A bicycle odometer (which measures distance traveled) is attached near t,rgm wj1230w5os wuto-;a zgihe wheel 01jwg ogatwrw-;uz 25 s,3mo ihub 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 pdoks5lz5 9(8 iv; xocvelocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increx8o 9 5kl(zop5s;ci vdases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of tm.lr(fguw2xp44r 0w ghe angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque tf-u1bq4cg: ph- s gpx,han 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 behind your head t pmg+q;( apnej2n3r5dhan when your arms are stretched out in frmd2pe (n5 +na3gp;jqront of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and tc2c wvfl,tf,jb:zt5 8hree at the pedal cranks. In which gearfj ,:fltbtwvz2c85, c is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being aby-r5 u+*lqzi dle to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribuyl-+q*5udr zition of mass is advantageous.

Why do tightrope walkers qx-fe z1 my o(6;gtalhr r5:/lr*f2sdlvqs/)(Fig. 8–35) carry a long, narrow beam?

If the net force on a.qeguspzm,i4 2;ekfep ;. 08qd ivql* system is zero, is the net torque also zero? If the net to pl uqk gveif8*20sdmz e;eqq;.p,i.4rque on a system is zero, is the net force zero?

Two inclines have the same height but make different angley-wddix r1,g 1s with the horizontal. The same steel ball is rolled down e y,x-1dgiw r1dach incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an incline. One )7vv5 vjkbnc 1pg8x.wm7i j8 xsphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the botvj1j xbg v577 .ivmxk c8wpn8)tom?

A sphere and a cylinder have the same radiu(.o:l -+vr8)o8jo1qqa xyezah, mhims and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the gm.mq yzi-vox(hoa:o881r a ) jq+ l,ehreater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular mo+bt jgm39skn0lb6w8 x mentum are conserved. Yet most moving or rotating objects eventually slow down a+38mxts gblw k9 bjn06nd stop. Explain.

If there were a great migration of people toward the Earth’5z,q3u mgco61 x jk8wcs equator, how would this affect the length of toczjw1gc5 um8 6 k,qx3he day?

Can the diver of Fig. 8–29 do a somersault without having any initial r9uluhuj z i9o 8qb5lm 5j)a4o6otation when she leaves the boardluuhzbl59m 845j q)ui 6ajo o9?

The moment of inertia of a rotating solid disk about an axis through itspk:1e6q vd)+z k5vl kj6k blwl5(x;s v center ofv 56 )pw6l1skevqk5 :vz+k bxjld(lk; 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 rotvz3wq5f n68yn mzv6;qating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, w 8m;5zfw 63vyzqnnv6q ill your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the)3lx2r+ rk y0qbl 8z0tjao o/v same mass. However, one is hollow and the other is solid. Describe an experiment to determine which2 3+ xla0tkj/)olq0bvryzor8 is which.

The angular velocity of a wheel rotating on a horizontal abu6mn, +u -dl)*buw jaxle points west. In what dirulu+ jd,bbu)a n-w6m* ection 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 the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turntable. Wheov-.*v qw7a qat havwqea-q 7.v*o ppens if you walk toward the center?

A shortstop may leap into the air to)rlft6w ei2. q catch a ball and throw it quickly. As he throws the ba i62. w)efqrltll, 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, discuss whyc9.e tle ,d-ex a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propellere.c- e,e t9dlx can operate to keep the helicopter stable.

  Express the following angles in r,igt,a3 gi :mkba*3ty b2l)hbyuxa*0 adians: (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 Eart/i 3rb0 mvfn3g l3f;xjh because of an amazing coincidence. Calculate, using the information inside thl j;3ff /g n3xivm3br0e 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. Th(u3kz thfow. c)bil8rsxb4 35e beam diverges at an angu3( l)bbfk.c5 o z348ti shxwrle $\theta$ (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 awq1y .o *k:k4rzf+f in rate of 6500 rpm. When the motor is turned off during operation, the blades1f.zir*n+k: fo k yw4q slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to e7(vkeb-/6n9mlzy 2 qx hd d)xanother child. If the ball makes 15.0ebv dkl(7n9ym x6h/ dq2 -ezx) revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How i:vpk 6: :srqwzp68yqmany revolutions do the wheels ma:: zv8wy6skr6ipq p: qke?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Cabn r7/+ xpbq:vlculate its angular :+ vpb/bxq nr7velocity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.0 s jcvcnw2j ) iu,cs7 *t-(Fig. 8–38). (a) What is the linear speed of a vjw cuts7*i), c-j2nc 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 u r/a8zi)z6ipthe 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 e(valktle 4n y c9 b/su+w802xof a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0 cm from / johv.k zeiq(w 3w ;hm,a9ct2o99snj the axis of rotja(kh3snjq .m 2h9we v9;ctzwo/ ,9 ioation is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates9dpqz 8u o;z*7. kg5x ohp.jsf uniformly about its center from 130 rpm to 280 rpm in 4.zo; f.gsupz*pd9.o87x5h q kj0 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 nuu q+ 7e t+ wta;twj+q0(ja2w$R_1$ 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 xi/3xinv.pv xc8 t38gd c*tr,spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.*nx ir8g8 .ttvvicxdx3/3cp,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 accelerat 9/rfbatg1v o9,8 l9 ypwv3yhoes uniformly from rest to 15,000 rpm in 220 s. Through how many revolutions did it turn in this timy t9h8 3oy1lrpwv ,f9ov/ gba9e?    $rev$

