A bicycle odometer (which measures distance traveled) is aoug*,34nj wbvttached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on o,w v4b*uj g3na bicycle with 24-inch wheels?

Suppose a disk rotates at constant angul5 :tp;jfpa8hpxq(lbvn)*2+l nm/a b dar 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 tangential acceleration? For which cases would the magnitude of either component of linear acceleration c ;)bnvlxbq:2t8pm5 dfahpj* p( na/+lhange?

Could a nonrigid body be described by a single value of e1d; uf +ccwf1the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Eb: hcl4i 2 4mlwl4t;zzxplain.

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 z6k08 eb ujy:q:7r snt6/rnna your hands behind your head than when your arms are stretched o7n e/tnaqbj60 y :nrzsru8:6kut in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprocketuc*mbx9.i) wp psl+0k s 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? Why? In which gear is it harder to pediu9*c ks wpl+b.x m0)pal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to 9jjdhgw zp 9jb 43ek+1e64pd)oyb-h krun fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advant4 h94 pkbp3j 1j96d yj)-wok+zeh gdebageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow beaul85 xjm-r:j3rkls* fm?

If the net force on a system is zero, is the net torqww( 5llsd54;l; eba cbue also zero? If the net torque on a syst l ;cb(l5w 4abe;ld5wsem is zero, is the net force zero?

Two inclines have the same height but make different angles with th g5epckr;k-ps9 a2c s/e horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be g kpaeg 5/-2 cs;cprks9reater? Explain.

Two solid spheres sieabi0 hy 9n/1f*)ag nemultaneously start rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the b/yfi1 e hn*n ag)a09egreater total kinetic energy at the bottom?

A sphere and a cylinder hxhl zrvx/90 )lave the same radius and 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? Whicx 9hz)v lx0l/rh has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most moving or /qkgp(l8jdo1r;p5q rtcsh6te :0 w e2rotating objects eventualgq :tjd/ wkrehe 5 r8(1 p;p26s0cltqoly slow down and stop. Explain.

If there were a great migration of p 9 57uhjlm9jgheople toward the Earth’s equator, how would this affect the length7j jh99luh g5m of the day?

Can the diver of Fig. 8–29 do a somersault without having any inix 7- by.kqfra5tial rotation whbyf 5k-r axq.7en she leaves the board?

The moment of inertia of a rotating solid disk about an axis thrl+eo s .ekk v5zhabbn 3-o, r8-*szsf*ough its center of ma al-seh**vn es+s ,kr3z5.z-fb obok8ss 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 houj0rb s-p. 2m1oo5-16pg*gvcdg oycslding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the sbocs65.ovg-gd 2g-0op m1 r jc*yp1usame? Explain.

Two spheres look identical and have the same mass. Horqnbd- djw 7u zzf8i-1sq 965xwever, one is hollow and the 1zi dq 59z6qj- wus8ndfx7b -rother is solid. Describe an experiment to determine which is which.

The angular velocity+e5hj)-g ymp brc 9m6r of a wheel rotating on a horizontal axle points west. In what direction is the linear vrygp+5 r6h j)e mb-cm9elocity 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 speed increasing or decreasing?

Suppose you are standing on th.hz e 0rfjjt.suhnnx04w 6l1k3u8: mae edge of a large freely rotating turntable. What happef1 kj s6wrh 8hu0xanmt .3 .juenzl0:4ns if you walk toward the center?

A shortstop may leap into the air to catch a ball and throw it quickly. As he thz1 xj u,xm+9hyrows the ball, the upper part of his body rotatesy ,xxz+1jm uh9. If you look 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 conserva o 0qv:8tr5tls )5e taldcd*9ytion of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter say*q:8 so5 9 )5drct 0dletvlttable.

  Express the following c-lpn/c67gi8xn w,qz angles in 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 becaus )0vg8exwm7m ee of an amazing coincidence. Calculate, using the information inside the Front C e0m7)x emwg8vover, 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, 380,000 km from Earth. The bea cy0h 7rmfxp8 )z4 -ucinorw0,m diverges at a c w r0 -8,czip7ouynh)04mxfrn 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 q7sz(i3c* lwh motor is turned off during operation, the blades slow to rest in 3.0 s. Whahc*7ilz( s wq3t 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 cw5u5 8 4-at zddejv2s;eah/lz d;uy k.hild. If the ball makes 15.0 r2l zy.;s-due;e8 uw5 v5/kdth ja 4adzevolutions, what is its diameter?    m

  A bicycle with tires 68 cm in ltsun+r5 ;k/ vdiameter travels 8.0 km. How many revolutions do the wheels ma ;rul5n/stv+k ke?    $rev$

