l:l4mq2kd :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 ofz1g5jhd;b m u5u..vz c 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 ofgxk-iwp-7 py h i4)0,vldd6vb our 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 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 ivbi 7k0-lw6i4v -x )hypd,gdps 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 stored2qaf q,dp36x bog2p m8 in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able toxq, dmaq2p8p2gfb6 3o travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrat drm8s/ )di*f,sdz7br es 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 horiz .b6a mt9z jl9p9bmwlunlq0t203svn1ontal 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 frivo/.az s,kp8sqqn61 tction) 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. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hin p8t.s vo,skn/q1q6zat: See Fig. 8–21g.]

A wheel of mass M has ratvpe xpvx(4 5r +y:bv.dius R. It is standing vertically on the floor, and we want to exert a horizontal force F at its axle so that ir4.px :5vpvyxvb+te(t will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling wa569,fifm iw0y9yw* ixvv+aeith speed v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist v9aem6 *, +fifa w x5yyivi0w9and 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 gqo:xumvn t2,wu9l:(0-kg rock into a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerati 92wu:xuql ,og: v(mtnng 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 thin rods, mak8.p2xby g.c(ff;7 gisocu.1 b)pb7na pwdw: wing reasonable estima7(wp827duw.icp.bg y;).ob pacbx w:sf g 1 nftes for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch atigv9h;j 2juk 2,9nsessembly which consists of two cylindrical plates, of masshs2jkvgut ,9n ej; 29i $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 loope21f3 vmpjn kf-d 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 poj3p-mn1f 2v kfint of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but dotjzu 6l0; n)hh not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions a n/*) l7eojpuvs the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular accel 7o/)*j plneuveration 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