A bicycle odometer (which measures distance traveled) is attached near th*dpu iwt61y w4e wheel hub and is designed for 27-inch wheels. What happens if 614ydiwwup *tyou use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velimp z x6o )o-dwd z8ipe07n8)iocity. 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 acceleration? For w8-di px08)e woi)no76 imdpzzhich cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be descr22e*mn6w w mfmgn5rx 1ibed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than ato6f ac0guke j4 m5e p6:*gv8t larger force? Explain.

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

Why is it more difficult to do a sit-up with your hands behind your he9vk-xd -s1iu tif-i/j ad than when your arms are stretched out in front of you? A diagram may help you to answer this iu1xid/s---jvkfti 9.

A 21-speed bicycle has seven sprockets at the 6b sh/ ve54gtjqu)t9wrear wheel and three at the pedal cranks.6 tbwtjhevs) / 9g5q4u 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 small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have stdl +l1 -(moollender lower legs with flesh and muscle concentrated high, closed+lmo1 ( -toll 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) carry a long, narro3bw4cm/ il:ki w beam?

If the net force on a+1clgj;1s nq w system is zero, is the net torque also zero? If the net torque on a systcqnl+; 1jsw1gem is zero, is the net force zero?

Two inclines have the same height but make different angles with the horizhf u7 *wwi7emzfvxnw1t 8(:) sontal. The same steel ball is rolled down each incline. On which incline i7sefx8w )1*w( m vn wutzf:7hwill the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultan zjs ivx2z3tu-y1 +bddwhz; 3v:lq(*f eously start rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the grefi+ udxb*y;jl :t1 zzw- 2z3s3(h qdvvater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. They start from8v jek0y17z4dyle:rgy z(uh0 rest at the top of an incline. Which reaches the bottom first? Which has the great7e0y dy:j 104 8( yrhzluevgkzer 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. Yet most moving or g5/t.thspi:j rotating objects eventually slow down and stop. Explainp.g 5st/ ijht:.

If there were a great migration of people + a53x :lcyd7gvl hte7sq*1s 1*ad xtztoward the Earth’s equator, how would this affect the length of the 1q7cstg5 1ae h+alz*x ysd d37*:xltv day?

Can the diver of Fig. 8–29 do a somersault without having any initial 5;rzoy7 jsenbxx -9bv fg48j9s;dx:grotation wh;-4x9nxvx8j7s5 :zb efs g;y9bdoj gr en she leaves the board?

The moment of inertia of a rotating solid disk about an axis through its centero50otm-;*ho d m59jebs ij5ay of mass ij; oah95om*o e5yj5tdm-0 si bs $\frac{1}{2}WR^2$ (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 mass in each outst(3 zmov+9l wydretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, orm39(yvzl+wdo stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hollowrt* 2 hm;pqss 8 g;,p d(sk,evh;h*dhe and the other is solid. Describe an experiment to determine which is wde dt; s*2;,rmgh*(;shevhpkpq ,hs 8hich.

The angular velocity of6wn;w vrecn 1u pj;:n kz)h8b5 a wheel rotating on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular wnn;ekv1z8)jh6up 5n;w c r:b acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotatingsl4 e/rz1 pk*h turntable. What happens if you walk toward the centere/pzl s rh41k*?

A shortstop may leap into the air to catch a ball and throw it quickly. As he c km* 509c4qihv 9ahcythrows the ball, the upper part of his body rotates. If you look quicklhc5cam4k y90hc9i qv*y 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 why a hv wix zio6v bg01/*(wvelicopter must have more than one rotor (or propeller). Discuss one or more ways the second pr 1/ i*vwxovv(0bi w6zgopeller can operate to keep the helicopter stable.

