A bicycle odometer (which measures distance traveled) is attached near th h-mdyq, e6 jf2 frp5.2fn8ukae wheel hub and is designed for 27-inch wheels. What happens if yqadr j m-n2 fpff8k.,eu65h2you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Doesy.r iuj.x-*y8zeap4.)am cm e a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does thmjayi4m. p.zu* y eec8).arx-e point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the angular vel3+cxxe(jx7h , y /nid*;c mnkuocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? e o*jrw,*c1vfnjpd0x*i m f-0Explain.

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).q ha dj9wp 3up)x1wj with your hands behind your head than when your arms are stretched out in front of you? A dujjpq3))a91 hx.wpdw iagram may help you to answer this.

A 21-speed bicycle has seven sprocke;+hsw;5axz r jz67yvc j,c -hmd xs4;jts at the rear wheel and three at the pedal cr;+ch; swmh c4jsvz yjr x, -adz7;5x6janks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legspx .vl8p d ;hw:9h-frs 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 hr.sw 8xfld 9-p:pv ;his advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrot q 2*df)eoqxvc o.,-pw beam?

If the net force on a system is zero, is the net torque also zero? Iff2 )b7rccv93hywfv a/ the net torque on a system is zero, is the net forw3hbf/ f cvv72)a9cr yce zero?

Two inclines have the sa 1z fm:gc9 k-ka 4i pp,u2et11qwywa0gme height but make different angles with the horizontal. The same steel ball is rocg e1y0, u1gkw m9a 1-ift4:wppqk 2azlled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rpqd0*6se mod(lm1g;()+v mlxmg bil*:os6rfolling (from rest) down an incline. One sphere has twice :mq ml (;f(pgsl*6d)1m ldg b* sir+xv m0oe6othe 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 greater total kinetic energy at the bottom?

A sphere and a cylinder have the sag6 mdt:* a6q.fpk e.xsme radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first g* dtpsa6qx.6:k. mef? Which 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 mhrt : ro8dl7(tomentum are conserved. Yet most moving or rotating r8t( :7drohlt objects eventually slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equator, how woul hgx(dy1-8v, congaa1 d this affect the length of8-v,d (1hc1 ngaxgyao the day?

Can the diver of Fig. 8–29 do a somersault with:gjg.gc rk0fy 4;;ixrout having any initial rotation when she leaves the bg r4:xfc.; gygkir0;j oard?

The moment of inertia of a rotating solid disk ab38e+v jl)7nhyh6j lufout an axis through its center of m)y l 78 vl3+hjh6fujenass is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg mass inx.h(s4j 68c;lawvtlj each outstretched hand. If you suddenly drop the masses, wtw sj;lc4l6(jx8. ah vill your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hollow a u8 lj/inw7cx4nd the other is solid. Describe an experiment to determine which is4 incx8l jwu7/ which.

The angular velocity of a wheel rotating on a horizontal axl p i64wsij4b::-rlbq 3q-rhbje points west. In what direction is the lineabbi qhi3 :4rq p4-6-:bwlj sjrr 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 speed increasing or decreasing?

Suppose you are standing on thhqh u+ucm1l* -e edge of a large freely rotating turntable. What happens if you walk towar lmhhu cu1-*+qd the center?

A shortstop may leap into the air to catch a bdkp*59ck1b l jall and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hd bkc*59jk1lp ips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of consere7td uv) 2jn/quw*ef7vation 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 helicuj7fuw*q)e/t2 evdn7 opter stable.

