A bicycle odometer (which measures distance traveled) is attac8jh;.d4*yvwu eep)md5)ozilhed near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wlewjm;4dhdy 8v) o5ei) u *zp.heels?

Suppose a disk rotates at9z fe hfp e)838xe6ty zvrjqn1(aiy03 constant angular velocity. Does a point on the rim have p6iaxnehzq ejf3ry9t0)e8 zy 38f1v( radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value xlumf 9;uaay4nn1-: aof the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque thpy*x u-a)7myqw4pmf, an 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 -oqlmrm yw;(1to do a sit-up with your hands behind your head than when yourqyo1rw ( ;lm-m arms are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the jgys)0 ibrk3 6 83x2q7a85dblu sb mfkrear 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 ig3 umb7ibdsl3 s 85bk)yx8 0ja2rkf6qt harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on beinj sry)*bt9 p+wg able to run fast have slender lower legs with flesh and muscljr)yp9+bwts* e concentrated 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. 8–35) car:,t*whj :bkh2vwg8u vdk(o8 *jewl)xry a long, narrow beam?

If the net force on a system is zero, is the net tom+ndfsor, x-//4qyj mbh z-znx/3jg5rque also zero? If the net torque on a system is zero, is the net forqzbf n//j--x/y,m mdz35 ns4r xgoh+jce zero?

Two inclines have the same height but make different angles whe ()tt:h lewd/r7j,k(o*h bt ith the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be gwhd*l,((tkebr htj 7:eo h t)/reater? Explain.

Two solid spheres simultaneously start t mlk 2ywja:wa)14:khrolling (from rest) down an incline. One sphere has twice the radius and aww:ky:thla4 k2 )mj1twice 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 bottom?

A sphere and a cylinder have the same radi (8vlxj+th9 twus and the same mass. They start from rest at the top of an inc8 v(+xw l9ttjhline. 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?

We claim that momentum and angular momentum are conserved 1i/umnt+9p 4y+dix3l/ j oewm. Yet most moving or ronodlx 3wm+p1ey9m+ //i ti j4utating objects eventually slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equator, how would 2 (6q.b vursbx fo,h)gthis affects f6.rbu(b go,qvx)2h the length of the day?

Can the diver of Fig. 8 f1ph.oc6 x:arfgow 3y8gab1( –29 do a somersault without having any initial rotation when she yhf8 1af 3ao(ggw.cxr1o6 :pb leaves the board?

The moment of inertia of a rotating utfvl6gdmg(:rp.9 n ./jx4dr solid disk about an axis through its center of massfgrj/. . xtdurnlm6gd4v:p( 9 is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holdi ci6t,ws m*b/ionjc-4ng a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity incrsii m-j64nt, b /ccw*oease, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, j9bqei )upa:f m-znx k214(osone is hollow and the otherp ua -izbxq)f:1m2ons9k4( je is solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a hori)1iod.1y o qqizontal axle points west. In what direction is the lind1 ooiy1i).qqear 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 +qoh2g lce9w00 ovs4ion the edge of a large freely rotating turntable. What happens if you walk toward4 0qcgo9s w+ vhlei0o2 the center?

A shortstop may leap into the i1t ikdj o9.u: 9*bwsyair to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rk9o 1yusid:w9.*t jb iotates. 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 yb5+m2p sn 3)*t fwt 7aeull0xwhy a helicopter must have more than +epn502*b aylw3sl mu7) fx ttone rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (w1o7z a /5vc)w k9ddcsa) 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. Calculate, usinhaqv*tym;i: 9 g the information insiv*a9:q ih y;mtde 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. The beam diver* 12axix:q55l v1ko(gaeum qlges at an an1 xqu21(5o5 :e*gmik qaaxllvgle $\theta$ (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. When the motor isan,/c g /qr3kjrgxs 0+bh.t+u turned off during operation, the blades slow to rest in 3.0 s. What is the +rb0 ,rjkcs+ 3au./g tnx/qgh angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level flo,ex90hohv8mx or 3.5 m to another child. If the ball makes 15.0 revoluti,9 m80vexxohh ons, what is its diameter?    m

  A bicycle with tires 68 cmtu, u md,dkn6) in diameter travels 8.0 km. How many revolutions do the wheels,,k6u d tdu)nm make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 250 x9g1hysu5 bo,)uh7ae0 rpm. Calculate its angular vg5ubu7 9,)shx oyea h1elocity 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 completpq5/ rma93g ice revolution in 4.0 s (Fig. 8–38). (a) What is thri 39/pgcq 5mae 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 vz4tl6ndqva2)h45wpm 9 eat5:och5wnl ;2a gwelocity of the Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear sp3tyf kwa 3--rteed of 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 xe54cievg jm l: 4q*z+rotate if a particle 7.0 cm from the axis of rotation*4lqzg : ejxc4 e5iv+m is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from/l5ppdh* n 8/2rar3 ip8h uxx2-oux 21omdyzq 130 rpm to 280 rpm in 4.0 85hi r/*xluyrpp3dad2ou 2/xhzxqo 2p8-n1m 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 ksovin07vl-azr ,2clr7; 2 ty$R_1$ 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 ) pgm cb*9m9ouaboard the Apollo 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 1.0 revolution every minute during a 12-min time interval. The spacecraft can be t* pbm uomg)c99hought 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 uniformlq-kncw / .hw7py from rest to 15,000 rpm in 220 s. Through how many revolutions did it turn in this tiwc.7nk pqw h-/me?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.2)v av6-scxmg2 zrv 4x5 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 flyin;5gv-v 6 xlnkatx or.6a k+2ohg highspeed jets in a whirling “human centrifuge,” whichrg66vk ;loho+atvnx5 2.kx a- 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 fromr. :gw lahcw/2 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheha wl2c.r/w :gel have traveled in this time?    m

