A bicycle odometer (which measures distancw dzfpm0(ww2sa5 7y 9le traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle wit f m2zw9ls(apwy7d5 0wh 24-inch wheels?

Suppose a disk rotates at constant angb f r,vgo) rbo.ek6i/,ular velocity. 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 tangentia/ ,e. i) r,gbvo6bkfrol acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body uy /(6fhkxrb3be described by a single value of the angular velocity $\omega$ Explain.

Can a small force evle7mjrtd - )isel) t03er exert a 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 a sit-up with your hands behind your head tha:p wid6 79 mgigan 4nymm--.orn when your arms are stretched out in front of you? A diagram may help you to answer tny gmw6-i.9nr4i 7dm:-moagp his.

A 21-speed bicycle has seven sprockets2y 1 beu/*il5dts v0xm.ela l6 at the rear wheel and three at the pedal cranks. In whichi2ulds1b t.m*e06laly5e vx / 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 slender lower legs withe8w 2dm i:5cduhsm e67 flesh and muscle concentrated high, close to the body (Fig. 8–34). On t86iw2mm h5du7ecd: eshe basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a lrx,s o iq:x2cy:e1 q5*vwphz1ong, narrow beam?

If the net force on a system is zero, is the net torque also zero? If the net to 3+5)j)qx- dtho mxywlrque on a system iwxd 3-yo jxml)+5htq) s zero, is the net force zero?

Two inclines have the same height but make different an12.1trbz rrw jgles with the horizontal. The same steel ball is rolled down each incline. On which incline will the spwtr2 zrj.br11eed of the ball at the bottom be greater? Explain.

Two solid spheres simultane()e5wuutd ; ggously 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 theeg;d5(u )gwtu 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 an2ph0 fn hk-3oz49db lbd the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic ener4dboz 2fhp k lh09b-3ngy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Y4f4roa b29/cak0yzdeg uk0:n et most moving or rotating objects eventually slow down and st9/a ug:0o4f0y2kcba r en4kdzop. Explain.

If there were a great migration of people toward the Ears2i to .h15z9l/ qqo:bpbtfj+th’s equator, how would this affec 1fqh5jl it29o .b:+ot/bsz pqt the length of the day?

Can the diver of Fig. 8–29 do a somersault without having any initial rotatione5w62l483yhm-posjl/erbhrm 5g- af wh6yr fbr--l58em hjsm/ wp4 e25log3ahen she leaves the board?

The moment of inertia of a rotating solid disk about and.):vptfn x 9m axis through its center o.) xtd fpv:m9nf 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 rotatin9a ;m g20ee)c viytcs4g stool holding a 2-kg mass in each outstretched hand. If you suddenly dropegaccmyi2 )9t4v s;0e the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is ( nohk(m, ,ckwhollow and the other is solid. Descrin(,kwo c(khm ,be an experiment to determine which is which.

The angular velocity of a wheel r**h.ix49ykvbk oe k0rotating 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 acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speebvhk roi.ky0x **ke94d increasing or decreasing?

Suppose you are standing on i.xn o w03 :xwoe3eudnzo i19*the edge of a large freely rotating turntable. What happens if you walk toduozwi xniox *eow9e0 .: 3n13ward the center?

A shortstop may leap into the air to catch a ball and throw it quickly(5ti spwk;2 zd. As he throws the ball, the upper part of his body rotates. If you look quickly you willw2ks( p5 itzd; notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the la+u- nb5keilld,b w5f;w of conservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or m5u+ 5 b;fll-eibwn, dkore ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radian+e7 izjo +4+ ymwn0zhfs: (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, using the5wpah rhji:zf c-z;14x-lf* l information insi - pi1l4-fzxzl:hj f5hcwra*; de 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, 3oi cy* /./wf2yo(lyufx b4 a2p80,000 km from Earth. The beam diverges at an/y4 .yy2cf x (ooib*aufw p2l/ 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 rpm. When the motor is turn5l uij)e4k g.dyeg40ned off during operation, the blades slow to rest in 3.0 s. What is the angn elk5u4 .eg4yj)dgi0ular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball gyjv ippex8,9ce7w.dg*w. e , on a level floor 3.5 m to another child. If the ball makes 15.0 revolutions, what is its di .g7pe.jvi,c p9 yedw*xweg8 ,ameter?    m

  A bicycle with tires 56p9s (epsg*x8 nda nh68 cm in diameter travels 8.0 km. How many revolutions do the w8s*eg hsap(x9dpn 6n5heels make?    $rev$