  Express the following c /i3 ec4onul4angles 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 because of an amazing coincidence. d 9l d,wyddpwe jq)g4 ):ye);gCalculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Eyd:)g4, p ;eq))dew wgldy d9jarth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is direlv4 3wo/ :kxz3lf69bfpgv9 llcted at the Moon, 380,000 km from Earth. The beam diverges at l94llwgv6x:p 9b/3fflzv3 ko an angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. When th7kz ;;g8 mvsupe motor is turned off during operation, the blades slow to rest in g;zv 7k8u p;sm3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3. uxa7jr*t)e11g r2zftcwr98q 62ssn j5 m to another child. If the ball makes 15.0 revolutions, what is its eqctg*t )j9 rr2a1fnj 8x7s6ru2swz1diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How n7h.:7 l.bi/nhsic7 epw k*ge many revolutions do the wheelsib 7.e.ng ip:n7ewl7/ khcsh * make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate igvs.s7. hec7 gts angular velocishgv7sg7c ..ety 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 comple;ksuwhj(eq5hx15 t y 1te revolution in 4.0 s (Fig. 8–38). (a) What is thee1 5y 5sw1h(u;kjqhtx 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 velociy g *+bo78eylhywz8sd0e 1qf -ty 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 speed of q13:5e.tbbl p rg;p xsa 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) muuszglox4a) z- hq6ysv*c9 /xb1l1 ad(st a centrifuge rotate if a particle 7.0 cm from the axis of rotation -davc9 q1yzb1szllu as 6x)o( gx4h*/is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly aboj1y atu03rymnq01ne uqx ,u7 8ut its center from 130 rpm to 280 rpm in 4.u uj8x 1te0n73a0 uq1nyqry,m0 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 radiust :g)d/enoe/ l $R_1$ 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 themsenu yxt07zrtx )2a5 z-9h;ml falves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no 5l)n2hmuaa9- zzf xyt0 ;x 7rtrotation 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 acceler 5dhv3d8szkr/5 q/q,sldh o4 sates uniformly from rest to 15,000 rpm in 220 s. Through how many revolutions did it turn in tv /ks,54rdd q8qzh/os d3l5sh his time?    $rev$

  An automobile engine slows down from 4500dyj(44i psw)n rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for +60x dmciq g7;jbd2ogthe stresses of flying highspeed jets in a whirling “human centrifuge,” which t2cxm70i+qg dgo 6j;bdakes 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 3rw tl9y0kjm +d( +hm4r;qkx+t60 rpm in 6.5 s. How far wid t+ym(xk9j;0mlr +h t4kq wr+ll a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It cb3xncy egy4+; sce7,0z qp8nturns 1500 revolutions y74x0g np;be, e8qcyc+cs n3zbefore 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 coed 28b475 ek*k pffwas the car reduces its speed uniformly from 95km/h to 45km/h The tires hkwcfd bk42e*oe78pf 5 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

  The tires of a car make 65 revolutions as the car reduces its speedfc+muo-u rpu a/nvoq7r -tun5(03nj4 uniformly from 95km/h to 45km/h The tires have a diamet5/-4uavu ro-nof jn t7qmu upr3cn+0(er of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight oxymyw v;sfud jc.9:51ok-, chn each pedal when climbing a hill. The pedals rotate in a circle of radius ,k;djcc.s5my1 ywuf o9- xvh: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 odokkuro 6:5v ,s4jl o +rb*;sxn the end of a door 74 cm wide. What is the magnitude of the torque if the force is exertovs,+k4;lduxb o r5ks:r j*o6ed
(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 thzuwyb-*(kvi rvm7r:2 mv5j0 ve axle of the wheel shown in Fig. 8–39. Assume that a fw7 u mi*r-0vy:b m5jr2( zvkvvriction 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 masssvsju/;7y 3gqi4ri1m less 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 net3rjq14;7uivmy/ss ig torque on this system.

