A bicycle odometer (whichl; wk:-k2rr(jf wd lf1 measures distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycl(kwr:1f-dj fl r;2kw le with 24-inch wheels?

Suppose a disk rotates at constant angular velocphvvjz6r ;-d,ity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does ,ph6vzd; j-vrthe 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 descriz*x ujbykoqe-*3k ,i*bed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert .k3 p lyjfvp7gcx26s 7a greater 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 ahd *y5. x+nsxl sit-up with your hands behind your head than when your ays*d 5n.+xlxh rms are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at:svrnax y jxa1:i7 6:o7+luoi the pedal cranks. In which gear is it o:: ixnis7 1:xoj ara+vu76ly 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 slenus8u+q34n sy2d 0gaflder lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantaga f+4u3d2snsugqy0l8eous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow bpw4:i*giwx58 /lnqn zeam?

If the net force on a sys+l( xldb51m u7fb9l sttem is zero, is the net torque also zero? If the net torque on a system is zero, is the net force zerobs dm9 tllu75+1lf(xb?

Two inclines have the same height but make different angleb)ax+n54ws) teg ( npdz8jg3 y54 gcvfs with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be gr 5asgg(3npcw)nzb5+ g4fyej v)4 8dxteater? Explain.

Two solid spheres siof.6gjjt*g,ph v6 r.kwp :jb lk :1x1gmultaneously start rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the botj6:tvp1f,kg r g.1g:lxk jj*p6who .b tom 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 samk 4fues8*tljd+j*caj8* 4d 3 ctcc)vqd (g3u he mass. They start from rest at the top of an incline. Which reaches the bottom first? Wddsj+* t38fl4d(jjc 8uhcgtecu3 *)*k 4qcav hich has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yn9 bo(7 0mbdgoet most moving or rotating objects eventually slow down and stop. Explai d(0o97bnmogbn.

If there were a great migration of people towardux6k6o6)pzueca(kbe-: 0 u q k*pjbu; the Earth’s equator, how would this afk6(xbzk0o6ue ppqu6c e;u:* ub- )a jkfect the length of the day?

Can the diver of Fig. 8–29 do a somersault witho 0thnw2db (h;jut having any initial rotation when she leaves thth w0bd( n2j;he board?

The moment of inertia of a rotat/ah(8s:)ejh dw3y 7 t ixi(friing solid disk about an axis through its center of mass ihrys 3atjhfw( /i)7x:ei(8 ids $\frac{1}{2}WR^2$ (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 e) tmf)a7(jtc-j6as tvh7xf /m ach outstretched hand. If you suddenly drop the masses, will your angular tc(tm m7-/ fh 6xjj7s va)ta)fvelocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, oneu3 d*cdta8 z6e is hollow and the other isde3 8cta *z6ud solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horiz 8aiace(j8gg.bc :s 73yoyqx :ontal axle points west. 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 linear acceleration of this poy xyo a. b:ejg8a3(7s 8i:gqccint at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standin,zay*6rnn artxw*m ;5g on the edge of a large freely rotating turntable. What happens if you walk toward th,zx5 nn6mtra* a*y wr;e center?

A shortstop may leap into the air to catch a ball and throw it quickly. As he thd8bh4icc2ba7 tg).th-8 ezokrows the ball, the upper part of his body rotates. If you loo2hg tb8)dzi.8h k cact e7b4-ok quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, dis gw/53t)psn9 cwtwg-cvm m*m8cuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propellws*5pt-ctcw mg3nmwg89v / )mer can operate to keep the helicopter stable.

