A bicycle odometer (which measures distance traveled) is atty(,o/7r xt7gy i*lvs6ilbpj7ached near the wheel hutiy6x*ysr (7vlp,l b/jo7 7igb and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on the rimx hsq,sr/8 :uu :4.ohpamn ix;ymzk-- have radial and/or tangential acceleration? If the disk’s angular ;,qu:x8 .hoiys/u mmxrpn:-ahk-4z s velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the angular vel1t.u3hp b3z lqb k4* djxd, tgnf63;yhocity $\omega$ Explain.

Can a small force ever exert a greater txyi n9s3687 jq :hcvbvorque 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 yoab1;tx) 9(l i sz6 1btkzp e1tr8i7syqur hands behind your head than when your arms are stretched out in front of you? A diagram may h;ez1sbp8)1xs9qzka (r tlb71tit 6yielp you to answer this.

A 21-speed bicycle has seven sprxht9 gc/07h(ccf 3 kmnockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Whkc 9gh73h c/(xcmftn0 y? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able kjh;id 8 0:uw5a wvv6lto run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8h68dvju50 ail; :wvwk–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow b4yvnue8nk * 9neam?

If the net force on a system is zero, is the net torque a9k lix,/bzxf48*,jr : hjofj clso zero? If the net torque on a syoi9,/lxcb : *,jzhxrjf f8k4jstem is zero, is the net force zero?

Two inclines have the same height but make different an4tzy7x8 u eih-gles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of x7z-he4uity8 the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an ivql 3c10z3ou5 w5sajnncline. One sphere has twice the radius and twice the mass of the other. Which rz51 aco53ulvq w3j n0seaches 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 start f0j+1htmdl-jitto / q1rom 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 tmt/d1o-l hj0q1 itj+ kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most movigkrd7n:ez3f oe su;ms+)(sp2 ng or rotating objects e 3o+e7 (pr;sfe sgmuznkd2:s )ventually slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equd 8;8+j cwjotuator, how would this affect the le8t c;dj wju+o8ngth of the day?

Can the diver of Fig. 8–29 do a somersault without having any initial ror:m k)5m4;xdsc v 6wtttation whwsc m kr 6dx:4t5m;)tven she leaves the board?

The moment of inertia of a xzrm+nsxuj m+k7d9rp7)7 4hmrotating solid disk about an axis through its center ofhd+z j+7m49um)xpsrx 7n7 kmr 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 stool holding a 2-kg mww ss)aad/f a6)gj)q7ass in each outstretched hand. If you suddenly drop the masses, will your angular veloc asaaq)/j) fs6wd7 w)gity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is ho z jfs:*u7 wm9kl)2oacllow and the other is solid. Describe an experiment to determine which is 9f kcm7wa)j 2:l*szuowhich.

The angular velocity of a wheel rotating on a horizontal axle p*4o:p4 hrmrvxoints west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleratirvh or:4pm*4x on 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 rotating turntable. Wh;1e tej. k:zjy g yzcg;;1yp(cat hap :gy1ct;k ejy jp z(1;yc.;gzepens if you walk toward the center?

A shortstop may leap into the air to catch a ball 3vh;22wuq -ynwd.a*kwj 4 agfand throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you w-wqwhkyn34fa.;d g * w2 avju2ill 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, dis ) r(d5l(fkglecuss why a helicopter must have more than one rotor (or propeller). Discuss one fgdel l5)(rk(or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (a) 30om l3dxntifzni g./pa2;t* p+ cd5q)2 $^{\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 coin6wht:dbk*w ;b r/: rj2n2n lkmcidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon,jbl:n2; ndbmr6rw:h wt2* k k/ as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km 8d* 2sgb7 5p awz9tqqhfrom Earth. The beam diverges at an angl p8s g5 qz92qbt7hwda*e $\theta$ (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 turn5nq m:,* mf dt/ae.xpred off during operation, the blafpq e5/rmn ad*m, :.txdes 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 another child. If the ball makeslyipr*y a8md7 bg.*0c 15.0 revolutions, what is its diamd*yg. b 8ypm l*0i7acreter?    m

  A bicycle with tires 68 cm in diameter travel sb )-5mjlir/ojk3szm9)1pm9skli4 +l; pjte s 8.0 km. How many revolutions do thelrb j ts5jm1);e o -j/pm9ml)k piss iz+l3k49 wheels make?    $rev$