  (a) A grinding wheel 0.35 m in diamet 3a1cb4ev t*auer rotates at 2500 rpm. Calculate its angular vele 3uv aabt41*cocity 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 01 xl je8oh3fkmakes one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seatedlex08ho3kjf1 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 angula:0qn9 )3bsf.e(gt vgeig-u,v xh ,jncr velocity of the Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear sp4oh b0y(e-ugg,u4lu geed 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 7.0 cm from thets 5,v q u50gy-g*)zjnm,nzdj axis of rotation is to experience an j*ug dnm 5qy gvz, ,05)tjnsz-acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center fr x*5cki-/how.m2fc u/aw)w zqom 130 rpm to 280 rpm in 4.0 sh zw) w5ocwak2*qm/.i xc/ u-f. 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 radius6+0xf dr z5yjt $R_1$ 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 Mo )fm2;1qh.o:bkp gtbton, 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 bmkt2q o. f)b;pg1th: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 fromv7+d z)u sn mb1y2qgmx9s0r k. rest to 15,000 rpm in 220 s. Through how many revolutions did it turn in thbx n21m7ks0q)zmsdry. v+9ug is time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5pgh x5rqk n xa*0)d kmjncky 6x9a)d-b+p584d 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 wqpwutz9b.*,hu 9yd v/vdzv *;hirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its fina ,zvyqu;w/ 9* vp9du .tdvb*hzl 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 accele ,.j4gacyf7ajpi q6q .rates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have tr7fa,cp q.qa46ij y.g javeled in this time?    m

  A cooling fan is turned off when it is running at 850rev/mik7 us q.df-y)w2 *pzjan It turns 1500 revolutions beps2wdfaqz *k)7 .ujy-fore 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 6wwrl o g8 -on3(q/zuhthe car reduces its speed uniformly from 95km/hg w o-6r wo(3z8qln/uh to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces icvn: ):* fhrjyts speed uniformly from 95km/h to 45km/rj c):: yfvhn*h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts axq rwq pv9ag-d;g, j61ll her weight on each pedal when climbing a hill. The pedals rotate in a circle of radiagg x6q91- ;, rdvqwpjus 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 55u1/n h ,l,nx cxa+)n 2 cg0sv8ozenu.z N on the end of a door 74 cm wide. What is the magnitudeczvg0l h+oc )nx/zs.e,a1,2x u8unn n 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. 86 uj 7kmh8rn5f–39. Assume that a frfhu5r7 8jm kn6iction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a massless rod w cru/cn( 7)h tx 3m.xsol+w/mehich pivots as shown in Fig. 8–40. Initially the rod is held i xcwr(m+/cxm /3ueltno.7s)hn the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an eng (mywx23c,efa ine require tightening to a torque of 38 ace xf2(mwy3, $m \cdot N$ 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 in ht22z1qtj)e*ug.v ngertia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is through12ezjhvg uq.2tng *)t its center.    $kg \cdot m^2$

  Calculate the moment okof op5 4ri*1ot*ma/ , ygmt;rf inertia of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined ,mipmo5o1r ttk/*y* f4 ra;gomass 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, ligh kmkieh-stc. 9v;o;e3t rod is rotated in a horizontal circle of radck k3;es .e9 oi;-htmvius 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 pottsv+;rk(o5hmk x 6 2kbfer’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between herf65m;(rh2+kkvs kxo b 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 objects shown in Fig. 8–43xn m2 8* /ft(uni- u0sfh0eooy abohx8ef - n imo*0 o/utufsy2(n0ut
(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 atomscaj.u .g d9j54is0eek whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylind fd3 o/8-zzznpnv1q ,hrical satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach -fho,n3z 1znd /8zvpq32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uni3 aasmk -b+w8vform cylinder with a radius of 8.50 cm and a mass of 0.580 kg. Calculate -b+ kaam8wvs3
(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 batpb 3n2p:pnviw 8ws.pa892 aus , 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 tae l /7rwl2atbcz,t s2*ngentially 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 2c *slzt2l /erb,7tawthe 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 rotqq. s 4ar hvd4;,gu8 db.qfiy+ating at 10,300 rpm is shut off and is eventually brought undf,4b qvuq8 y 4idsgh.q +;.ariformly to rest by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a 3.6-kg balppipb6zgj nw.9x c-bf;0 0 w5bl 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 actsybc3 b:k f4-ehgr7yi2 yt (1dion of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelyyr3ib-hc2 : (b74e1kd y gsfterated 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.r+.sakhqzfz(4 v/ x; g 8–46..(q;rx+hf4zzvag ks /
(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 con,-v: .mfxqnmp sists 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 f,9xjoo.xru/ from rest within four full turns (revolutions) and releases it at a speed , uj/oxxr fo.9of 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 miu(9yatg*t*g xgv t0 c)9c*l goment 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 pi :/7x mj1la)ssqq9u 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 withoy e-2ah( w(eanut slipping down a lane at 3.3(a2e w na(y-eh $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun astuu kp)(ff)2 j the sum of two terms,)) ujfpu2f(k t
(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. Ho z t4 ;m,li-27 3i tnjlvxb/dppc(xh6aw much net work is required to acce t- zvpnpcd4lxlx 2j h(btim;7 a,6i3/lerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg st m)pwf3o:tfb pizg) )+arts from rest and rolls without sib))gp:z otf) fm p+3wlipping down a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is balanced vertically on its tip. It sta4qpo21 c vqfl8rts to fall and its lower end does not slip. What will be the speed of the upper end of the polpclv4 foq2q 18e just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.2 eh:l d n3e1;qbh.c+uy10-kg ball rotating on the end of a thin string in a circ+.e 1lb3decn :y;qhu hle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum *jx:mak m (h qpz1a9-dd;mjh3 of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotatih;d *k(qmpm: hja3xmd 19az-jng 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 8svk 6)fo+ 1rxo9qtj xi82j wqis rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases to 0. qf rsxv o)wi+x8k1oj6j8q2 t98 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduc: b*tvyre;mx1 gaw y*6e her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular*y be m1v6yag:*;tw xr speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increi 40e-ve7 vy3r;pea d.j-sacj ase her spin rotation rate from an initial rate of 1.0 rev every 2.0 s e37 0r- vi-;e4yescj .avdaj pto 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 c 1yithf 85h-tbm *c1ybenter at a frequency of 1.5rev/s The wheel can be considered a uniform disk51y *hcb1h t -tmbiyf8 of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinningv/z3pp8xmdt 7 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 momentum of xg* nl 32okp5aqj94cae)l xg.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 onyn)i2(*fspz8c 4orm ; .exmhpto an identical disk r2rm*e(zh)y 48of. c; nmp spixotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns s:.4dzh0z5;hzi i pve at 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotating merry-go-round p3rjbh 61sbp x)latform of radius 3.0 m and j1bpbxs )hr63 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 rotating freely with an angular veloci1tj/v qfx rnlctxf7:c, puj5l,c;o7 (ty of 0t 1/fr 7c5,tjxq( :vu ,clloj7cpfx;n.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 fgjy 9shpf:(o ( o8t*o+z+zpginto a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotzspf9t o 8p +gf*yh:z+og(o(jation 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 udt)k3it3 0cxwinds in excess of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass 5zw-f)k t/tdt;ebv)4zusfd/ tohzm:h 3 1tz/$ 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, initiaw7 :7ynfh zr7nlly at rest, that can rotate freely without friction. The moment of inertia of the person plus the platform is n7wnzfy:h 77r$I_P$ 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 merryjcubu 8 /jb:-k-go-round turntable that is mounted on frictionless bearings and has a moment of inertia of bju8: jkc /bu- 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 th,h;c,aic32 i l db(a y/uys uw:v:oc8pe ground with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. 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,cw icl /sp 2v3: ,dy(ui:c;hyouaa8b move?

