A bicycle odometer (which measures distance trave0rtj 6w5vol 75+9svb noncfb1c;l a8m led) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch tnv l1 c9j ns0a7v r8 ;6o5fmb5+wblcowheels?

Suppose a disk rotates at constant angular velocity. Does a po r/bp-,cikukgrx.. b0ye1gj 6int on the rim have radial and/or tangential acceleration? If the disk’s angular pkuxgcr 0e.1 bk ,b6/gj.-yir 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 describeo*l g/3w8ysowd by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greato els/ 5: )4cmiupc-e6 x+zsemer torque than a larger force? Explain.

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

Why is it more difficult to do a sit-up with your hands bn/4z owxc :v/iehind your head than when your arms are si :nxovz/w /c4tretched 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 at themfx;)e7l naxxl*n:b3z9g toeo ig n3d,4-i7s 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 pedal, a t3ifeiszba7-d ;on4 x : gx ,3lognnx m9e7l)*small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to jysj2+ a;ca(mqof ei g/m8+u4cbet++ 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 distribugja;+e2cm i jf+couya+q( e4 8m/+bs ttion of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow beamkb6 xm5jx/ (fkdp2g-skerjh76+whg )?

If the net force on a system is zero, is the net torque also zero? If t0vjh8i1mg 4 jihe net torque on a system is zero, is the net force z4j0i8m ijvg1hero?

Two inclines have the same height but make different angles with the horizont 3p )vhn0bbov+r k:v,zal. The same steel ball is rolled dovrkb+o,0zn:p )v v3 bhwn each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an incek7em 8(6:) sp.-s )ywfgj g pjrwsu4vline. One sphere has twice the radius and twice the mass of the p8 j7j- :4gvf)r(ee)6wgw kspyussm. 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 radius and the same mass. They );gprr7m l;tastart from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the gremr;p; ag)lrt7ater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum arr qiuz9gb3(x*+ 1ynx ie conserved. Yet most moving or rotating objects eventually slow down and stopyx1+3(i ru *gbi9qxnz. Explain.

If there were a great migration of people toward the ofc zx(4d( kioj8dmf6o 3cs6 82u ta :c*;dszhEarth’s equator, how would this affect the length of the day?io c 8f2: 6d;c3uoddxt8k(jfcm z*hs ( 46oasz

Can the diver of Fig. 8–29 do a somersault without having any initial rot;gv3 mtmdee6 i61szdn1r / l2wation when she6iz gnet2rmmle;d3d1 6s /w1v leaves the board?

The moment of inertia of a:,b :sz :ip. vwupl6rv rotating solid disk about an axis through its center of mass ip:v 6 bv,.sw:iz:ur pls $\frac{1}{2}WR^2$ (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 holding a 2-kg masl;nlo xqk-rm:i 4vqakmd;88i:;td* ws in each outstretched h * l:rq; ;ktvn;d:w4ikmaqd m8-xo8iland. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hollow an. bvrb05 0m pszn3gmt(/: tswj b-xa3ad the other is solid. Describe an experiment to determine which is whtmbbb r0 p3s-awtas:vmjn0.x g5/ 3(zich.

The angular velocity of a wheel rotating on a horizontal axle points wes 8)4f*ovik oxoag*y*lt. 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 ka*io* ogo l*y)fxv84linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge /f +;tfpij.8s4xehxtp +q ,g0 p*huakof a large freely rotating turntable. What happens if you walk tquip h ejg/,;hxk +a*fp s.8+p 40xfttoward the center?

A shortstop may leap into the air to catch a ball and throw it quickly. As he thxn g xjlnpa3g85 +w5 g;p53sfhrows the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite x; n3gg gwnf3558+ljx s5apphdirection (Fig. 8–36). Explain.

On the basis of the law of conserd5gg) x,i 38qewl v:txao 7bh+vation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operv d h:3 qw)ao75ggilte,xbx+8ate to keep the helicopter stable.