  (a) A grinding wheel 09jg.q cz h)v8z.35 m in diameter rotates at 2500 rpm. Calculate its angular velocit)cg9j8 h. zzqvy 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 compwnr)f ) jia:a,lete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child sfai):r aj,n w)eated 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 Eart2. q8ygmu(k txh (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speeg kbw:u+js -oteuj3i72(( a,h q)fxcld of a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle t9 +lrf:n y* i0 pbv zodo-oj8,mgr*7v7.0 cm from the axis of rotation is to experience an acceleration of 100,0t oiy-*vpo:l90 odb8grm nr7j *v f,+z00 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelb p;.hr/nwyyw1k(1 s yerates uniformly about its center from 130 rpm to 280 rpm in 4.01.w;h/1nyy spr(w bky 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 radiuskva)a3fhp rux (:3f2l $R_1$ 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 Mhuewy8uzs 9/;0+tr bjl 2u/s xoon, astronauts aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a d sbrxut j/elz+9hw02su y8u/; iameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm in 220 s. Thx4,lpwoi,) qu pmby./ rough how many revolutions di,wpm qb o,/yl)p.u4x id it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 5aqdab7lzi06sm; g7urpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed jets in ofkpj6ulv. -cjo00qj x-l c:0 a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching o60cpq c-:lk-xjl jvfjou.00 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 :;9sj4rd :/uwp3yr5fmeyqk o360 rpm in 6.5 s. Hufm ke;d9y/pr3r: 5ws: 4yj oqow far will 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 85d o-3e*;t6x(zg nzonrc -- ty35jvkul0rev/min It turns 1500 revolutions before it comes 3ogy-n-t-6r* xj3zlc d5n eutzkv;(oto 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 redoq, ujm*j,-. ep30 iwjwz2fenx( t e1:,dlyfvuces its speed uniformly from 95km/h to 45km/h The tires havnwimueyw2,pdevx.lofjf30-*1q j :e,( tz , je 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 reduces its speed unifoewgxgwf.65 z 6je xq w+.) l4ek(mx:fs7eqa 5hrmly from 95km/h to 45km/h The tires have a diameter of 0. xs6(k4 )xf.5g7hf emweg:qaw6lz+ j e wex5q.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 we 2w6qq z7x 43;j.s qhji5zocdright on each pedal when climbing a hill. The pedals rotate in a circle of radius .zxhjqr34c5 7o qwzq d;6is2j17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cm wide. What is the m*c6qn5w)0*s uwz e1bhz+ lgtsagnitude of the torque if the z zh6 *bnulg0s we+wcs*5q1 t)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. Asqf5m6nn rya v5 htx9v7qza5f+jw;(4 wsume tha 7wwa6marnxy5 vt5(ffvq94;+ nh5jz qt a friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of n1j, cmh(mtz)mass m, are attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculazm1nm ( jh)c,tte the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tds2*x a u.ue+rightening to a torque of 38 x*u+ur2 a.sed $m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of isry66jop py j+ 4oy.-radius 0.648 m when the axis of rotation is y-oyj.py6 6rp4s i+ojthrough its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameteyzgfn nh+-;:p r. The rim and tire have a cog-hzfn;y:n p +mbined mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rom y5 /o;+n 1twe8gxfqztated in a horizontal circle of rad5+zqy/exmg fwo;1 tn8ius 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’/b x8e9x zgbm3 jf9o1j 3tg2etzbqk :4s wheel rotating at constant angular speed (Fig. 8–42). Th3xe1gbo9jmzb3f 8zj/4e 9: gqtkx2tb e friction force between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objects5dj ep a t0w.c*u+fwwuw1v/b8 kyqy 8pz6;br. shown in Fig. 8–43 aboazyp /586cw0*qyvp.u8b . ukwdwjf1rbt+ w e; ut
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen atoms whose total mass s2od, azq aj,havug29u (j/6 lis $5.3 \times10^{ -26}$ 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 cylwnx7cm,():dbk 7 b -r*a9. gpadvwwuoindrical satellite spinning at the correct rate, engineers fir( r 7au-.pabwx :ngwb9,c)*7mkodd vwe four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with t 6pdlk4,e gw2i*t9jpa radius of 8.50 cm and a mass of 0.580 kg. Calculatk*gtip4wd,6e9lp2t je
(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 swingsb 4 t/hixgpy05 a bat, 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 hand0o)l/ al9 dz7 iqtjl-b-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 mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each t)a7ib q /l 90dljlzo-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 shuteym 9 /50l8hw0cy0lui -)sphwhcwx9c -sd yx: off and is eventually brought uniformc 0xw:mh) e-ws p/ly0 i-dhc8y9w 5s ux0yhcl9ly 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 a0e10u 4v.hsxza/dr ,vkib sp6 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely si:aa igx. .2cby the action of the forearm, which rotates about the elbow joi2 :c ixs..aaignt 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 blacaa09s+rs 6mnde can be considered a long thin rod, as shown in Fig. 8–46an6c9ras+ m 0s.
(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 consiku , zjc)n99ywxg (,d+r(gpjasts 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 four full turns 3,s6j 7j k engb6a9* ennp6zbd(revolutions) and releases it aebban9j* 7e36j knp6,gdz6sn t a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor ha/,wm t(hi2 jfts 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 i/:xfaet)f q) 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 without slim :kuhg; nii-ren ,5ce,5 zuj;pping down a laijhe :,gi-;5e,nur ncz 5uk;mne at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of thcn3+ cukqrrw1b+b 0zokux3)3e Earth with respect to the Sun as the sum of two t1uwcx q+br0o+k 3zr) bnkc3u3 erms,