  Express the following angles inu* 7 dt1fb6 9hnll3a dqtd,r+l radians: (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 becausb*fu77 i6v9;;tzyxu sw l:d ove of an amazing coincidence. Calculate, using the inform vbx*svuzyod97i;t6l:f w7 u; ation inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, / /1tsiwq kh* j7dbg/r380,000 km from Earth. The beam diverges at a h/stdbw ikr1qj//* 7gn 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 of 6500 rxrl c(l+o/wxix j+n*q;* vf) wpm. When the motor is turned off during operation, the blades slow xf*; )(+w lcnwrv+li/xxqo*j 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 yv4 +0:zyttwnm to another child. If the ball makes 15.0 revolutions, what is 4wt0n:+ytzy vits diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many 5diw7oqlk n,wo s)vf*d; m7ri8 b0pi*6e8g qxrevolutions do the whe*8 *s0nlpidq5w;x rm77 bv6wgo k oid e)ifq8,els make?    $rev$

  (a) A grinding wheel 0.35 m in :5vcl7c7 q: vyhggnuz+0m s g.diameter rotates at 2500 rpm. Calculate its angular velocityng5u7 mg:qz0 gvv: y+ l.hscc7 in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.0 qie8vpv h;j/ 8-e o(vds (Fig. 8–38). (a) What is ; v vvh(i/dep8oqej-8 the linear speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Eart ddovm- q27xc+h (a) 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 point u fholx50;uz uv y*zp.83w(r)nv kj*k
(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 7aja uv/whe:fsyhg; z*p6ob3x*m2 0 j2.0 cm from the axis of rotation is to experience an acceleration of 100 g:ej;b06f j/as 2a wuoyh*h*3xzp2mv,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center fwe/ygc ,8g: fjj:a,t*h(meqg rom 130 rpm to 280 rpm in 4yqctgeg gw j,h/m8:(*j a:f,e.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 zsrf* wo yfk7d::)(xc $R_1$ 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 put themj,* rhcc531 yqnjyf)lselves 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 mi,c)hnc y3jqr f5lj1y*nute 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 r- uokgnrx: ja89r d5z0est to 15,000 rpm in 220 s. Through how many revolutions did it turn in this uox8g9jr :z0nard-k 5 time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200i 2ctia/;n9t (uxec)pg: u*jj 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 flyixsp9f0mu 7 3xcng highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 cu m 9xpxfs703complete 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 accelew8+pi itmc69var-v 6prates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge ofv9 6p a-ipr68cvw+i tm the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at fo) zinx0(ju bj3qz18 850rev/min It turns 1500 revolutions before it coibqzjn1x)3 uof8 (0jzmes 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 revolut ta zox9 (q1wmu.:x7utions as the car reduces its speed uniformly from 95km/ho w(tmuaq z7x1:9 .utx 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 5dek8 ou.jm-5w6ymfh revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires hk 5 8my-owhem.ufdj65 ave 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 we kudtegd6v1ba /c 6/z*ight on each pedal when climbing a hill. The pedals rotate in a circle of radius 17 cm6bkg/a6u/ z 1vtdcd*e.
(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 ie)c ajeh5x8 n (vlvq43s the magn38lv(j qexavchne54) itude of the torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig.e(h 8 wx6pcclwtq8; d1 8–39. Assume that a fricde h(wq88wc1 plcx6t; tion 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 end9 vks0b znnlz841my; hs of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on thn0zvy; h8b9s 4kmzl 1nis system.