  (a) A grinding wheel 0.35 m in diat1;zhs50 rmm3w , cr 6g*maillmeter rotates at 2500 rpm. Calculate its angular velocr whca3*mml5 tz,i;l16r msg 0ity 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 comp ul,ho4 -yf/1tg fx 4yjvck/)ylete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from the centeoyfg -kuyfhy , 4xvt4 jc)//1lr?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Ea6oaksp a37l(: q8i whjrth (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 p/s q3 9qs94ppk5eciexoint
(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 pwjwike, st w: k.e-e7,article 7.0 cm from the axis of rotation is to experience an acceleration of 1,jw-,7 kwe w.eetsi :k00,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniforzgc/:pf,sh* g zg9vt6 mly about its center from 130 rpm to 280 rpm in 4.0 s. Deterv:g g6cps* t,hz/ f9zgmine
(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 radiunsm4qdz q(3 pmub98w;n319zmf j/fb n s $R_1$ 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 abb(uk 7 d-/rqeroard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy eve k 7(u-rdbq/renly. 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 diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,0z9h uih)k+8o agon-0 czm6,kb00 rpm in 220 s. Through how many revolutions did it turn in thicohkk+a8uzb6i h-9 m0gz ,)on s time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5hc;-7cv(j mrmcmu pp,,1 hcy. 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 a whivc 7qa qsp;4r:rling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolution4rqp c;vs:7aqs 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 accelerat4 ql:hj bx,:xhes uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the whehhx 4b: q:x,jlel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 150fn/z 50 ml.i,vec:ilj ms vf*.0 revolutions befoj5/mn.mfsfi c0 llz*: vv, e.ire it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reducfq 7 oxu6a /bwi0((kwues its speed uniformly from 95km/h to 45km/h The tires have a dk/qu( w(ai7wx60fub oiameter 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 l vwix)s,u1 k-as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of )s xul-,1ivwk 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)xk a6n1 ahla3phh 3y: puts all her weight on each pedal when climbing a hill. The pedals rotate in a circle h3hh a6p :3ay1)axnl kof radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of gs0rgeeyk - 27a door 74 cm wide. What is the magnituyg0eke-27gr s de 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. 8–39. Asak dom38u,d 5n) iscj4e1 c/vksume that a j13)v m ceus,k8ad i5kcn/ 4odfriction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a massless rod7s hodl9 k)hy6 which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal p yd7 6)skolh9hosition and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine rik tw;hbh(2u 8rnk4gx;u 2bi .equire tightening to a torque of 38ux.;2r (h;i ubg482h iwnkbt k $m \cdot N$ 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 inertiap3x-ot qgc f0:bo (4mi of a 10.8-kg sphere of radius 0.648 m when the abfto 4po0cg:(3i -mxqxis of rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bick q-6mym kwe ie3qaq52+rq;b+ycle wheel 66.7 cm in diameter. The rim and tire have aqmq36w-5iqa q2 ;bemkk +yer+ combined 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 rotated in a horizontalx -pdzye v0jh-gr qz1(q)m7*b circrvmz-* xdh1p7(ej qbz0-) ygqle of radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating at constactm;v7sijce13ru s2i73 /thvnt angular speed (Fig. 8–42). The friction h3;rvec1ci3 /isv s tt7m7u2jforce 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 inerv 4f5./ ijgnfitia of the array of point objects shown in Fig. 8–43 ab/i n ivf5.fj4gout
(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 m ej3k:( iyww(vass 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 h:x9v v-h8hbirate, engineers fire four tangential rockets as s 9vhxh b8-h:vihown 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 ar- bp d;:oceg57 d(cnof 8.50 cm and a mass of 0.580 k: o; geb(dr-ac7dc np5g. Calculate
(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, accelerating it from res/2w6cbwe byr:t 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 mer1c((rbaim-ka;vr5tbwb- uq/ry-go-round and is able to accelerate it/i (b b(;5r-akq- 1 ctmuabwrv 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, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10o5tkva-*n(r0np* 9l6br kqh d,300 rpm is shut off and is eventually brought uniformly to rest by a fricanb k0lqrkvp(t9 o-** hndr56 tional 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 accele1u 1o8t /k5v *b-zey/oc hcsesrates 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 the aempqqsuv,. ny:-vq3 *ction of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. s:q qv*up,3ey v.qmn-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 burup))a/r+fsp 6n0k de considered a long thin rod, as shown in Fig. 8–4pndka0s+ uu rf6pr))/ 6.