  The bolts on the cylinder head of an engine require tightening to a tor5 svhs3d3hi 5xque of 38 5hid3 s sv3xh5$m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radius 0.648 m w9b7.jev;zu; yahr 6wh+(d ji xhen the axis of rotation is t; zjbi yvh7+xj. wh dau96;(erhrough its center.    $kg \cdot m^2$

  Calculate the moment of inertia o,vs5)ddp9 7edk5( dd3z zckpzf a bicycle wheel 66.7 cm in diameter. The rim and tir3p7 (c zsevzk ),kpdd55zd9dd e have a 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 zgy+iwsq0,v w 0e2w6fthe end of a thin, light rod is rotated in a horizontal circle of radi620 z0vyswg+ww eq fi,us 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 ro t-w vo)kg1ildkzfme/76my +:tating at constant angular speed (Fig. 8–42). The friction force betw7vm f/dkomt:ei -g)k+6 yz1wleen her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objemp1jvrb / mm2t r +yvei/1v)a8cig .6jcts shown in Fig. 8–43 abo8r6.g)+eii m1pt2/vrab1 jycm/vj vmut
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen atoms whose total mass o* 9ocl qur0j;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 ratewkaw 1le*d ayf :*j0z+, engineers fire four tangential rockets as shown in Fig. aaw+lw k1:y*e0*fjd z8–44. 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 witdrxt(yo3ag b)fl)/. d(i vkwpt5*o0 msrwr4 3h a radius of 8.50 cm and a mass of 0.580 kg. x5bsa. lvwm)o3w gd/4 otrk(id3rtp0*) (ry fCalculate
(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 rem)n)j(w 6f inlst 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 smallcc(,,6kb9.w hprywqx 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, negw,(r w9ch 6kq,cb x.yplecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is eventually ,by 9btvuc6s*d9ctd7( i fy 6vbrought uniformly to rest by a frictional torque of 1.2 9tu 7v(,d *6d sifc6ycvbb y9t$m \cdot N$ 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. kk- ilm5g973y x5wzijn 04 qcp8–45 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 the action of the mw+b9c s,w,f3-xozcq forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8q sf,9bw,wx-m zc3+oc–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 ro0 c4 zpumotqo8228ckkd, as shown in Fig. 8–46.kcm2 qpk4zt88oo0 2c u
(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 cfzgh2+mx1 .sr 4enk3xl3;iuof 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 fromd vxzr,j0 62xh rest within four full turns (revolutions) 6 djhzx, v2rx0and releases it 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 o9 smjs*a)nhw3 r h83w:fnf9t bf $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine devel2e1.9x,e aeq gb5npljops a 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 radiuatrpdj7djk:r.k38 fwl/u(qq uq1 n4 (s 9.0 cm rolls without slipping down a lane at 3.3 jdduklun q 3w7. rt/4k(fqj 1pra:8q ($m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with9g+ .xhlsr b4f respect to the Sun as the sum of two9h4.r gsxblf+ 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 raff cww1yk8qjp(w2 1e( dius 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 i fyfew(kp81cqww 1 (2js a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg star3vk-s ( kahz-wtw*oh: ts from rest and rolls without slipping d:ko swh-hk(*wv3t-azown 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 stalxh7ez 3u;. cvrts to fall and its lower end does not slip. What will be the speed of the upper end of the p.u lxz7 v;c3ehole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotaa,pas4(36.lj,oifj zh4mfu wr.f,a ating on the end of a thin string in a circle of radius 1.10 m at an anga4 (6s z4hpj,mi.3lrf ,.fafwoj aua,ular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of+pd1 eg)s0pxh a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotating at 1500 rpm?0 p+sg )dehp1x    $kg \cdot m^2$
