A bicycle odometer (which measures distance travel. tywyw c0-*4 xggvmr;ed) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch whm4x0- ytycgwwv.g*; reels?

Suppose a disk rotates at constant angular velocity. Do81n:bqgj k f: 8xsj1kzes a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial8 jj:81 :f qb1nkksxzg and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described wl+ : ; -vxjjxw:n5k4yxew rjmclj/ 92by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exeri+ o,p:k ,zj by0qsm*it 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 ye pq /ws9hrxurtvr8lx.s. .1 .our hands behind your head than when your arms are stretched out in front of you? A diaxlh8rpw rq...t9 xurs.se /1 vgram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rearitik/7q 2mvf2ob .( r,dslfy, wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket?ts2,f2f ,dq(ii ym/br7ol .kv Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being ab.ooo6 yv6n- wi;unx29*0ksc0ain az bgyq6 *xle to run fast have slender lower legs with flesh and muscle concentrated* q692 wi avo u0ny-xo 6kcnx; .ynozi6b*gas0 high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–cmqb5 k (a6jl 34y u1cxnt: mdq8lkq1435) carry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? If t 9*l5/mnrv (i17vhyy m:ccslx he net torque on a system*(r1 yx/l ym9sc5cm7lnvi h: v is zero, is the net force zero?

Two inclines have the same height but make dif.ie87cyc ch97q;s wu dferent angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of tu7csy8q7 ;.c 9w ecdihhe ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an incline35v bgif1/a oc. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom ofoc1i3gfvb / a5 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 fs. 7haeug7bf:993 l1 lyrgzjmass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?bg9zrlfeljs779g 13 ua :h yf.

We claim that momentum and angular momentum76dd(0 dacwd vnfvn30 4xo+ro are conserved. Yet most moving or rotating objects eventuall ad6 v74nd0(r do +3xnvcf0wody slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equator, how wou97okd 6.zs sk 8jmv;jr ;auq/sld this v . ;uqz9sd78kar/6 sm; joskjaffect the length of the day?

Can the diver of Fig. 8–29 do a somersault without having ib5z 9 nxsccgk+1 zi4svw1j8:any initial rotation wvzzwj:s kixcn15 8+gcb91 4sihen she leaves the board?

The moment of inertia of a rotating solid disk about an axis through i kqrb-x0ixrdc0wem3; o91 1 mrts center of10bx-d; 9ex30wirr km corqm1 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 rota:-nm.9cjzb s;rzh yp4a( vpd )ting stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, ohzs-p n9mcy)j .p(bz:ad;vr4r stay the same? Explain.

Two spheres look identical and have the same mass. Howes0(b;nes h jegu) 1dy ;elc+)hver, one is hollow and the other is solid. Descr)e 0 (dsh lhc;bs)+enu ;yg1ejibe an experiment to determine which is which.

The angular velocity of a wheel rotating on a)7f z0ij/y may 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 eam fz iy7)y0j/ast, 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.i lp+fvzuf mbyi )7:-0 What happens if you walk toward the centermf0bfi)v-u+ :yp7ilz ?

A shortstop may leap into the air to catch a ball and throwe3w )k yy..jp fw)d5oj it quickly. As he throw)o5w fpdjy ..jew) 3yks 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). Explain.

On the basis of the law of conservation of angular momentum, du wq:o6 ybog7k,k,u+36e+vq s yg gsw8iscuss why a helicopter must have more than one rotor (or propeller). D6 :,gy+ kv g7 oy3wk6,u8gwsqoube q+siscuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radiz8d,lbil6 bb*ans: (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. Calculasx-42b wd hx,xte, using the informa-2s bd4xxhx w,tion inside the Front 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 from Earth. ile -6t iw5j8vi 1h;gyThe beam diverges at ah ej;t i1ivyl5w-6g8i n 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 at a rate of 6500 rpm. 3p-gh2c f j51hcnfvz 7ct1*ndWhen the motor is turned off during operation, the blades 1j -n21c tn7 dhp5z *ffvgc3hcslow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the bahl899toox ( zsq(b l3ull makes 15.0 revolutions, what isl8 zl 3(o9xb9otq(u sh its diameter?    m

  A bicycle with tires 68mpc,xeb.i b6ha vm5* fpx4 3 6v1o4mic cm in diameter travels 8.0 km. How many revolutions do the om i 16i.vac 643c5,b*xfpv mbpm4eh xwheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. v4d9ej -a,anqt5;enp5 o,m fcCalculate its angularn5m-o qat n5edve4ajc ;p ,9f, 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 compjbmrc+s io w+8r uuxmg8p1). y h -)e+/j0qxyblete revolution in 4.0 s (Fig. 8–38). (a) Whaxoj)+xg ueyb8c 8u-qi)sr 1+m r.mhy0w +jb/pt is the linear speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbit arou e8qhp, up32e ckk+*cfuteg9 8nd the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of af6cpy 1n sjk)7 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 vayq/e:a3 k78;rzen rfrom the axis of rotation is to experience an acceleratir: eaq/v 7yzna8 3kr;eon of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its )ta4d h9gbq:+cp- eux center from 130 rpm to 28g9px-eud:cb4 +)thqa0 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 radiu1l xyp . 1owmo-sb(vg9s $R_1$ 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 themsvq2 omm9/q bbkcgc( r3d pu.7;elves into a slow rotation to distribute the Sun’mup m.2kgbv/r b(q cqc;37od9 s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a 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 u+qsy-axlz /2y 6n kh.nrolp(:iao2 0. Through how many revolutions d n yzl2.+(-xra02/psaok6ni: o yhuql id it turn in this time?    $rev$