  Express the following angles in radiansh (xen00n:c: sa.ruao: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidence. Calculate, usin rewuc74 ite,0g the information inside the Front Cover, the angular diameters (in r4teic7ew, 0uradians) 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)+byj:p nbkz6 . The beam diverges at an angle+nk)zj6 b:pby $\theta$ (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 is turnedhqujj4 ee m876h vw-l /8ik8dr off during operation, the blades slow to resm4qw8- vle 6idj/j8rkheu 7 h8t 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 e u18l)q aq.cr3.5 m to another child. If the ball makes 15.0 revolutions, whatrcl. aq e1qu)8 is its diameter?    m

  A bicycle with tires 68 cm in diamet3z jyh d1a5rx:nnj8f. er travels 8.0 km. How many revolutions do the wheels maz j5yn hd.an:1f 3jxr8ke?    $rev$

  (a) A grinding wheel 0af*,70 nq bbpd.35 m in diameter rotates at 2500 rpm. Calculate its angular velocityf 0d*pb,ab7q n 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 one4iba**u c7nal complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear spee4*ci 7lan b*aud 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)f k cpnl2 v*:gyut)tw 2/if.y( in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a sp xpf .nll;:,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 a0(k4d+ uox 8li stw)m-(d.x wlcw *lvm particle 7.0 cm from the axis of rotation is to experience an acceleration of 100,000 lodvx+4)ksmw ld0mc8(u -.l txw (i*w$g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpm to (u va)wh1.y e6w lz0sq280 rpm i)v .y1us6(qw0 zealhwn 4.0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radius rrb0u4lr:0-r eh b-f k$R_1$ 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 2 z nopp+8; ayfj;a9(nhubw1o put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of tjzypwoa 9(no a fb+u;h n2p;18heir trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to *:a ztcna7 vtmie(m(:+ q n2el15,000 rpm in 220 s. Through how maa *2 v:zq+ l7mec inantem(:t(ny revolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Cc .vmrh; z0z4;mfgllqh4u w ig0u1)k)alculate
(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 stressu0lzg*ym7y5 k : ejpbf ue5o/6es of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its finob0 pezl:u6uk 5 e7yy*5gmj/fal 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 3npux2x/t: q j960 rpm in 6.5 s. How far w/ q9txp2u x:njill a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 1500 f2y+f5pxa :hgoz2p) wrevolhypw ) f:zfp2x2o5+agutions before it 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 speed ude5 ns2*ws8+y1 x2py5gp,ac f9h ene tniformly from 95km/h to 45km/h The tires have a diameter of 0.80 ,y8pd5yt p2hw1 ef2 na gecn+es*5sx 9m.
(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 re+.5g h5i l yy*jjory3yvolutions as the car reduces its speed uniformly from+.*l5j 5jrgohyi 3yyy 95km/h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each pedal when climbing a hator 0 e*ql:5li5t :a*0 lleorqll. The pedals 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 55pi ,n9d-,n5u;yc5ejz jyb w- x N on the end of a door 74 cm wide. What is the magnitude of the torque if thyj5p,;wn u9- x,j-c dnye5zb ie 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 torqueicad( 1 fu.zcb+++ean about the axle of the wheel shown in Fig. 8–39. Assume c(iu+nz1e+a b+dfa. cthat 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, are attached to the ends of a massless rod which phkm xhcq(wlzvz - il7-t+p*8(ke ,jwg-sg :v;ivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and di*zm (jsk--v-w8lk7h qg:h x ;w el( gcivp+,ztrection of the net torque on this system.