(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 worh vm;tmwl.9 dt qouvi0wj**15l:xb(g k is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it ism1tdx*hvug0i(w: oq9;l w. jv 5*ltmb a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg sh lmc*s;jpo8:k 7e :wxtarts from rest and rolls without slipping down a 30.0jhw*8kep x:o:ml; cs7 $^{\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 vertical:,pizn9r -iyd 1vph0uly on its tip. It starts to fall and its lower end does not slip. What wilnp:ui0,9yrdp iv1z h-l 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 m4x;2qv yegcd) omentum of a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an angular;dx2) 4cgy eqv speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-km4g2 ub; tl bkyq emqkh5r (l3ech6f d19gd;)(g uniform cylindrical grinding wheel of radius 18 cm when rg khq;93 gmdkym 2ctflqhbb(e5)(l4re ;16duotating 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 xx 4pv) e):pcwside, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig.x:4 w xcpp)ve) 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 ra6kzh((jyb f :educe 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 in6:(f bhka yz(j 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 1.0 n7h8- n sdnc;xrev every 2.0 s to a final rate ofndnh;78x-sn c 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 ezp1ldi,.p-a4 c3l/1 aqu rvl its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.4ql ev 1cp p-ida4a3l 1,/r.uzl0 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 wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinns d0sl*o mw4y75 tgqp2ing at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momeqrl/x6cj3 3gjntum 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 droppe -e cn(tlvw9ucce+ 6bm,jaw(*d onto an identical disklvwc(b *t,j-ew(e+n9cma6 uc rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4wbvdz,9 b -de0 6g0 q+vmv9hne $rev/s$ 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 merry1e yo r1qwi.(s-go-round platform of radius 3.0 m and moment of inertia 91sq ori1(.wey 20 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angular velocity9(cgtpir3u: yfx8w 2:hkwl6e of 0. 9p6:xu (2lye8ktfcrg:ww3hi 8 $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, losing a1x9feq .w7dc5qg/r1ildw g 4 dbout 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 thwd gxc g.17eq 9rdw5i1fd 4/lqe 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 winds in excecf i(2v96vt-7 ypkcs4lrw6kr,s2 swy ss 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 fj6makp-7 4k z$ 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 lcqlsz vfpk)c334j 7. rotate freely without friction. The moment of inertia of the person plus the platfor34k.lq3sj cpv lfz7c) 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 stands5,+awm /li +xd9t dzpo at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and had d5/9x wpo ,ma+tz+lis a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of the rope lying on t3w6h3 mpcop . lxeg)9the 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 rollsecmgp o3xp9 wlt.63)h 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 4,lyvx( .dh0mc/ qoexside always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon lc( dxo 4yeh,xm/q 0v.as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate ofq* i6gd*t (y(kai.ifq 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 unifoax,*0w0k uq q+fhbi0x rm cylinder of radius 0.20 m acquires a rotational rate off0q0uq+b h*kxxw0a i, from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of )he v)ecgty/p )v95 xumass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its5g vxpv)tceyh9 )e) /u 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 iyhg34 )ag13o g)maagns the angular speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as great, werd(ec j(c- m4,6wepvhtsbf7 w7 e rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be?cp,v7bjw-wfmee4c 76s((tdh 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 l lak,c5lgwy)3 ow-pollution 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 uniform cyliwc5y3g)kl la, ndrical 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 illustratessk,f.k . hk77tj 6ddyp-.gh;en thd/ n 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 ekfq jn1n.1o +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 ignore fricr/te *myzotigvu9*2+ e7o5fg tion) about a hinge attached to a wall, as in Fig. 8–545o/ e yzt*em 7iu+vtgro* 2fg9. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is st 5ju5q th(i/flanding vertically on the floor, and we want to exert a horizontal force F at its axle so that it wiltf(hql/ iu55jl climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is ma0y /0vynrtrdrs9 c;; . cl.msrking a turn with a radius The forces c;ct;syvs0/ydlm0 r .9r.nrr 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-h3 v;v+z 3vvdk. nzz no:,vk2lkg rock into a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. do:;vvkkn+ zlzzh .23 v, vv3nWhat 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 armsp-i*o2oo:8 tq oqdxr x;b qpg,:( tl4i as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held ol:*o io4 x xpto;p8( q- :dq,bqitgr2tightly against her body.   

  You are designing a clutch assembly whiclk*gkmwy v)y +s9/)yvh consists of two cylindrical plates, of)l9gsykwvm yk*v )+y/ mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped rough track of i5 db.prj,r.z;gxj ,goq iv;2Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest.zq;2ixj;bvdr. ,oi5,rj gp g point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, bm3ezg)46+dgc uotk, u2d gtv/ut do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the ta0rcv gu2;)+r+eq)agit,aucar reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0ra2 vei;t+at gaq)u0,r+ cg)u.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