  Express the following angles in radians:,uxckbi2 akp: (m r,:aw12+fkp9szos (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, usnk;/1yoi -pase :(hf.m( b/jry4h u( ptxmx)uing the information inside the Front Cover, the angular diameters (in radians) of the Sun and the ( yyx.at(/i( h hp1pxfk e rsm)n -:;b4/moujuMoon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 kmg89;kd6 em db-7nd pe45)x/vj azz fiq from Earth. The beam diverges a;d g-8x 7z4vai/ pjdkdf5bezmqe9n) 6t an angle $\theta$ (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 v8xc3(c6c utn6 dbpq4r7 j9 hwe5hdb9 of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as t 9d 8937 e5cbh( ncup 6jw64rhxvcbtqdhe blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the ball makes irb+y 8 6.p (fzxz;iax15.0 revolutions, what is iizzf8(iy.xp; ba6 x +rts diameter?    m

  A bicycle with tires 68 cm in +gq6l*ae,f it diameter travels 8.0 km. How many revolutions do the wheels make?e* qlgta 6+,if    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 r+ z;pn in0wy;.a j9o-m)mxbsqpm. Calculate its angular velocity in m9;. y0m ;s)+-no iqxnp zjbwa$rad/s$ $\omega$ =    $rad/sec$
(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-rorx:6w8 -lfkkr und makes one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from 8 l6wrr:xk-f kthe 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 orbbc*e u7lik 59*;l/zntv +uco sit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed od:;.n s0fh+b i v5znz0ob 6 +pop3wgazf 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) mu;.-)hw8mw z 6b t3yboy 7 psgue,xcj,yst a centrifuge rotate if a particle 7.0 cm from the axis ofc)t,;mhy7o3w.b8 psuy6 xwbzj ,y -eg rotation is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel acceb3h.zuz o+w2*+mj njx lerates uniformly about its center from 130 rpm to 280 rpm in 4.0 s. Dowm2jb . j*hz+u3zxn+etermine
(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 radiu(bj,jwi1y mq-i p;jb8s 3h6ids $R_1$ 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 aboaezdc 4 dx 36yq6fj/*snrd the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerzje6fq yxcsd63n* /d4ated 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 jyh.1ft h,q)v- 9 gobn nk-l+muniformly from rest to 15,000 rpm in 220 s. Through how many revolutions did ioh1mtf l- gn+.n9h)-yv jkb, qt turn in this time?    $rev$

  An automobile engine slows d vu- fkcx9xtr(6b( 0d9l; nxqdown from 4500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying.hf u1ol:u rg0 highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to u0ofg r1h.u l:turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 360 ezd4+h q7ec5c z4b :ux6y x:zfrpm in 6.5 s. How far will a point on the edge of the we5eqb:uxd: y z4 fzxzcc 76+h4heel have traveled in this time?    m

