A bicycle odometer (which measures distance traveled) is attachent xknbp .2l o*bvi( -ltx49:5j9zyb zd near the wheel hub and is designed for 27-2z*xnyb 9: kz -ilpt9 jt(n 54bb.xlvoinch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular vjv lemc iflxs7l;* v00 2d *(kwhr,r8melocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceler hsflvik(dr0,8l 20*c mj;lwxm7v er* ation? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be describi(*pyl6p 1 tt y5auy-ned by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a giobv h v z9r6/ a/9i/9sr+rvr25bvfanreater torque than a larger force? Explain.

If a force $\vec{F}$ acts on an object such that its lever arm is zero, does it have any effect on the object’s motion? Explain.

Why is it more difficult to 6m-/v pyg v9z1fhuql3l.zhj9lflc-- do a sit-up with your hands behind your head than when6-.mlvh h93uyz91vlqll -pcz f-/ fjg your 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 rear wheel and three a,uk7,+ x 92(kxn;2 r zj9i cdojri axht/zp(jat the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a 9j(z;pxjnx o 2aiukd z,rjkh/, 7(2ixacr 9t+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 bvij m(l*n/,s 9rhlnfii6 +,v meing able 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 distribution of mass is advani ,r +jfml vin6lmn,/vi9(*shtageous.

Why do tightrope walkers (Fig. 8–35) 29b,w+t ryjsncarry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? If the i mkam,5 :fdv+net torque on a dma+: f5kvim,system is zero, is the net force zero?

Two inclines have the same height but make d 8oet+kv,gfj h+ q8l8l8zew,g ifferent angles with the horizontal. The same steel ball is rolled down each incline. On which incli,g jeo 8,k88q+ +lelwt8zvh fgne will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneou(*k4)k z/4iktuzb jwdsly start rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bot4dtk4j*i()z kz/bw k utom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. The sk:zs*/ 5 eou,unhf8(bzs;vfy 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 ,z5e(/hno 8bfsukf * s;:uvsz? Which has the greater rotational KE?

We claim that momentum and angular 7z*)dss2pebe g dg/ x1x+lug2momentum are conserved. Yet most moving or rotating objects eventxpbz+ 2e* s u7lx g21dgeg)sd/ually slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equat/z, ptnq*3nt tor, how would this affect ,/qz * tnt3pntthe length of the day?

Can the diver of Fig. 8–2b1qz3tgo4s5 vm3 pt:q 9 do a somersault without having any initial rotation when she leaves the bo 1zt3q4sot p3v 5mb:qgard?

The moment of inertia of a rotating solid disk about an axis through its centocdh ;n hxjukz:oz/3dqdi* *) +8 bfi4;o/wjqer of8hbhdozuj i x: )qcfowi3 *d;ozjd*;4//qnk + mass is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stoolzxg-p/p:do u4 6lh xg- holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or gz-pl4gxp h -/xu6d:o stay the same? Explain.

Two spheres look identical a4 m-ywu. nmrt4nd have the same mass. However, one is hollow and the other is solid. Describe an experimeny4r4um .ntw- mt to determine which is which.

The angular velocity of a wheel rotating on a horizon685 al1lk1c xodz35 8qvzzb vktal axle points west. In what direction is the linear velocity of a point on the t5 v6dz8xlzab5lvc 8o1k k3z1qop 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 ed575 aafig8qajpkl(oh ,l 71c kge of a large freely rotating turntable. What h7jlqf,c p(io k18l55a7a k ghaappens if you walk toward the center?

A shortstop may leap into the air to catch a ball and throw it quickluydkb d9p .l*;y. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legudkl9 y;pd .*bs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discuss why a h (1yf-c *m:ymvob 0y. hazz9anelicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stableyc *y (h zv m:91an.-zoymfa0b.

  Express the following angles in )gywhj2*s;m )d .u ldkradians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing /z/-frevv5-5 ne yhlf coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as se--zv/ lfve/frye nh55 en on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directe,dbz t/4:kxs;3 hx+hu. tmal nd at the Moon, 380,000 km from Earth. The beam diverges at an angskdb3 lu,xt+nzhaht4m: / x;. le $\theta$ (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. Whe+r8 d9yp)1sbky4fz v u1vi/vwn the motor is turned off during operation, thev k8pzvvw9y/yu4ri)+ b 1f1sd blades 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 anothe21 ig i90scez6b.iwanr child. If the ball makes 15.0 revolutions, what is its dia06gn 1izsbwei9c.ia2meter?    m

  A bicycle with tires 6poez:5, wb3wvd(p- nk8 cm in diameter travels 8.0 km. How many revolutions do the wheelkep d,w ovw5(p:n3zb -s make?    $rev$