  The bolts on the cylinder head a- st7n rpuf71bc/v:r of an engine require tightening to a torque of 38 -7usra/vbfnpt r1: 7c $m \cdot N$ 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 inert- i3hm/lds-6vr 6uv cm twb5p6dc)*nvia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is thmdbvldic s-u cn6t/ph 3*rvw-m)6 v6 5rough its center.    $kg \cdot m^2$

  Calculate the moment of inertia of o,caky:hwdr 7l4:gi.a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the huor,7:.4:iy k lwdach gb can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on th09:xnux7mg cr0y/w lae end of a thin, light rod is rotated in a horizontal circle of radius 7x u9lnm0:cy0/r x awg1.2 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 at constant angular speaci .( o(hk68p7s d*neh f -fij6fq4njed (Fig. 8–42). The friction force between her hands and the clay is 1.5h (o.jhn8 -f6ji is7f6(adne* 4 pqfkc 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 iner bpg+;ufa1j6ws7z)l ny. ua-jtia of the array of point objects shown in Fig. 8–43 abouwb-j6.+u 1jn7 a;y)s gfapl uzt
(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 z)8xrvp:hhq2uxs4 :d9awvy - atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindricfz 4kz0b4-jz ) h wfsldzh(;;xal 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,z(b zfh4x 4s0k;;z wh-dfljz) what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 cm a0pw ci:xl* o9fnd a mass of 0.580 kg. Calculateopl: 9 0cwf*ix
(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 teupnw2(vc oqe )l(8-usi.n-, accelerating 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 hao g5(h2x1a tgynd-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 has a m tay1o5h gxg2(ass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is er) 2bt - 4y ldoqh.xmwae2 in6qk20)lshut off and is eventually brought 4.qnrm6ex e-22)oait bh lk )0ql 2ydwuniformly 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 3.6ox;/ 16zvchwhb:) lh j-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 sole yon8jcohkn +2mpxn -v1 y(0x9ly by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelem9yxkonj+ xyp1n8no c2h-(v 0rated 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 in Fisv8l eo q(u) zm):*px ),kigym, vfp,xg. 8–4 e,yvfz x)oum) ),:* p,kvmqspi(8lgx6.
(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 clp5 egnz(l4p) a 8i2qeonsists of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest within co ul47o h0 *+2qwdwdufour full turns (revolutions) and releases it at a4cw0d o+luwh*u 27doq 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 ham+x4.g9ceimq7 uh *z us a moment of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque or ,k00ow9r kj 5xrcw+hf 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass7i/w)/hlfu0q5yhzy n 7.3 kg and radius 9.0 cm rolls without slipping down a lay/ y57fih hz)/n qwl0une at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth wi p1 jlj6q*;i gu :9go+x5th8sxql2uyw th respect to the Sun as the sum of two :8;x5o6qgwj+ut2qj s*p l uxlg1 ih9yterms,
(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 lq yw6lt c.i wv((qz,(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 is a solid cylindew,wlz (qq6l.(cti(y v r.    J

  A sphere of radius 20hr4 k*6.b0i qwn3yz ao.0 cm and mass 1.80 kg starts from rest and rolls without s nyb.o0 a*rw6qh3 iz4klipping 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 tipho 40u3ph. at.wp rz( bgd)v2-tcju. a. It starts to fall and its lower end does not slip. What will be the speed of the uppe.t j)v.wtup4-2 3r( bhzcpd uo.a0ha gr end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum o5z2k ag0w-rtko b8t0 cf a 0.210-kg ball rotating on the end of a thin string 8 2wk0tt -5z rg0cabkoin a circle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg unih m/buen /t-v,form cylindrical grinding wheel of radius 18 cm when rotatibemtn/ -uvh, /ng 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 a0zdeid- yr55 g platform that is rotating at a rate of 1.3rev/s If-iz y05edgd 5r he raises 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 sv5f* 6 k,o2jdij)iig ps::j p ca65dwshown in Fig. 8–29) can reduce her moment of inertia by a factor 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 tij:dia26o *p ) pw 6cfj5is: j5gsd,kvuck 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 init vnncjr h6 6.lz-*yp.yial rate of 1.0 rev every 2.0 s to a final.. vz-yh jn n66*cyrpl rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis through its cente765 kocv2cg ga,qdc -hu.h vyfw6twb7i4o, r -r 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 wuvkv-5yoc dcqr.,6 wo4cg 6a,b g2hh - w7f7tiheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinning at 3. j+*yk, bcmy)r.(znhi*lrk h 05 $rev/s$ 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 mugq.f1 36 bcjzw8ceyip6wx 1, a(9bsbomentum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$

