A bicycle odometer (which measures distance traveled) is attached near the y+n,rx) 6 ycazwheel hubr x) z+ay6nc,y and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotatewg/ 9n-tyxv euq0 v;o2kw 2i7 vy:md*ys at constant angular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radi*ked v27xt-;mvynuw:y wq yvo/ g i092al 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 the angular vel9 2 p f(*-gxeb.xitkuiocity $\omega$ Explain.

Can a small force ever exert a greater torque t ( b-jjai++ u(;sdzgutugk4lkm,o;2al 6*sedhan 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 ui:hi sm bu 13;wx/km7than when your arms are st3ihs/7u wx mi 1ub;:mkretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sph9bzt c8(7l.cytum(z rockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to uz (c8yhlbt cmz(.t79 pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being a+mr40 a(f phw/ yywndkb7t9j*7wj9 xk6la+ icble to run fast have slender lower legs with flesh and muscle concentrated high, close to pa9+9/rtjwwba4ylxj0w6hnf(y7m*i k7 ck+ dthe 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 a long, narrow beea vxnx7tm)(;am?

If the net force on a system is zero, is the net torque also t vs0: r(o mce z3g2v,y;kr4kjzero? If the net torque on a system is zero, is thg0ztkk 2m4vejrcsy;o (r: ,3v e net force zero?

Two inclines have the same height but make different angles wi6baii 8jxp wyj6*96hgth the horizontal. The same stw6 pj6y ibh6ja9ig8x*eel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolli:n )czzo-y,mgv h.f 2 uvrqol(p b+*d,ng (from rest) down an incline. One sphere 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? (fo z)bcz2+vgrquop .l,n*m,y-vdh :Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder havtmul b3xtk6ya7pz ag :/9e n)2e the same radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?xug:a2 9atketnl)7 ypmzb3 /6

We claim that momentum and angular momentum are conserved. Yet most mok43i ysw* s2 0iren axua+8oo/ving or rot/8* yo na40k r ii+xweuass23oating objects eventually slow down and stop. Explain.

If there were a great migration of peopleps/(r vt /oarwu qo.jm*sg70+ toward the Earth’s equator, how would this affect the length or+vw 7a( *o/ t0/s.gqujsp romf the day?

Can the diver of Fig. 8–29 do a somersault witdzxs h/ j-f5ta69uto5hout having any initial rotation when she leaves t5 9js /t- atofud5hx6zhe board?

The moment of inertia of a -+vws h0ux+czbb:4fh rotating solid disk about an axis through its center of mass i4 hw+:buz-0cb xshfv+ s $\frac{1}{2}WR^2$ (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 ze22,m -rqe )/gnpxqq /m /ifastool holding a 2-kg mass in each outstretched hand. If you suddenly /i gx mqqn /mpef rza2)eq-2/,drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. How p:tw- 3bzvvk6,po*dm ever, one is hollow and the otvwtdb mp- ov :,63pkz*her is solid. Describe an experiment to determine which is which.

The angular velocity of a dgwybmo +25u 0wheel rotating on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the topmy+g od52w0b u of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of jvan i3 uweouo3d9.)wtag6tx09b. +i a large freely rotating turntable. What happensdboit0u6oe aw.xji) 93wv9.t+3uga n if you walk toward the center?

A shortstop may leap into the air to catch73ji/ dzwf kz, a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposid/zw zkf,j37 ite direction (Fig. 8–36). Explain.

On the basis of the law of conb58x-fh ux0e28ko/ ci; h; zwab:eqll servation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways exkx8l 82u ho ;zbqe-b fwc0a/;:5 ilhthe second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (a)+lmtm3;oq f;4oj 6jju 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 Earthz9xfapn4b5uoscy8 e b 6 6:qf: because of an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen opb :zu966c y4ef xfq: o8bn5san Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beam m a(z+yaydpk7 2d1f2e diverges at an angle d2aym+y7 pdkae 1f2( z$\theta$ (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 rat,w) 6rvjbqc)zij6: pj e of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration a, j)q)z v bicjr:wj66ps the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to anok2 cic j+,*hks0gwy.hther child. If the ball makes 15.0 revolutions, whwcchkjs.2 0* igyhk+,at is its diameter?    m

  A bicycle with tires 68 cm in s1ld1stua(vn6 /.ale7 oau/r diameter travels 8.0 km. How many revolutions do the wheels m.eatn76l1aud(l /rsauvo s /1 ake?    $rev$