  An automobile engine slows down from 4500; d khfyykzp8jt 72qqac5005 r rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed jets ta t )c6zh)io /3:udux6 xfg+tin a whirling “human centrifuge, it3)f hd)u:+oa xgxczt 6t6/u” which takes 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 uniformly from 240 rpm to 360 rpm in 6.5 u;d:,r0uf nds6 1cr kcs. How far will a point on the edge of the wheel have trave nu;fc 6skc1 dru,0rd:led in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 1500 6go 2fju:h -jdrevolutions before it comes to a stop2d-h6o gfujj:.
(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 revpdbd- h9j,58 :hcfdbm olutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diame5c-f,bd9 mp :hd dhj8bter 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 unifos5cqa f 7l07t/;sv ngxrmly from 95km/h to 45km/h The tires have a diameter 7n g;tvc x0aq fs/5s7lof 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 pum2pttwyh8 -c(ts all her weight on each pedal when climbing a hill. The pedals (yw2m -tt h8pcrotate in 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 casu8l8nj 1s 8N on the end of a door 74 cm wide. What is the magnitude of the torque if8s s8lj1an8uc 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 th8p4ran6hsa v ;jp*9xbe axle of the wheel shown in Fig. 8–39. Assume that a f 4 p*p89v s6xhba;arjnriction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m,i (1pz7m3nahb p7 k:j try 0mfpb3ds18jx -m0q are attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizok031m3p aymbx7qi8 pt (n j brj0-s1:mfzdh7pntal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening to a torque o+) ,k8;rbx rj(h v pct)ebeema95( b xtev6e0cf 38+vb9 mj xbp c5(tr eh,(eerx;c0)6e ka8t) bev $m \cdot N$ 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 bdivl7iatfj r v: dn)78:tx)9of a 10.8-kg sphere of radius 0.648 m when the axis of rotation ) v7:)7ii8td9ljvdbxt : afnris through its center.    $kg \cdot m^2$