  An automobile engine slows down f m/tjowdh9 timmw3 . 6f936vgsrom 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 t,fmdo.r4 u+gax- d c;xhe stresses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its ;gac+ xrfmod.,-u4 xdfinal 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 uniformly from 240 rpm to 360 rpm in 6.5c zm0,bno7w a9ww:13cx 9t0 cr6inh9y our kh( s. How far 0zc y 3 bwxnhwri97,1 w:6couhc t99aon0r k(mwill a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 85*- uq i1yj.. 4e ixy,ek4ioiob0rev/min It turns 1500 revolutions before it comes to a s4 i.*-ikqb. ,oy uxeioi 1ye4jtop.
(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+b gs khlr2(:r reduces its speed uniformly from 95km/h to 45km/h Thrh r+:g2 slbk(e 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 rede v7icy./6 lxt/sg)0xur5k9vp;d zuv uces its speed uniformly fr /cu;z9 /rpdg )xv6ltik0v. usv75xe yom 95km/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

  A 55-kg person riding a bike puts all her hk7iz nukn( zls,16kqb*6, cuweight on each pedal when climbing a hill. The pedals rotate in a circle of radius 17 ch, k(c6nk1zskbni, q*6zu7u lm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of v2j2r* k /qqhqv8-b ff55 N on the end of a door 74 cm wide. What is the magnitudf2 2vf k-j/hbqr8 qv*qe 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 about the axle of thed * (quj+ *2(,7zmgprpf/gne.ulkyok wheel shown in Fig. 8–39. Assume that a friction torqufk(qru mo/gp (,uk*dj7z .p e+ng2*l ye of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, alv- p1oq*up q/ rukw+7re attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initiall /-qowq u*+1vupk r7lpy the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening to a to m8/ -h6 soizdw8kza)urque of 38 d z /)khwuiz 68-a8oms$m \cdot N$ 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 ob/+5v r5/ rdkfpvah)w f a 10.8-kg sphere of radius 0.648 m when the axis of rotatfrdh )5b+ wv5ar/ /pkvion is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. The rbxeh ovpag h/:8 *h.:i7rcyp5im and tire have a combined mass of 1h.y5re:vca7 gx8pp /bhoh:* i.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, lig a,bnvhw4od/ a1m;8u -r+wz utht rod is rotated in a horizontal circle of radius 1.2 m. Calcu-mwa/ 4ozt;ur wh8 +vd,abnu1late
(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 angular speel1/cg f2pdmi*.cxu6x6, jn pod (Fig. 8–42). Tp1j.x/662 nu lomic *gd cxp,fhe 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 iedd )0r68wwelnertia of the array of point objects shown in Fig. 8–43d 68d)rel0ww e about
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen atoms whose t23gu-j bm -;w7gn ssocotal 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 sz vzr*r ra+-iq-rg 8i,pinning at the correct rate, engineers fire four tangenti qr+ r-8a-vrzgiz *ir,al rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each 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 a ra5w(2r hf9gs7u((gvbx lj chg4dius of 8.50 cm and a mass of 0.580 kg.vrcu j x 2hsh7wl(fb g45g9g(( 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, accelert):glh 1cx w-yating 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 hand-driven merryfhw5g9.oa hha13na tss5 o(e, *4unvt-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 a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, uesho3(tvh*aa5 ao451s fn g.9,ntwh neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is ersnk02pz8 7yl ngnd15ap)2l gc9g rio,4k so1ventually brought uniformly to rest by a frictional t9 ,go snglrk p54gz71p0r2c1y8 s) in2ankodl orque 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 ac 5xnjz q94pq0tvo,;tm+(7 sfeb dmd8x celerates 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 6 ifd5pubw 7uvu*hv54thrown solely by the action of the forearm, which ro 575uh v6dw f4pb*uuvitates 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 thionj80e5ds 6j hn rod, as shown in Fig. 8–45n0o 6s8djhej6.
(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 ol vg(y -j,j0gwh4hw x p x6teo13q2r9consists 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 within four full turns (revo:ebn)ndy 4ge9lutions) and releases ee4ny)n 9g b:dit 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 b47sc0te zrp0inertia 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 torjdb4 ofg6um1/p u.x3cque 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 dqr9j .bl6:egg2dqh7 down a lanq7el b h.2d9gq 6rg:jde at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as t+a1ibm p,: nbkd:j )gohe sum of tm :pa)+jndgob:,ib k1 wo 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 kg and a radius of 7.50 m. How much net 1:r e93n7 d.xmia,b oa0vhuze work is requiredo:u iaemz 97hb.r3dxvne10 a, 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. .lf7;sziy 5wo80 kg starts from rest and rolls without slippingw.l s5z7fy io; 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 i 0o r)yk*33u-th 7ntbuqff-dbts tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole justtb)u0qyh- 7f-t bud k 3ronf3* 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 rnu w umt/.:h vs(r ):0yu)fx6s5lmaw string in a cir.)yvnruuu: xlmrmha fss )wt /5 w0(6:cle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical gr-fv)o(vnoyc ( inding wheel of radius 18 cm when ry)vofcv (n o-(otating 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, hand f3.tf(yxmyl;s at his side, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, theyt( ymfl;.x f3 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 reducdb2lrgoog,)w+ 75rkbe her moment of inertia by a factor of about 3.5 when changing from the str ,go2b7)w rold+ grkb5aight 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 herzklt:xkh0 o .,(xf bs2 spin rotation rate from an initial rate of 1.0 rev every k,2:lxkfs b0zh(o xt.2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis t g - 9l3fx3,9tamtf,xi 2/uw9zef hanfhrough its center at a frequency of 1.5rev/s The wheel can t9ftm,fa3-x/a 39,iglfz e whfx9u n2be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular mome+pe*m sa x52dintum of a figure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of thybq;1qu /nucm 3 t:i6ke 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 momt 2w6+(wnlp yn6p3w6 dl ihwa4ent of inertia I is dropped onto an identici 36hwnlw d+yp6 w2ta6n4w( plal disk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4 j2eimdd0 h15k7 d/nhi$rev/s$ 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-ro jv)e35wz zx dc ogq11:xox(2wund platform of radius 3.0 m and moment of in:o2o3excxwz jdzvg q)1x(1w 5ertia 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 vec pm6m61xi6 qylocity of 0.8pmqi61 xyc6m 6 $rad/s$ 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 event3t7u*2cfcz ptghs 6 /pually collapses into a white dwarf, losing about half its mass in the proc*s3pzt/ghp ufc6c 7t2 ess, 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 winds in 8ybu:vsj( p,8q tlvm9excess 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 oe ka.nmy q;a2,1m)hk$ 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 at6ik ke0 9gxyq5m3f(p j rest, that can rotate freely without friction. The moment of inertia of the pers3 k ep9(m6f5xigyj kq0on 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 ygau.+0dte qaht ex5o2(w; x+at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has0dhetx.x+gw(e;ou y 2q+aat5 a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of the rope lyingph)ahgk jx m2 +*.4eum 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 behiam 42ej puk g*.xm+hh)nd 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 always faces the Earpd2+,a3c)iy:r*vvq ,n 7xs kdyg2 ) zfzdt3kyth. Determine the ratio of the Moon’s spin angular momen2s* y+t yr zi: 3dgyc)kqx)3dpf,d,n2av7k zvtum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate (+r 2ywk tymc)g9b7hbof 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 unr)6jykl b3 od9iform cylinder of radius 0.20 m acquires a rotational rate of from rest over3y lkr9ojd )6b a 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 dis d lfy-b4w 3+b qe gkk4drw:co.*pl-1zks, each of mass 0.050 kg and diameter 0.075 m, joined by a (conce--*r31dg w4df qy+e :pbb. ko lwzl4kcntric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of g dnhwq2s;/8+mz)i vgthe 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 grv bln-:aaag*bc e)9 3veat, 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 i-l a b*vcaeg:n)93bv ats mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use o97wu hydsmy/+ 8nm.m energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be as+ymnuwh.o7 9 8ymmd/ble 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 -o3gt*p .7fhslbb8 qpan $H_2O$ 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 p:q e4kq,jz*b-h t3 pvhorizontal 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 lengts1al xarw s(8/1 / ccj+kbiw,xh L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity actsxbc+ wxw8a//kss(a ji, 1l1c r 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, yv., vm f2af66z:xcbmasm0d 1and we want to exert a horizontal m0c 6 mfz.a sm2xa1v:vd6fy,b force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4m)8e 4j 90a0/ tb othvbyshe+1r;q xcy.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist and cycle are the normal for0e)yys;bc er8o4mtx1qbjthh+/ 0 9 avce $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ 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 rock8z)byj - j+y32iqrmho 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 rpmzby8)qr +hm3 2-joiy j 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 skate-5,lr:k*oltivpeyxg * j( ,ovr’s body as a solid cylinder and her arms as thin rods, makinltpi *oy: (j5*v rve,-kx,lgog reasonable estimates 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 cltnpqhy1rn-31d;4j0gw3*f y yq8yug5 0ubl rm utch assembly which consists of two cylindrical plates, of massuljtfpb5wm qgn30yg 13*n r1-ryu 4 q yhy80;d $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 rzf uad)m5) h 9ea4chm*9ldi5g rolls 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 highest point of the loop withoutgu5)mz d l9h 4im5fc*dhaea)9 leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do notd pq--1 zvzc4k assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the ca2m6gpy;gv1a p qoms0 4r reduces its speed uniformly from 90km/h to 60km/h The tires have aom;y1qgmap2g04v sp6 diameter of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s