  (a) A grinding wheel 0.35 m in diameter nm* 7uw5nepy/rotates at 2500 rpm. Calculate its angulm 7 e/wypnn5u*ar velocity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.0 s (Fig. 8–38). (ayh97sg+da iz*) What is the linei yad +9gsz*7har 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 arows rus 0s ,kp:4o0o8tlund the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of + ro*l7hddzb 8a 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 f*wql21 pk*wgen 8k;cp29 u gluoj) +fvwk0o7 particle 7.0 cm from the axis of rotation is t w*k7jogo plpg28)ke1 ql;2u 9nckw uf0wfv+*o experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel acceleratesnk(c95 u9;ehmdhsy 3(/cojbw -em b7 z uniformly about its center from 130 rpm to 280 rpm in 4- 3/ be (c9jhcb9nz oy;hmke(57smduw.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 radiua( 6s9pli 6vrcs $R_1$ 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 themselveg,cnne4t)kgfmq+qr: 3 8m rwy)gna0) zz). pos into a slow rotation to distribute the Sun’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 spac:gga)qwqz+, mr8 r0e gfkntyon)cn4 z p.3m))ecraft 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 rp6uns6v l ci0:xz9i o8mm in 220 s. Through howl6zm9i: o68v i x0cuns many revolutions did it turn in this time?    $rev$

  An automobile engine slows down,5j.j7bhx4rmb wz0 y(f; eys c from 4500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed jets in a whirlinxbv /na4ck p082p0mzrg “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions beforkv 2 crmz4/0p0p8xbane 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+l7dwygv cbx5y2i jurtxo9ce0 / g5 05.5 s. How far will a point / g 5xvxjudl0w0 c7e ybrotg+c9y55i2on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running atg4.8uyvvdkn7 850rev/min It turns 1500 revolutions before 7vn 8vgdku. 4yit 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 speeeelnrty 1x3yr 8h9ua e l*2y- 0gn5gx;d uniformly fr051n xy 3xge-;n*2ayh etg8e l rryl9uom 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