  The bolts on the cylinder head of an engine uf7r(poa8 2*ald9uch require tightening to a torque of 38 *ufapuh ao9lc(7r d28$m \cdot N$ 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 ofn h,n wgy/w*;0ouuiz4 iq j5n/ radius 0.648 m when the axis of rotation is thzioyn5u; gq0w ,iu4 /hn /wnj*rough its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diao ,x4ixn bdu7;0a5xha/e+ )pd+qbfymmeter. The rim and tire have a comibqexmxu+,h/ +40bp 5) a;xad ofyd7nbined 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 q5)7aspvt ,7w l-obyqrod is rotated in a horizontal circle of radius 1.2ao , lpwy57v qt7b-sq) 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 ajdid19*wtj 6 ) oyw;ltt constant angular speed (Fig. 8–42). The friction force between her hands and the clay is 1.5 1)t6ww*;9djjl oy idt 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 inerb3k9h -f g7lxz8j)kx5 9dp ulotia of the array of point objects shown in Fig. 8–43 about h9xupdkk -7l8gzl xj5o b)39f
(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 tnq39dav8, 8d0/ishdy2wucw votal 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, el / ywu435m hmfh)9xmep98bxangineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steadm/ 5m 8u9lxxpwa)yh39hfb me4y 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 and a mhby-ho d 4*j9 zi4sjq*ass of 0.580 kg. Cal bi *hz4yqs-oj4 9djh*culate
(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, acceleratinjg4 eh j 8y1//imestkm/3g1rkg 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 tangent uglaea8-ex*epb)d5js 9 g77mially on a small hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and 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, negle7s d5p-8lg*me ax9e7)e agbjucting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shutmp j+j3by 0)fd (r(.(ogb jiuk off and is eventually brought uniformly to rest by a frictional torque oouk0 gy rjj.3 fp(d)ijbm+b(( f 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 um1p xe m21j1jkqk-u9–45 accelerates a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by thewa)a pa htd/.3 action of the forearm, which rotates about the elbow joip /dtwaaa) h.3nt under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as shown in Fig. 8k /yb9z4j3 :imc8)d(d2vj6vdzxo y di –46. z 82:zcjbiyx 9i(dvd6mdy34 kd/v oj)
(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 h rfw w8b3devc8x ;aw577czj)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 frmk0l w1x 1cec6om rest within four full turns (revolutions) and releases it xw0c6c elm11k at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment of inertia an;nfelma z*+9m7tg; 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 of 28p gxvhqpi/-;:9azmzgd5 +lt) 0 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and ra):,yyeyf4++o: siv ieiqm o x.dius 9.0 cm rolls without slipping down a lane at 3.3 e,ii x+ymsv)qf:ei4yo yo+. : $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with60 3m(rd m vm5gak ,w0ns2gude6l1) k3wn jewk respect to the Sun as the sum of two term5n le2 u6m0mdr (v6s gkwn0 3ww1,j mkk3ad)ges,
(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 andewy q (v tqs2 thtgc7z7v2su 10o;gz-vtq7 -(d a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a sol2(w;vztz2qs oq7t 7vcttv-gueh y1gq-s (0d7id cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg s7s 1;mzhyqwjjjr 8nf21 .baj:tarts from rest and rolls without slipping dhfy7;jm q1nr jsb 18az2:w.jjown 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-75mt s y go()dlc,bnh balanced vertically on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits ty,htsmg c7( -bodn l5)he ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating otq,qy4h ybl;grh9 ;s0px uj+r56x/ixn the end of a thin string in a circle of radius 1.10 m alqr +y5rpb4xq;0 xs,it x;hj/h9y g 6ut an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a z jd:hivq0b- )2.8-kg uniform cylindrical grinding wheel of radius )b hvq-z0jid: 18 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotatsfo ms99xr vaf.+ln, ms9 rpv:j,2v(p ing at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fsvm(2lsp r:s,n.9f x9 vfr vajo+9,pm ig. 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 m6ktv3h1ez f5ly j;6z.gmc j7j oment of inertia by a factor of about 3.5 when changing from the straight 5e6cjzjy3kmgj f. v7th6;l1z 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 ratebz7 g0 j s1k(ufb9gkw3yi.3gt of 1.0 rev every 2.9w1 7.sgbgy3b(k0i3 utfjgkz0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a verovc8jxs2 oa; t5qops y.l*9b;tical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel.c*l9 p;os xaq5j;.ob vy8st2o What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinning at z a(kkm/ vu uh,5ysmup,i8;3t3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of thelchn +t6opu /3 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 inv 82;ywp1 4q6liogedxertia I is dropped onto an identicag 4we 82l6ovidx1;ypql disk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns ;gr4( :spzv xsat 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 rotatingz5)ubxtz.ue pg c*4 h8 merry-go-round platform o 5e4p8zhuuz g c)b*t.xf radius 3.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotatin/rm+q 8i4p dnygckj( rs* h)fzt,(0fbh;y 5kjg freely with an angular velocity of 0.zd(j 8*fb /jsgmyk)kqp 4rh ,n;5tf c+hr0iy(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 eventua:js4ib w +lws7lly collapses into 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 angularlj:7ww+s4 si b momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in excess ofdgb:p 9 +r1 ggkrdzo*4 120 $km/h$ 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 z e oz-9wwrfqc 0d1ceijl8( 58$ 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 p 3c.us *esz3v(niwys +latform, initially at rest, that can rotate freely without frict +3n*s3visuy c(.szwe ion. The moment of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edg (wf 77gkn4(yt,n 6 x1cbgl( *qyinszn4clfl,e of a 6.5-m diameter merry-go-round turntable that is mounted on frictionlesn,i(f7ywlg*t f4lk 167zq n (l 4xcnnb,gc(yss bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the u2;j mfqf evdw(occ-d/:)zq+end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls bo q+/ z-:v;fcqfjmcu (wd 2e)dehind 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 alwah 9,m+mgfp8wbtma:;g ys faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting t8g :mf9gp;b+m whamt,he Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest aln1p8/xf t1 hot a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of rkvkg z6 xvov35j8dh5 wk*v3)f adius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Cv* kdfx8k33gk6 ov5 v zh)vw5jalculate 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 and dtwqe3di5kl;7ldig o w.:t*,q iameter 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 its 1.0-m3wd gt.k ,7wiedlo*it5l q q;:-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 tby 1s f2if *bc9+bgzb v)h*4 ncjl5b qsfag3(2he 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 w1a9 9e7 9m6ikngrw fykg3.rtrith mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravit19gk7tif ranre rwg 9m36yk9.ational 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-pollution automobile is foo34nt ,jtgxb x 2tu *vdtm/+t+r it to use energy stored in a heavy rotating flywheel. Supposx,2d+ tnot3vjgut b+x/*tt 4m e such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates ank/wjc ll05 7he z1-m8apmz,nn $H_2O$ 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 )i;ywsz)4pz/ bes1 jz 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 zbp0j41(psdf k l*(lrxvxm5 6 can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54fx kz1sj b 50ppdx(r64m l(*lv. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing verticacroyd.)9;dal k n9w 6p80u v)ls q;doglly on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step ha8)l ad 9q. 9lo60yk grp u;vnsco;dd)ws height h, where h