  (a) A grinding wheel 0.35 m in diametero1i8shh t1+ rh rotates at 2500 rpm. Calculate its angular vhhh8 t io11rs+elocity 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 in3zmv)yox cdbs.:nli36g 7 2a l 4.0 s (Fig. 8–38). (a) Wh3z . m2dl7y:6v)aosigl3n cbxat is the linear speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its*m zpv/2y;bi q orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a poinpj/f 29pppq,j34v efkt
(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 centrifug eqomfr x01b /vcp/+p,e rotate if a particle 7.0 cm from the axis of rotation is to experience an ac/p1 f0m /,v oecb+rxqpceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130-;bc yysfxt30 ti5uk2 rpm to 280 rpm in 4.0 ttfxy 0c35 -s y;uk2ibs. 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 s41 skubd69pw$R_1$ 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.;aqn (9doj;zbwu r3x Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the ;zaj9d(. 3xw; nroqubstart of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly f 1vbnplj.2fv-h0k4k/,keg gb rom rest to 15,000 rpm in 220 s. Through how many v. 4gjk-2 pbfkn gb,h /l1vke0revolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rp,ju;ggt, :j z/wuhi4kc32bdyo* mr 8y m 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 iu hx5+d 72vx9x,ykaa in a whirling “human centrifuge,” whic i9xy2dax+av khu75, xh 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 unifortfhxh cz4h xqh 47v a5u5.gjr;d0(y5bmly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in this time? 0 ahh45vby;t (zqrfdc55 ug .hhj74xx   m

  A cooling fan is turned off when it is running at 850rev/min It turns 1500 revg,civqt4y */ +hv. kdnolutions before it comes to a stoikvvc+yt4q / n,.*dhgp.
(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 speed uv00)p8 xdcu ewniformly from 95km/h to 45km/h The tires have d8vecwu p)0 x0a 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 reduces its speed uniformlye hfk7djr7a7)cw1y(s from csh)7a k 1dj77we(rfy95km/h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her o.h3ukl4 j b5eweight on each pedal when climbing a hill. The pedaluk543b . loejhs 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 5+bb)20y xcw 0narl rn25 N on the end of a door 74 cm wide. What is the magnitude of the torqun0rbcbwl x2ya+ 20)nre 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 wheel shown in Fig. 8–39.zuw 2w*gdir4( 2 ab :6utrl y9y4qaoqf0p3n)d Assume thr 0r4nu6baid)l:duoy4a32(y w g*f9tzq w qp2 at a friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m,w5 osd8zrkm lh w;)6y; are attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initi)rsd; w;5lm 8k6whoyzally 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 hv7x :(d;pstm 1n)zq( gtkibj )ead of an engine require tightening to a torque of 38 );bzn vpstgq1jtm :7kx ) ((di$m \cdot N$ 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 (3pp7t sbj.2 paxemx9d6fjw ;inertia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is through its centera2 x3j9 7f ;dsb6(.ppmexp tjw.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. The r qgh7tnr4p/6tfnq, b ucg(*,zim and tire have a combined mass of 1.25 kg. The mass of the hub canbf7* /z,cr,gg6putq4t nq( n h be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of 3v;fce7cq rs3 a thin, light rod is rotated in a horizontal circlv resf3 7c;3cqe of radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating atj 8 c-rcb77eai constant angular speed (Fig. 8–42). The friction force between her hands and78jrc e -b7cai 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 7 zunvd h:k)9h5a*7i+vo ar9-5tlmmvf+umra of inertia of the array of point objects shown in Fig. 8–43 abouvt m759:-9 )ldhmnar7avmi+rkh* f +uazu5vo t
(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 co capsc3x yfg **./n2mass 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 correct rate, enqz4 nz8vz4vw af:4aj;gineers 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 tw8fvj:aanqz 4 ;4z z4vhe satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 cm and atr, 0tff)5iz t pn5px8d 0xo7n mass of 0.580 kg. x7n rt5 ,tfo) if 5x8pzpd00ntCalculate
(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, accelera/7h -s h7t: rofipjd;tting 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-driveago (05oq; o1nwd,dw tn 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 has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposit1nw 0 o;d (5goodqat,we 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 eventu2y2iwm gu./ff n*8 wro t2ujiyt,a,:p ally brought un2gt,ia* wpo2f : r/ytm, fni.u8wuj2yiformly to rest by 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 accelerates a 3.6-kg ball at h0nt dbrsc*h 2l2f ieh7l7)*w 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 th( r -8gx gjwfp0.xbs11 hqou7crown solely by the action of the forearm, which rotates about the el 1 xr .po8wc7gg0hu1-qxjb(fsbow 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 1xy golu80yaxn 1zb km**8be5considered a long thin rod, as shown in Fig. 8–4l1e1x5kbu 8x nmy*aoz8b* y0g6.
(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 massedq; blef0 y+39kfs o 2kl3 sy(mh./kqxs, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest within four full nmul3n.j f;m8 turns (revolutions) and releases it at a speed of 288u;fnlm.jmn3 $m/s$ 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 a9s0:ijf jwy-g 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 engine developskc zf81*a gygg:) jnijz5 hl*/ 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 witho)4s k6cmd6 vncco 3qo7ut slipping down a lanomq o dv7sc6c)4 6kc3ne at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun4vpzz ; /yoir/ as the sum of two z vo4izry ;/p/terms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg andu y tvdm+a:*mnb5;3 oj a radius of 7.50 m. How much net work is required to accelerate uadvoy3j: m 5t*b;n m+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 1.80 kg starts from rest and r1,x bbu5 efc*l18votvdj y,8nolls without slippit8d x* b,n,bvfc ve51u1ljo8yng 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 verpw8b, z d8nnga *y)ee-tically on its tip. It starts to fall and its lower end does not slipapg) ,nn y8d-be8w*z e. 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 0.21 ws5o+:d s9o/8jbhuq r0-kg ball rotating on the end of a thin string in a:orsswjd5/ q8hb+u9 o circle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momen)r8 y o+9msk36h5 xy :iayejqhtum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotating at 1500+y3 6yhi h:somq 9rxk8e5j ya) 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 rot5t6zaa 39wu lp 0.3vuy 3fdussating at a rate of 1.3rev/s If he raiayls3.9z aut3s vpdu0u63w 5f ses his arms 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 hs dr3 k/ej:/di5yawmo)0mtz3er moment of inertia by a factor of about 3.5 when changing from the strairdod tk3503/ey z:ai m/smw )jght 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 spin rotation rate from an initial rab1 co c4rzot g4v/)i-hte of 1.0 rev every 2i4zo tcro4 1ghb )vc/-.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a verticam q0hisqpv8rwn1g0-+ st .3vtl 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 tqv+mr.p08-v 3n ss1g0qti hw3.1-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 skater spinning 50qqfnhyp 7x8v.rrdnp7 9; jiat 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 momeesq1; kiae+t2ntum 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 Iedkau4- 4ixp9 is dropped onto an identical disk rotating ak9uaie -4 dpx4t angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns a1q2awwn d 7rrifr*b/ r7)si)nt 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 center of a rotating merry-go-round platf: xzaz6xg1f k j27;gstorm of radius 361 fzx tj;gx:kg2z7sa.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 s6cdfg0 :j-dgfreely with an angular velocity of 0.8 0cdgjfg6ds-: $rad/s$ 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, 223kl)jbwyegh :l7mylosing about half its mass in the process, and winding up with a radius 1.0% ofjbh:2yyg2w3 l) le7km its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve o2lcym (wkbpo/jnq /h)w :wa-0bg,,hwinds 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 n:*fzg1kf;has m8f5f n;(edn$ 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 ri zqr;o*0c .gw:3md yrotate freely without friction. The moment of inertia of the person plus the platfory ;wrzom:gi.0rq c*d3 m 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 tq c1*ui6y*cec he edge of a 6.5-m diameter merry-go-round turntable that is mou6c*y1 ceciqu* nted 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 wi:sgx f-, c.vblth the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far do :fl-.cgbsx,v es the spool’s center of mass move?