  The tires of a car make 65 revolutions as the car r 4.u6izknu2ik:ilw-f educes its speed uniformly from 95km/h6i2kiuz4u:kfwl-. in 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 weight on each pedal when climbing a hi: k/ uu6.libfill. The pedals rotate in a cikuf.6 b:/lui ircle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end :zn x;7 vf/cxzof a door 74 cm wide. What is the magnitude of the torque 7zv /zxnf:x;c 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 the wh.ay ; s;e8stkueel shown in Fig. 8–39. Assume that a f s;y;k.8eua tsriction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are at + y p:syx(mjbb8a3x;btached to the ends of a massless rod which pivots as shown in (s bj 3ab8:bmyxy +x;pFig. 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 a5g: wtja;xlv-n engine require tightening to a torque of 38 l;g-j5 :twa vx$m \cdot N$ 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 spheree i gu7czzd5ts7gjd96+ 8xe 0d of radius 0.648 m when the axis of rotation is through its cexi9tjes+d7gguz7dd 58 6e0 zcnter.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicyclex vx5h 0luv,a; wheel 66.7 cm in diameter. The rim and tir 5vax,0;xulhve have a combined mass of 1.25 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, light rod is 8vvsn r **ym5 0i;vnrnrotated in a horizontal circle of r50i**vnnrm ;n 8vysvradius 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 potq g.hgu*q9+ kater’s wheel rotating at constant angular speed (Fig. 8–42). The friction force betweenq9a*qg. +ghk u her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the arrbuk cc,nap-*u*vvas nzxb1a; 2g4 85tay of point objects shown in Fig. 8–43 a8a-bpx ck;* bt,g541u 2*ucvsnnv za 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 t(m7s z)yok7npp9qu c+ otal 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, unifox l59 0 eweqkal*:ds0irm cylindrical satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 35e09:d0e qlswa*l x ki2 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 cm and a mas4qwrwu+ r s6 7jhk0sz.s of 0.580 kg. Calculatewzu4+ s 6q0h.rwkj7 sr
(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, afdr 48dl:;dds- gt yxa+1 taw+ccelerating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merryv4lao. a0k8py zv e61q-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, an0v ez8q poa41v.aky6ld 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 10c( tipb7 i6hc z(n9vfry rcw)a0 d8 l*fc26di:,300 rpm is shut off and is eventually brought uniformly to rest by a frictionalc6f :ctid 7 (vz(0hic8w)cdrlnp69 riyf*2 ab 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 accelerates a 3.6-kg ballbwyscbya,g7y). : e 1h 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 for znfc3(v.su +oearm, which rotates about the 3vfs+nc .ozu(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 thin rod, as shown zxm1/jrjzd*o t* gv/8 in Fig. 8–46/zx* gjd tj 18vzo*rm/.
(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 o c7qitc +9e-b9mqs3qgsv7/x n f 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 witutq 49g5j qb a5)wm9lhhin four full turns (revolutions) and releases it at a speed of)hw q94gmt5l bja 9uq5 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 mom7 ufyejnd 4+zf9wh*6g7 5omogent of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine devel 4th1 /zkg0p(vg0)umutadp 3b 2;1 gh+wuthrsops 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 anb;(4 *hpvscccx 3mid-d radius 9.0 cm rolls without slipping down a lane at 3. xdb-vchcsi 4mc(p*; 33 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earthzlds z0lqbe;z1w :4)s(sgg9 z with respect to the Sun as the sum of two te(qd1)z 4z0 ;lzss zlggwe9sb:rms,
(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 o2d hzd5 wxjqlf(;s0t(p,qb )7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume jh)z l0sbw2to,fp(5x dd;(qqit is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls witz)x 1ce0flsc )hout slippinzc e sl)01xf)cg 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.k4nr12 s5y uxmi0 t)z -ob3s34glnzx t It starts to fall and its lower ely)5 x3zts4to n g4s12-3minurz x0kbnd does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rtt7n u:5b7ihrs ,9ua dotating on the end of a thin string in a circle of radi:9tnu,r bti757a uh dsus 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 ra+omu(jcn;yv e)9+ t uniform cylindrical grinding wheel of radius 18 cm when ro(;y+e omvu9cnta r )+jtating 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 thathy2r0ex q3 +a blk*qmg;g: 1ya is rotating at a rate of 1.3rev/s If he raises his arms*xle y0qb:y g+hkma2 ; 3rq1ga 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 moment ofi*rb jt d5i*;c inertia by a factor of about 3.5 ij*5rc*bd ti;when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase h 2tv)g+,j to 2kyv,rtler spin rotation rate from an initial rate of 1.0 rev every 2.0 s to a f+ ytklvr2 ,t,2gj )tvoinal 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 kjrhc9hf/9/1n ygb 9laround a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of /njh l 9ghkf919c /yrbclay, 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 momentpegh:ys. z) n9um 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 m3 19e6 l;++lsx yln xvonf+uayomentum 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 inertixhooc28l.e x2j l tban7+9bp 9a I is dropped onto an identical disk rotating at angular spe2x ollapethc7.2b9j nb8o+9xed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4 qblq r9s.ko w:l0g .x:$rev/s$ 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 a0s(z.x.so vbdzbdf * 9uga;5fas6i/e t the center of a rotating merry-go-round platform of radius 3.0 m and s9(o/*budax.0gf szz5eida.vb6f s ; 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-4o1buz,t g1 9b6ykw-gig6z yh round is rotating freely with an angular velocity of 0 bbk6 -ohgwz6,y4g9z1ygtui1 .8 $rad/s$ 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 *itty(dg 1 s6gwhite dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries awa*6sg t idgt(1yy 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 wg+ u7z*0u6 r /r ( auiadf*zz2e5ybkzcinds in excess of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass 4c+vg2c 8 it.pcu r*;4yh 5bvx.c:vvwyp: rgc$ 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, init78m9i u5- ; meyxuloetially at rest, that can rotate freely without friction. Thu iyet589ouem -7;mlx e 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 diameter merry-gp q*954 wozoivo-round turnt9 4qpz vi*5wooable that is mounted on frictionless bearings and has 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 lying on a 4cvv. em. tl-p3naz0lnsg08 the top edge of the spool. A person grabs the end of the rope and walks ze 8. .c0p l4mlasn-3tva0vgna distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces the Earth. Deterqa p0c26n (anoya2 uoem4)q9xmine the ratio of the4xna26qn9u0aopom)c q(e2 ay Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from re-:3gtn)q0l qhm f - d8 6vynabwf myc;aa,rc(+st at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder oi,eyny wxpg*7vhj3 +:f radius 0.20 m acquires a rotationanw y v+egp3 :jy*h7x,il 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 cylindriclcbup ,-gt1e5bme-m -0 de5c aal disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo wha t,epb 1-ucde0mmgl-5 5bec- en 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 /6t ql.i yvbphy i2cp/87b :pithe angular speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with;4q7rinx/w lz/jpm +p mass 8.0 times as great, were rotating atw /l4pq7 ; nr+i/mpxjz a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for gnks7 j4 yssqx-8) 4rcit to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywhe r)-s4kc ys78gxj4ns qel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates a odw.ns8nx;/ mqu,1atg, .b zfn $H_2O$ 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 al vog/gh;99k4a1 yjf horizontal 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 fricti kz*va d9eh/0dq wvyc0f.8fw 1on) 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, andayeqdw*k w1hv cvf/0908f. z d (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the floor, and we7/1yc:fz:nt da s ;vuh want to exert a horizontal force F at its axle so that it will climb a step against wh:z7 ts; vcnadh /y1f:uich it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is z7vs-ofy zb*a t9dc /qc (dd.5making a turn with a radius qazs5/tc9bd (dc * zd.yv-f 7o The forces acting on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg 1x8 wous*vk5z -jha1m rock into a sling of length 1.5 m and begins whirling the rock in a nearly horizontj*vw a51hx8om-z 1uskal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid 6hbpj udh85t 6cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the rt5d6uhjh6 pb8 atio 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 +w q+vxdu6 xpv2 2i1a(gl;zov two cylindrical plates, of( ux lp12awi+;dovv+ x2q6zvg mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r r8 f ez48mxxc 2j0 jy:t,;jffeeolls along the looped rough track of Fig. 8–58. What is the minimum value of the verticaef80xj ;fej yem, j8z:ct2 xf4l height h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but doz2ad*3 cwtlea o9 x(643sscca not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutia1/ 4wbuq6k(dhjaf9aons as the car reduces its speed uniformly frou(h4a1qw6 baf9j/dk am 90km/h to 60km/h The tires have a 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