(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is d /r fy90eqyur/ropped onto an identical disk rotating atu yef09y//rqr angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns -kdto ij0x3y ,:ni*k tat 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 v f fm6q+g(nn7;umok,cc2(s;zlt cu2 platforl6u;uct fgmzqnno2(; ccm2 v+ (k, 7sfm of 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-rod,m dg *83h4zjj).ad six2i yeund is rotating freely with an angular velocity of 0.8 2yizm d84hjsegjda.,3 d*x)i $rad/s$ 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 ob pya y)12g;jclw(n .its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no .c)pybwo12 j(g la;nyangular 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 involvem ouvl jd7/v,9 winds 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 lu;2.k iu a,h w,;tswmb wwy:c/ ty:s,$ 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 ro7dgt(49,6ik oteaa+yuk ;l zn tate freely without friction. Th6( ,kk+d;y7uoit4an te9azl ge 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 stard3q(tu/0 r wxh2r+eszv y0xmxq1h/c8;/mqwnds at the edge of a 6.5-m diameter merry-go-round turntable twu rers1 0;m yrwt/+/c2q /v(mq 8xqhz xh3xd0hat 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 roq g0p-2v obwo)v0n8: pyk8l( i: fybemlls on the ground with 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. 8–508m2(ybpk o v-pf )q00nowl8eg:by:vi. 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. e ,uxn ,gect s3vozyap0cy1.ah 26 28tDetermine the ratio of the Moon’s spin angular momentum (about its own axis) to its orn x1p ys.ce0agz3tu6ov ,8c 2 teh2,yabital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates frqor6fn4n 6l 8fom 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 t 2m1bns*rj za8he shape of a uniform 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 motorb*z8js n2ra1 m.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical d d0te(x 7b+iuhrzt7; fisks, each of mass 0.050 kg and diameter f+ txr z7t(d 07e;hiub0.075 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 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 ofd5a fa0d0z2*x ex dm.j 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 6(wiecy *v m0 2(dt(uqobis5h our Sun, 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 gravitationa m woh*(yieq 6u0b2dtcs5(i (vl 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-p4h ea/(.aq uceollution automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniforea a(q ./ch4eum 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 8o6lok1bj9x deir4 d+$H_2O$ 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 ong2(btjs i(kpx:ia k.lv3 td1- 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 r;/k i;ew* tit9 ,htjz2n6g1eyg. q vf(i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the tgv;/t ;fkyw6in* .9jzig,21heqre t 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, anmo+taxq h97frkk(50aq j +n7kd 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 heightk9qnajq5h7xo(k a+7t +mfr0k h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is maku o4j .h*c notrea3i)(ing a turn with a radius The fooe 3oij ha*(4.cnru)trces 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 tk(7 wz)vaad;9z-pwo5 pnnv /rock into a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his hez kvwod5ptnazn)p/w7v9 (;- aad, 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 bivd f;7(iik5yody as a solid cylinder and her arms as thin rods, making reasonable estimates for the dimf k;(v7ii y5diensions. 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 pl5 r sjun9( z4xf/j-5)km ioemcates, of masocs9nm5m/j ife -x4 ujz5)rk(s $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 roll ,k;znuiy/p f0s along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to re;pyk, nf u/z0iach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but 2 t8;ahyuw d78gfj jk;do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutionstolb 3 )r4,q ejgc;l:u as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90g4lu, trbl;q3j o)e c: 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