  The Moon orbits the Earth such that the same sid i75l;:kl7xt4 tmsbwbe always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to it bklwmb7 xl;5t4:t7iss orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from, cgsfo1;*9fnq-w a2ofztx. b rest 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 of radius 0.2pa4 j3k ud32dfw*sfg4hj;fm 10 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque4dkadjfffj hs*14mu;w 3gp32 delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 kg and d*uriw b6cutt1*hlrd3 r.43tsk)d v g28qto6jiameter 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 ofutr1jsr ctgk .t3l o)3h **r2wq4dbvt68ud6i 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*i6 +9y ;kuec/ejqrho 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,qel;wnp- k*o9-jd,5 ydcu x3b but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would -e;y 9qcd bpd,ouk nj-x5wl*3its 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-pollutiok+5emi6 8tyj hn automobile is for it to use energy stored in a heavy rotating flywheel. Su my6ijhtk8e +5ppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able 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 an+z qgxd* cb6gwsq4 n(( $H_2O$ 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 (ho) w)tr vrofn5i 7qu5,iop) is rolling on a 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 ignor;zbvksg: 2x o*e friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the:;obk2vxs g *z 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 on the flp:a0m/zqhov9vfa d70 oor, and we want to exert a horizontal force F at its axle so that it will cl0qao9m z:7p/df0avh vimb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4u/im5vqd+ m tz 0fuk;,8st xr6.2m/s on a flat road is making a turn with a radius The forces actf; /m+i uxzusv,tdkqm 586t0r ing 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 s 5qz:;nm jkqv( yuf:e+ling of length 1.5 m and begins whirlivq5jq m;z:k(:ynue+f ng the rock in a nearly horizontal 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 asa j bkuzzo5q9.o 3i4g+ uc)q7 (i8eckw a solid cylinder and her arms as thin rods, making reasonable estimates for th +)kw43(g7q8ie9q kuiz.uo ob cc5 jaze 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 cylindrical pt b8dzeprw8 *-lates, of mass8edtwp-z b8r* $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 looped rough track of6ceqpi+ a2(f z Fig. 8–58. What is z+(6p i qa2fcethe 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 the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but)b vn2)rh s;uz do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car redgxsex22bq j,/* x1u;hzi2 abduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular ac2de*x/ ;sx z1 q 2bbjuag,2xihceleration 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