  A cooling fan is turned off when it is runniy lvi,g/z:mc:ng at 850rev/min It turns 1500 revolutions before itvm, y/zi: g:cl 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 r/w s klci,)k c5 xqs3lra+;ab9evolutions as the car reduces its speed uniformly fro3 ki/5w;9,sbql)ac r sk clx+am 95km/h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduc7o0,kmocp;y p)3s4lc- cgqk m:r ifp1 es its speed uniformly from 95km/h to 45km/h The pf,-c rq 0psm:km 4o 3ki)oc; 17cgpyltires 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 ea-v/) -rz gprol/qv3wb ch pedal when climbing a hill. The pedals rotate in a circle ofzqbl -) /-gv rw/rv3po 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 9qq s76 0yx2pkqc z,e /w+itnvbb2 ux.of 55 N on the end of a door 74 cm wide. What is the magnitude of the torque if the force is exert9z2 q.bb vse xn,0ykwqi6cq 2+/7x tpued
(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 aboutv7nj j, p)c5,a +htekw the axle of the wheel shown in Fig. 8–39. Assume thatn c +vhp)jke5,j, wta7 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 e9c7awav e qyod,3o ;y-nds of a massless rod which pivots as shown in Fig. 8–40. Inia 7o3;d9ywa e,-o qyvctially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening to a torquefc e)m0k 67txo scr*-n of 38 n*0 7fets x6moc)rk -c$m \cdot N$ 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 inejg 3 ydh6+j)f-:n wszlrtia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is throughz6:n) jhw+ slf3jgy -d its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle whign/ wfn8r20qeel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ig 02qnif8 nrwg/nored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is 0mwu3wix knw1m2bdf8hs353p rotated in a horizontal circle of radius 1.2 m. Capm dn0 hb8wki13xu2smw f33w5lculate
(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 ahg3vhdynjq1 s a-c;k (clx/8 bo:dx,1ngular speed (Fig. 8–42). The friction force between her hands and:;b3xq x1v hhjnkgs1(c,od-8 lac/d y 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 Fi2a// qaq-6mujy9.whiw1p xxi g. 8–43 apa2 y6amxi/j hq1/xw 9u-.q wibout
(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 atomsa/ cy:wb.k*/x/ ytfwe whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the correct rate, engin* v9no8x+ lskqeers fire four tangentiavsko8*qxn 9+ll 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 with a radius of 2gbp5 8r )e lx8azrbdsdbycb *4+p, 0lj2ey(w8.50 cm and a mass of 0.580 kg. Calculatex(ep eg b*)rdpbzybrj2a 4dc+wls8yl2 8b05,
(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, vtcm09v6kmj:k ;az 4daccelerating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merry-go-z* ;;gt4*puq 45 cvrpg xao*.ubkkwe0round and is able to accelerate it from rest to a ;.uogx *eu w*prkg4tk5 *cqpzab v0;4 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 opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is eventually brouh;vf8 jsw0 4nmght uniformly nh;4 s8mfjwv0 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 bm7235ljlzd qball 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 sols lm vj;rp3g:2ely by the action of the forearm, which rotates about the elbow joint under the action of t3plrvg2 ;j:m she triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as shown in Fig. 8–45w8d 6kztmq v*qvnqjzm3+ 8a 335*dq8znq3 6m v zjt8 k+amvwq6.
(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:lrv s p;f+qh- 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 3/ 7xeh6;aa kzym(b fafour full turns (revolutions) and releases it at a speed of 2/m a fa7(kye3 b6zxa;h8 $m/s$ 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 inefrrm z6tz, r ada-*2ze4*+bkrrtia 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 torqucz7c9 ycv(fv/ e of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls without s+ l)oci*tad, hf m z2koq.5d)ts5khe/ lipping down a lane h kl25ceso/k5o hi)tqa*zd)f,.t+m dat 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kineticy o6 6*z3wsuj :/amdfx energy of the Earth with respect to the Sun as the sum of two terms,xsf a*y3mj6/o zu: dw6
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius of 7.50 m. How much net work3gv zor( .dl xyl)6j)h 7ru3iz is required to accelerate it from rest to a rotation rate of 1.00 revolutioh v36rlz)ol )xry3 7udiz.jg(n 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 rolls witz ry ni,bq( ve;f9):rjb/ -anuhout sli ajur)nbefyzr -qn/: ,b(i;9vpping down a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

  Two masses, $m_1$ = 18 kg and $m_2$ = 26.5 kg are connected by a rope that hangs over a pulley (as in Fig. 8–47). The pulley is a uniform cylinder of radius 0.260 m and mass 7.50 kg. Initially, is on the ground and $m_2$ rests 3.00 m above the ground. If the system is now released, use conservation of energy to determine the speed of $m_2$ just before it strikes the ground. Assume the pulley is frictionless.    $m/s$