  A cooling fan is turned off when it is running asp5r;a zjl:6x ;b0bm ht 850rev/min It turns 1500 revolutions before it comes to a stopp6h0bm;5bz x s:jal; r.
(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 lgu k3 4sxpzt9q ne5w5kc5029c unw7w the car reduces its speed uniformly from 95km/h to 45km/h The ti 5 xk szl0cwc5pkq249 7n9wuwgntu e53res 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 ye,r6ah h )ydl: ia.rc-x13iqreduces its speed uniformly frol,hi6a3d1 xrr ) eyhayc :.i-qm 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 puyy882r urf4 q*;v 11 vbl7jlpceoh8h pts all her weight on each pedal when climbing a hill. The prlhqv2*puvc1;8fp8e ro h yjbyl 1748 edals rotate 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 N on the end of a +rhd8gl0l5; fry f kg6door 74 cm wide. What is the magnitude of the torque if the force is exerte ;5gy0f+frl8rdhkl6g d
(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 wheel shown in Fig. 8–39. Assum4axp;;q ind4amf 2 wm7e that a frictio;42;mafim7a q wxn4 pdn torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the z* .wgwsmzb np/ 0-ifvdo hv;yhn4/+8ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontalp/. d+ v/h-0n4 w vgw*ymnihsfzzob;8 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 torqd-g n qfl-6lk1m05ivkl8u( uhue of 38- (m0gkn -l815hquiku l6f ldv $m \cdot N$ 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 t*ncu6-1q jd*nl6t hcof a 10.8-kg sphere of radius 0.648 m when the axis of6qdt c6jt-c* u1 hln*n rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in dimd1g*r rv nm69ameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can b9rmv6nrdg* m1 e ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a th q+qc vaj hj9:rg1+gd6soi1, c9zk:wg in, light rod is rotated in a horizontal circle of radi:johg1gi c6 c9 9w1az ++qg,rds vqkj:us 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotatinfxl38b4i;hrd 8 +ycwgyi5n( lg at constant angular speed (Fig. 8–42). The friction force between her h4(;hr +ibgyx8l8y ifl 5n3wdcands 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 array of point objects shown in Fig. zmq;).uqvu3t 8–43 abqqzv; .u mt)u3out
(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 total mass is **4-f5s6 ;e hezl3q5jppann2vg baub$5.3 \times10^{ -26}$ 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 cylindri6z)gig(wn f0xr4fu ;vcal satellite spinning at the correct rate, enginnfgz6 );x0 irugw(v4f eers 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 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with5fi+u 4gz ybf8ka6g -t+i,xn d a radius of 8.50 cm and a mass of 0.580 kg. Calcul xt-u+5fn4g zbgik,iy a 86f+date
(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, acceleraa7k -, nx)zlysting 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 tangenj/tr*4p cwnde7z t q2.tially on a small hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and hadt. 24 *ptr/ejnz7wcq s 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 rrsh( a:txtl y w.r re*:8bte18otating at 10,300 rpm is shut off and is eventually brought uniformly to rest by a fri8r:t:rlsbte t.e r8a wy*1h(x ctional 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 az qhe0pb 3x-87ue kfb- 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by th)ex-g+3 o nas 2-hpmtpe action of the forearm, which rotates about the elbow joint under the action of+- eg3-o2 pa)tpsmxnh 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 conside2d+ uuzj2 qy)fred a long thin rod, as shown in Fig. 8–46.q jdu2+f2)uzy
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consists of two masses,j 3y)jzv h9vi*4*b;pz s8ejrq $m_1$ 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 withki(dk tae ;m 4+17 6vehtqhh;oin four full turns (revolutions) and releas eha+eq ok1td vh; m7(tk4ih6;es it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment of i. f.eo+v nyfhv-e x(wxscf) rat)w7hu u-,70pnertia 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 6oa+m7htk4l tx1n xw0 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 k/gxzbb7i7sie- 38ojf 6sa2o without slipping down a lane at 3.3 ab7f/-6ijesx2 o 8z7bs3i gko $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy ofh:w0umf yt*(iz i:3:yk r px9.lhxfm0d x.x-f the Earth with respect to the Sun as the sum of twof p- l3*m0xixfhf:x9yi. :0wm.k d (:urzytxh 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 ks4np vrh)7af(98qyjwfb4; v 1640 kg and a radius of 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 it is4sq 7(fv jnfyvk4rh)b;p a98 w a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls w xw7)oh qid( su awhh1*0-6lrxid9b (cithout s(ho61-9crai 7u(w*d0 )hh ixwb d lsxqlipping 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 verticallyqd2d ,ik-2ahrd6ap g / on its tip. It starts to fall and its lower end does not slip. What will be the speed of -hq6 dr,2dk/ ad g2iapthe upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of iwx6 :nntor5- a 0.210-kg ball rotating on the end of a thin string in a circle of rant5x :iw6rn -odius 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 unifor17/-ewpx -ipf5f iw* iwapd2lm cylindrical grinding wheel of r5aip if1wwx-7p-il/2*df pweadius 18 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotating at a rap:k.g rh)p/bzte of 1.3rev/s If he raises his arms to a horiz ./:)zbhrkpgpontal 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) wokur3 /5pii1can reduce her moment of inertia by a factup rwi5i/k1 3oor of about 3.5 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 her spiob5vp 0nm9n7z 5yal)w n rotation rate from an initial rate of 1.0 rev every 2.0 s to a fina) 5wbvaylo 75p90m nnzl 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 aq sc23w-:mrp/++q qisu- umaits center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wspsi :+cuqq m-au/+q-ram2 3wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater sp dfr aq(bc6,*einning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of the i o1*q7on olo9Earth
(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 vw32*s4yni) ;08r5 b qqj yjei8uazmuis dropped onto an identical)8qms3wiy2qy 8ei *u 4j5ujv0;ab z nr 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 a+x zljno6l4zg0 pu8l ;t 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the c g+r;/;f* 9kgaxi(j ljy6 dzbventer of a rotating merry-go-round platform of+yj(g d6 if j/gzk 9x*av;;brl radius 3.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angub-z b1s5jzsb 2s/qh( alar velocity of 0.8 jszss1 5 (qh-/azbbb2$rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, losing about half its mm+okzdo ri6bxl6-9 7gdo 8 ww7ass in the process, 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 rolwr k z+oxim-b7odgd86o79w6 tation 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 excess of 12 w xh:64deo;cqms7wc :0 $km/h$ 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 d18tes2/fsn(js + ;bpv c; *kera8g mx$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, that can rotate freely withuc+4h n7dwlq5k/z i( iout friction. Th lqw +zk u/(i54ci7dnhe 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 bfrb +1od0+ var51nro edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia off 1b+0+ a o1rbrdn5ovr 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;4ua tkx10x6 r 5cvyts + )axdgo:jm0w the top edge of the spool. A person grabs the end of the rope and walcd ou)s 0gx;tm5 0 6j +t:xa1k4wxyavrks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earttbg 5 f--oa3jvh such that the same side always faces the Earth. Determine the ratio of the 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.v5jat-3b- fg o)    $\times10^{ -6}$