(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 tw g *f/,3l tfe4ktk.hhgo 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 hammegma(ixah,7q; p8pn *ea4c)z p r from rest within four full turns (revolutions) and releases it at a speed of 28 a p4 mhpecqp(;,x *)niza78ga$m/s$ 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 momruxl rj7gqgl8a 5a r(+r0y1n 9ent 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 torq 97 phc-eovrwu:y; p5que 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 radiusc4le og,j z)bga+ cd)s4szb 2: 9.0 cm rolls without slipping down a lane at sz oe)jc4s+g b:d4g b2a,lz)c 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as th- 8bbyoak+jkx rkp2q (d*7;2 .wcg hlee sum of twoe;kr+kyqwg(2 l2ja.c bhod*kbxp 8- 7 terms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius of 7.50 m. How much 8 +1auhfxz;um5fvvuc-l5eb8 5 bow* wnet work is required to accelerate it from rest to a rotation rate of 1.00 re58h uwfufzvv1bxo 5+8 5lbmc-eu*a;wvolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg startsom1 urk+0xw-srja l 02xnpa(1 from rest and rolls without slipping dowjn -x wrx u0am1+(sa0kr12pl on 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 utzpqjx 2a8 b2**.awj8c gws1nbh 6w*its tip. It starts to fall and its 2 zp w88.h1*xgjqaw *nt6j cws a2ubb*lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the++9gsiw,d ).e p owbc4zj rv+e end of a thin string in a circle of radius 1.10 mcozi +geb+rpw. wj ),d4sve9 + at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular mom*1vbod(wm oc ls5 1d/mentum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when w ocbm5v/1*ld (ds1 omrotating 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 rotating at a rate ofgkbio30gm. /7lg xi.k 1.3rev/s If hi/ixk.m3 o0kg.lgbg 7e 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) cyzt 28sx23sbtan reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1 2s2t3tyb8szx.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 f/* nbntwc fwy,85 +oxxrom an initial rate of 1.0 ,n/c 5 woxbt*wyx8+n frev every 2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis through its cbe2anb3m9h0n 0rm jn:*,ks h b jjfi2(enter 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 clayr(jemfjbann3 0:n h b2 *m2bk ,j0h9is, 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 mp1clscnu 9c,gz ;)4/db0mtg n omentum of a figure skater spinning 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 moment, qbg nv:t1cw8 +pu,dl f+oe/ *qh(hnium 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 o 0wie(pz4fu l.c2*g wyl+jr:k nto an identical disk rotating a lgfiep. :ur2(4wy +*wzjkl0ct angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns af eeza1l; 3 7ldw0mmy7r9qp;gt 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotating merry-u 5br7 ,6 -emzus1pzgjgo-round platform of radius 3.0 m and moment of inertia 92g 7e 6spj-5mzu b1rz,u0 $kg \cdot m^2$ 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 velocity o- g)r,2gc0+a qg;x qn8 zadiycf 0.8gqig +)8n,dc-2yx0 ;racz ga q $rad/s$ 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 wujm-5a a9bt5 fhite dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming fj9b5 -amu5ta the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in exz7;y(crz hq8op; z-gu cess 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 ki5 nyg8ng j;z kf6n,ag6bh z7p0x1/n$ 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 rotate frk 6och:g3y 9g-x9tu kxpkx:-geely without friction. The moment of iny-u9xg tk93k 6: cgpxhgx -:koertia 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 edge of a 6.5-m diameter merry-go-round t)dw4 ;+kdc3/gz4yed hway 6mrurntable that is mounted on frictionless bearings and hasmay +wy/ c64zdd dwgr 4hke3); 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 rop-hw 37jqk5m58 g 611 f9hlvaakfacz bye rolls on the ground with the end of the rope lying on the top edge of the spool. A person grabs gwf6qyk8 1j5 a1a5vzfc3-mlh9bha7kthe 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 :9w kzaobs7,k,arn5q Earth. Determine the ratio of the Moon’s spin angular momentuqsa,o ka,k 9n:w57zbr m (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerategt6:swp) n)dil6s q-qs from rest at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of radivwm- 4if2 mmexunzj7e 64b;3aus 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angulaa47 mejfivx4ue 6zwmbn2-; m3r 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;)rzdbe 1m5mj and diamet e j5drm1z;)mber 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-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of the rear wheel lupv(w(zdpulq 4i +4+ ($\omega_R$) 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, wera(v zl+*w-yq i65e- y bfaidv8e 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? Assume that the star is a uniform sphere at all times, and that the lost maeqa6ili+v-8f-w 5y*za(dyv bss carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is 4prnt3 hduy+0 for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 14+4hrnu0 dp ty300 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 anfb/k 6hu:b2kd $H_2O$ 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 w.)5ym) rpk uo*pue8 9ulmr,ion 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 frictioc6n 8ky4( fefe c li*mta*a8s;n) about a hinge attached to a wall, as in Fig. 6tle(; afk4cnmcayisf88e**8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the 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 verticayczts a axt,6pn6 2+a:xbx 0af588 ciplly on the floor, and we want to exert a horizontal force F at its axle so that it wnta: 2 x6baspipx+86cy c,a05ztf8xa ill climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling:d 7u2 yoai5 f8lt4ikl with speed v=4.2m/s on a flat road is making a turn with a radius The for4tdo8uyl 2ka:7l fii5ces 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 begin /w 2es;a/epvws whirling the rock in a nearly horizontal circle above his head, accelerating e/w;p /wsea2vit from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as s4 )m 1rprva3lx4euh e8c+ 3ulthin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and witau4 m14u s3pxrlchleev +r8 3)h arms held tightly against her body.   

  You are designing a clutch assembly which con5e 7ibkbio0e/ sists of two cylindrical plates, of mass /ebo5k7b 0ei i$M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 F nem:ri+w:z4s ig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leavinz:r n:4 +mwsieg the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assuu3sp q*j z(hl +r+xna3me $r\ll R$ h =    (R-r)

  The tires of a car make 85 revoluta4jtm g*j(rp 8ions as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90a j(r*t j8pmg4 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