  A bicyclist traveling with speeozff2 04ns1h-b,/fd;g7b f s c:sbr6i3xhsx d d v=4.2m/s on a flat road is making a turn with a bs/ro-h 7 nh1bg,3z42;sb6ffsdf dcfx 0six:radius The forces acting on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5 m and begins whirlivc ci3i,p3x0l d: kckq/2ju5o gq/nx1ng the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, 1 p l3:0co 5/qkugkidxn2q xic3/jc ,vand where does the torque come from?    $m \cdot N$

  Model a figure skater’s b vaik0t svqyi2/ akuji; 0afn6- ,:v-kody as a solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, a: yn,-; 2 /6ajkivikivfu-vqst0 k0aand with arms held tightly against her body.   

  You are designing a clutch assembly which consists of h6r*vbx0po yd 794ded two cylindrical plates, of masv0dy9ed*pbr76dh 4o xs $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 trp*r 7rcku *ii72ghqf 8ack of Fig. 8–58. What is the minimum value of ig*k7 7pcfrrh q2u8i* 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 assume2:jeo 6wpjiy; $r\ll R$ h =    (R-r)

  The tires of a car ma yy rr,5idxnq8 bb0/if dq.f9:ke 85 revolutions as the car reduces its speed uniformly from 5i9qq dn8 r.0ffyy, x bdr/i:b90km/h to 60km/h The tires 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