  A cyclist accelerates from wkcvt))oe3 xl1+c v5jbn/- zzrest 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 unjxq)(rjf q. s;e40c umiform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor qscjmuje(fq;4 x).r0 .    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 kg andsba- n9d h0rv,u;u.lz diameter 0.075 m, joined by a (c9 .s,;lnudzr -h0ubavoncentric) 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 ang 3zmrdk;y5x 2uatose7/xy 0+h ular 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 w zti1:0d. dfvjz7o.kn ith mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost massi v0jdz tfz 1on:.7.dk carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use energy 9x47frbf/t cx lufsln7-um:.mc1x 8h stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindfxn h8-b ucmc1f7l4/sxf mu7 .lx9tr:rical 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 illustra1 ydysj7pji(.tes an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horiz/kj/ p*6eihp zontal 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 ,ael5c/d/0qx zrk/kyon) about a rk/qlaz0kc/ ,5 e/x ydhinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically l/;dtxj /6 mh2 n1zkmq pe7-/,rwkeb von the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, jtq-//d xrn he;w7z 1vkbem m2,6p k/lwhere h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a ftucq). . *1:i a-tog*8fbqv zkyquf;turn with a radius cftt.fgiz .yb)*o fau:*81q; qvq u-kThe 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 rock into a sling of length 1.5 m andnxp6w6ijk n eq707+gr begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to + j6nk 06iqwprgxe77n a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and hef+)huzec ) enmcl,/ y6r arms as thin rods, making reasonable estimates for the dimensions. Then) emez/hfu6, lcn +c)y 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 assembt8 ces;u7eb,w ly which consists of two cylindrical plates, of,we bs7;u 8cet 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 rolls along the b/d (. xjgy:mslooped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it ix.mdb/( jys:gs 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 do not assc0fr :hggw wkt902)vu2 h:dseume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reaubb yn,sc ),w1 ioad,v/) (luduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What wasuua,(o dcwsb)il/1y,av, )n b 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