  A 2.30-m-long pole is balanced vertically on its tip. It starts to fall ,,cjw 5ty 2er*in nil9nu.q3aitf:+hand its lower end does not slip. What will be the speed of the upper end of the pole just before it fyq+twi:nia lejt5 2c, i,9 urhn *3.nhits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ba ipp6xrwj:w o ;qj76q)my/e3rll rotating on the end of a thin string in a circle of radius 1.10iq wx7jy6: /m;qerrp op6)3wj m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grinding wheel -yzw c+sxoywj r:) l.3of radius 18 cmwj 3z-o:ryy wx+.sc )l 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, ona2qjz l,zg(ui.ke qqik 7d4 8) a platform that is rotating at a rate of 1.3re i(zilkk82 d7zj g4 ,quea.)qqv/s If 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 shown in Fig. 8–29) can reduce her moment q og6u1ue )pnb2w8 m7c)5wu;vkn- lxcof inertia by a factor of about 3.56mwbeg xc)p -ncuw u7 5u1o8;q)vlkn2 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 spin rotation rate from an initial rate of 5y7hhwl .uo7 (wxfw-b1.0 rev every 2.0 s to a final r lxh- 7.7by wu(5fhwowate 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 center at a fr fgs w1am(px93equency 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 ( g13wmp9xsfais the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinning at k+r)liii7so* ey.k 8q3.5 $rev/s$ 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 momkn52y,mhmrigcc .v1 pef87 k7entum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dropped onto an0 nmshi2l sj,j, nmr:k0rkbnw ; m4.+a identical k,ms0+m2j knj,w4.i0lm: sr ;rnbhnadisk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns atub9m3ai l*/ vikhe2 e3 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 tqn 9bdqe:+b s;:ov8xche center of a rotating merry-go-round platform of radius 3.0 m:+bvqd c:xn9s o8e q;b 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 merxc(cv eul48:ury-go-round is rotating freely with an angular velocity of 0ecc uv(4x 8:lu.8 $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses in;q w74o0vetwdto a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rota;ve4dow0t7 wqtion 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 excajzud-stk8 93 ess 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 jeir 1wh98e/f$ 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 pla f;0/u;5lzc hrg(sqhdtform, initially at rest, that can rotate freely without friction. The moment of inertia of the person plus0g;zh hfqlus/ d (r5c; the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diameter : e4w 4bx8 paf07u4faplhe wz9merry-go-round turntable that is mounted on frictionless be4xhfab8 :9pwe azw u 0ef74lp4arings 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 grounklxs a-c /f+-jzngi .*d with the end of the rope lying on the n jxf-g/*ca.+sk-i l ztop edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces the Earth. 2:fa2dy rdnk5:q9 uw tDetermine the ratio of the Moon’s spin angular momentum (about its own axis) to its o29 n :wtyqkaurd:f2 d5rbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate jp3lryvp: g8tkr;utk6h:e2 ; 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 ,h*qq 5vii r2dthe shape of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at consv ,q*ih25ir dqtant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 kg ,:e3 vtb j7p 9cwgjs8lq2(bfland diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and w fvbjpl3,:2897 tlsqe(gb j cdiameter 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 of th+ha 6;xjcku uv63khb :e rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with h (mu:a6c s7yqmass 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 speedqa:(7yh 6su mc be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use eneco 2p4 n,wdkg-rgy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240,o kgcpdw-2n4 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 illustrao,r kco50qcwwe , x6.dtes an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal surface at speedgbf4vbg/m c: tg5bfuo..rs 0 8 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 ighet/a08l0wikkfz l c1(w gi -*nore 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 accelerationk/(lz aeckl itf1wi h* 008g-w 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 standkfy ,+ .u9bwi uy6l+.zbsv0v ming vertically on the floor, and we want to exert a horizontal force F at its axv+f9w0 6+u k b. luzybvyi.m,sle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist travelin y thwnlw1d1 47bfb (1iy-guf+g with speed v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist and cycle are the nh1w+gwf7 ytyb1 fd4lb nui1-( ormal 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 infyiti7mml i. 4xeny :/q+ *s(zto a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from :y(4zy fx mq/i*smin7.etli+ rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thin rods,8mj:q6p nn:of cz2nc. making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning s .8m:ojpqznn: c fn6c2kater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindrical plates,sew7; clde1r0 of edlwr1;ce70 smass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 t g c(h;/zy0ntahe looped rough track of Fig. 8–58. What is the minin 0/tyz (h;agcmum 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 do not assume mxay zpu1w:k -rc 7y/-vy5-a c 5sx,qk$r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed6+n pk)kv z og.e/sy(d uniformly from 90km/h to 60km/h The ti+ z .edk6 )ogv/pynsk(res have a diameter of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s