(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 28b-z.1 kxqlh,hcyf l platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horil. 12-xlc,8bk hzy fqhzontal 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 i51 gqn3hy:sykml/+bdn Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight posy+hdy q/s3lk ng51bm:ition 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 initiap)3ea za( -cb,xj v*g1oi3vxz l rate of 1.0 rev every 2.0i-x( 3 )vcag13jazox*,bezpv 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 throu -yq k2y.s6gvdgh 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 gy6vqk-y.s2d 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 quj0z,y(;ou *0 5p9vtk+pr -dkcvyje 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 momentum of thegnkcl2c arv35h*elxc+r0k4) 0mev;c 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 isdyt6*.wzp : 4xvcj(t m dropped onto an identical disk rotating at an :(tcpzd.tjmy 4x*6 wvgular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.xlkg p /n 6dp ,el5vk+:b5cw4u4 $rev/s$ 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 lj69r/e3vx(i6ud4 ob h1 idswstands at the center of a rotating merry-go-round platform of radiuelw66i43uhbjid/d1s 9( o rxvs 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 rotating freely with an angud6 y78t0inv+:qfjet*3v qzs x lar velocity of 0.7vqn fyi* 8t6+:3vzt qej s0xd8 $rad/s$ 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 eventualjw a+ 47u dk+pzg;9vmoly collapses into a white dwarf, losing about half its maopm;4d9u gzv+ 7+a kjwss 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 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 excess0qn;h c ;.gz3g:h d+.i lxc y2spfi-dg 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 atbrrr o a +6b73p6lf5ni:gj3$ 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 frt;vag;i pd5 hs4v+ v8sagcr2+y 6e0kneely without friction. The moment of inertvn 8p++gvc kyaa 65ig02she d;vr 4;tsia 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 edxn +54rq bg ot0 vx+8;nlhoe2ige of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia of 1700t5x eoox8qn+rlv+b2h4g 0in ; $kg \cdot m^2$ 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 ther9 454xvancnwvl y 2g) -hokl* 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 thelknx) v -4h95y2vca4o* wgnlr spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces the Eart;e1ex(aenc (qg0ve ; ,qhcg;x h. 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 particlee cxqcqhgne1(ee ,;v0(ga ;x ; orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at4qnvd (*:n nf 5bz+o ojv-rk7x 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 thes4o .rle/w/fb shape of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s inte. ls bf/4wor/erval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of jhtpfq, ign jp,+/n3(;mf n:j two solid cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindri,h3mjjiffp/n+p:n t; , j nqg(cal 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 wheam 7w u4q+kb6mel ($\omega_R$) 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 .tyy0(rzwkw0 q /gx2d8c:w.5hmyen eas great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losi y8yt ./. qymh(e05rkwz2xw:wdcne 0g ng 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 isj n,x z;9uyg5z2usvt 6 for it to use energy stored in a heavy rotating flywheel. Suppose such a car 9xujus5zy6gv,;2tz n 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 ap9 3 )nm;sfpp2a6drhgj )vl:in $H_2O$ 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 rotq+9u mas:y7 ylling 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 -:ke ia3w 2zgxh 7yk)nand length L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. As khez7:w) y2k3a-ginxsume 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, and we etd6o rip/lv2 t: b;k8rh2vl3want to exert a horizontal force F at its axle so that it will climb a step againr6 lkvtile p2h2;8t:bvo/ r3dst which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling win m2uzd1uoj +t 1ox11tth speed v=4.2m/s on a flat road is making a turn with a radius The forces actin11 om2u dou 1zxtj+nt1g 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.f1ivve4 t*ve5zg6ur( 5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does thef6vv*(5tveeg r 1i 4uz torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as tm r pb,3+t1yokbgl 4o/c6 7pxphin rods, making reasonable estimates for the dimensions. Then calcpkbpoxl7/o , 1cb+ 4yr gtmp36ulate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly -r( d,9fa wogwr/.mxlfg9; uswhich consists of two cylindrical plates, ofxl d-m(/wu9 arsgf;.fwo, r 9g mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped rough track of Fig. 8pqat0t 2 .gwj3t3yxk7–58. What is the minimum value of the vertical height p3 g x3.j0ttqkt2w7ya 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 dokoe - ed8q6nrtsk9 5i8 not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its oa4.y dunh*/aos 2oc1speed uniformly fron4osyo/a2d.uh ao c1 *m 90km/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