  Calculate the moment of inertia: h,ecw :v :fiu/s1 ovuv,m;we bal6-j of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hubv,o:bus- 1ul:w/vhw,f;cea:iv j6 em can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on fm8 ey3ig .vg*the end of a thin, light rod is rotated in a horizontal circle of r y.ig*3v8e mfgadius 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 xwq,xp) e :/kton a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her ha:)t wep,kqx /xnds 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 azh*6.5ye bw u)9ab kye5 w)cwinertia of the array of point objects shown in Fig. 8–43 ae9b z)6 5 .ea)wy*ubywchwk5about
(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 9ta e 3xcoci j/-p*hopepc,9: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, uniform cylindrical satellite spinning at the correcti4/l qs 0ibd8z90exgw 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 the requiredwl0eb04qsd zi 8i 9g/x 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 cylinde6xff su ;f8w.nmn 6+mjr with a radius of 8.50 cm and a mass of 0.s6xjmn. f wfn6mfu +;8580 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, acce .s9 *z g3pfyju f1xpjj)y7fo7lerating 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 smallfs,4 2lz)5fv vkt 9itk 6cd/vc hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm iikks6 dv)9 zt4t 52fcvcflv, /n 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is eventual6bha lbjx,a3/ ly brought uniformly to rest by/x ,b3ba 6hajl 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 acceleratescfe / rqah6 ra42ru 2l-/9pxcq 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 y)wy38vjjbd+fz;6a at /mko3 jrou2. ball is thrown solely by the action of the forearm, which rotates about the elbow joint under the action of the8aywa/kdm r3oz3j 2)yfjjt6o +.vu; b 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 considvu6 a2 g/0puxt9oh;pyered a long thin rod, as shown in Fig. 8–4 gx2yu;hupvo6a / 0tp96.
(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 p2n*6 4ymeeqkt3,caxof 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 azc9 wehaz6-f*oo 4a 8within four full turns (revolutions) and releases it at a speed of 26 4h-ca owea fzoz89*a8 $m/s$ 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 hasd(h0m /lpa ;;y fvax;(0pupxn90z fk5e ij4tb 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 enginerc kv1g4 d.nz8 develops a torque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls without slipping dogptc6jea j a5hqf3qj op*67(5wn a lap(7ceo5 fq j jg6p5j* h36aatqne at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth wimdvyzfu 11 g.5w(ch o7th respect to the Sun as the sum of two ter5w(7ch1muzg d1fov .yms,
(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. How13a mq.xwgzdotuzlt ip,76*c c0 .9yx much net work is r.a c0ldp t1 *3uq,omi.6 z7z9yw xtcxgequired 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.,+txsr 6hp*z ix u/,dw80 kg starts from rest and rolls without sliptxs +p h6*r,w/zux,id ping 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 its tip. It sta1thr(fjj .2 -snsz r5frts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the s1 snrj5.f2t( r -hjzfground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end of a tkub0/ ;3 y:mqy/l sqcahin string in a circle of radius 1.10lc 3/ qk:baqs/yym0;u 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 grinding whe37h ick7aj2;( eebkueo,b,oa el of radius 18 cm when rotatihj i;,beoka(7e k 2 o3eu7,abcng 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 pnnt h 6z62,alat his side, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fh lnz2nt 6,p6aig. 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. 9mkx j613m ann.6artp -n0oxu5bmc0u8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the03cm jx knaa69unur6pmx-t.5 b1o0m n straight 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 spr; vmo svklng):tj;( *in rotation rate from an initial rate of 1.0 rev every 2.0 s t;r:sklo)v;m tngjv *(o a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis through its cfix1bvu 26 0qpenter at a frequency of 1.506xviqbp 2u 1frev/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. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure syv9(lrcvy nh 9 4hqk0/hz05tsrfo+ m1kater 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 momb 2s6qno:dj:-fd f c)u4m3mrbentum of 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 dropped c r tw7f 7cakr;6j))c upx5-fqonto an identical disk rotating at angut)-5jrc7 wufafxc;7c6)r qpklar speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns adt 2 ( suuemo.,ombg61t 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 afv-wkx01 2ck; hqe wdj a8i;:dt the center of a rotating merry-go-round platform of radius 3.0 m and:-q02ia;ek 1 v w8cd;jwxkfhd 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-gylwt cqrx)9f l)1 :ry;;nq n(jo-round is rotating freely with an angular velocity of 0.8:()t yqw9rc ;nf;1jx ylqr)nl $rad/s$ 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, loswpw0/-ljh s16kpz4 wqu)9oiming about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming4q09-hwk jml s1 p zpiwu)/6ow 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 enl c/sp cxja36bc34 5sxcess 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 d o1z - rqy0,k0mc.xmkd9 u+jr$ 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 platformpbaa-u3u* f; x, initially at rest, that can rotate freely without fra*3abu-f p ;uxiction. 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 persp4nc2)v lxd4 5k :kmalon stands at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia2mdlv 45n4:xk)k cl ap 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 2(.* iese3fzjsq sr/vrope 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. What length of rope unwinds from the spool? How far does the spool’s center of mass movezfs/ s2er(vj*s3q.ie ?

  The Moon orbits the Earth such that the same side always faces the Earth. Det8 f tw:1m,4 de5urqq irn35fh eydn87cermine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treae 58m1e 8c3urd ynwqh, ff7:4qrd5nitt the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rt5/e9 :md zecfql7* epest at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniformc6o iyme4xk.ef3 j2y. cylinder of radius 0.20 m acquires a rotati3mo y4i.y.f6keej xc2onal rate of from rest over 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 disks, each of mass 0.050 kg and2hdf -.ns sf, 6zija;j diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use cons adh.s n2-fzi6s,j;fj ervation 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 the rear wheel r.de2z86qlkz fw e2kx9;iy1 h($\omega_R$) 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 timehg(b 06npx .vw4auhe ,s 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 thatp guha6(04w nv.xeb,h the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for icz -b rgo1bc;*kp,2 nit to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg,,12n*pb- iog; ckbzr c 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 illustraxrq- 7:g9f/+v;xm lmpu5z ,rrxwa .p gtes 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 horizontal surfj:/zleu2 c9/hu f09fl j/ xuig4b lcp7ace 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 fric :(p7r*7dji1xadvbp mtion) 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, aapipv7 bm*:rx17 ( ddjnd (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 thld,rcvx adud8j 26,h8e floor, and we wan,6a v 8dc 2lrjddxh8,ut 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 height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat /u1a8 f wde2prroad is making a turn with a radius The forces acting on d2pr 8aufw1e/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 rock into a sling of length 1.5 m and be/ 7i2y5cm, lwtx.b waggins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. Wha7bw.cy,i/a t lm gxw25t 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 rodssjq:tti- i4ui )k8e:a, making reasonable estimates for sa:4ti qiuti-) e8:kj 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 assembly which consists of two cylindip(/m h7: qgo4j5 xodmrical plates, of mass q dmo: hp5(/ijox74m g$M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 alozg6 ku-x8g- 7bh9 s 1loo2pcaing 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 without leaving theg- - bl81 o9xs7h2cik uzpoga6 track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not ask55fjri,gjpv0x6d h)sume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed 0a )fg7udamk9cdu .h*uniformly from 90km/h to 60km/h The tires have agaf* umd ha)0c9u.7dk 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