  The Moon orbits the Earq8iq ktuvkk3d+ 1nwmrq; 7+c(th such that the same side 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 as a particle orbiting the Eqk1vt;k7qkn8 rid umq(++c 3warth.)    $\times10^{ -6}$

  A cyclist accelerates bu x, od-4js*nfrom 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 ofvn zr0cp 0i;u6 a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by tznvu0 r;06pic he motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disb,,((f 6pry,e3t:xsr vg ni p19upptqks, each of mass 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 conservati9p3snb1:rrpp(pq i eutg(6,x,vf y t,on 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, ho8ybj 8kx.gr .ww is 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 wi: kp b(qo6xasw.c179jpks ) 3)iutmmijxre.6th mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. Ikijku csei6p:6q3xa17r.b(x jw ms)p9 )o.tmf 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 mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use energy stored inad kwjx*0rkz wr wka+:b4+v ga54qj* 2 a heavy rotating flywheel. Suppose such a car has a total mass of 1400g4r0dwq w 4a5av xaj+ rk*jkzkw*:+b2 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 ph2vgd*4f(jb 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 (hoojb l+g(r 6he4rp) is rolling on 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 o)o:qlu7)a 4 pee kp42sb7v bipivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horiz:pkoev72eqil 7 )4o4p bsa)buontally 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 vertically on the floor, pjt ,w4lmz5 l nw+.*qfand we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, wh,.qwj wt* 4p +lzn5flmere h

  A bicyclist traveling with speed v=4.2m/s on a 0se; vwu;p1qkh j-9re flat road is making a turn with a radius The forces acting on the s je eu ;0v-1r9pw;qhkcyclist 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 2 , 8uhze:dj(ecbue0rwhx.wdd .3p2g 9qq*resling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelez ,w2 dg:8j(h*u0r.2eddew req x3e9qc hbpu.rating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s bod0m-9ys crg82ld *0chdefgci + y 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, and with arms held tightly against her body. 9+d g-r0im0 cy2sg*e8dhc cfl  

  You are designing a clutcmbj26evu 0 jx1eq;:tfcfsq3 5 h assembly which consists of two cylindrical plates, of1:u6xvj c0b5;tme 3sq2qfj ef 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 q2/whfjq is62n9 afw1nd p1t/ radius r rolls along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leavinj i1h26pnqqwtsad1nf w/9 2 f/g the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not sf8;j 3pcv9qh assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces ita4k* mmdgej8e:(w.9l rj3hv rt;+ zgjs speed uniformly from 90km/h t(mr.d3z: 8ltmj+*g 4 awgkjvje; h r9eo 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