A bicycle odometer (which measures distance traveled) is attached near the wm-be* zdfi9cih m1 ed5c)0-cnheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-- 19ci5 fbc *-izcemmded)n0hinch wheels?

Suppose a disk rotates at constant anmxo7 a1vum v/(gular 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 ao71/a v uv(mmxcceleration change?

Could a nonrigid body be described by a single value of thxap/3 dz+k;3ckq s(fm e angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Explain.is /, diy:,hlrhuyg .pm-/k2k

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 gs.+op4p-s or9j3rsg your head than when 9jg - 4o+s prg.oprs3syour arms are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the pedal wxc .v/) idh6u0hsrffd )ialh/+* pa) 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 peda)d fcvh /h.h)+* a di)u6wfxlirp/as 0l, a small front sprocket or a large front sprocket? Why?

Mammals that depend on bei28xrqs h7xfh /ng able to run 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 adv x2qh /frs7hx8antageous.

Why do tightrope walkeumw0sb* 8 osx5rs (Fig. 8–35) carry a long, narrow beam?

If the net force on a system ism2n:)do pri0 q zero, is the net torque also zero? If the net torque on a system is z:0i nq)rom p2dero, is the net force zero?

Two inclines have the +h q( amtc,g2wuz:,kho )lyr ,same height but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Exgkc,z)w+lam2h ru,:h,( oytq plain.

Two solid spheres simultaneously start rolling (from resbrkc/xz 2tac */fv (z2t) down an incline. One sphere has twice the radius and twice the mass of t z/kac/2ctv2zrb*(xfhe 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 h/yoc84ik q)m/2zto bhave 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? Which h/coiz)yt/8o 4m qk2b has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angula.t 6zj)vy vhxzi6tg4 6r momentum are conserved. Yet most moving or rotating objects eventually szzixth tvv6 j6) 6.4yglow down and stop. Explain.

If there were a great migration of people towardre /w- c1potv1w+czc5 the Earth’s equator, how would this affect the lene-cz+tw1rpcwo5/v1 cgth of the day?

Can the diver of Fig. 8–29 do a somersault without having any iniz8 m198mldpk6 w7 c qzijc0p*mtial rotation when she leaves the bow*lz86k mpmpc7 1c z0id8qm9j ard?

The moment of inertia of a rotating soo*f1kzdyt;tmpe 3y35 lid disk about an axis through its center of z3pyekto5y *3f 1dm;t mass is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg mass in each outstre/tzue j: ugs y;7)(dyvtched hand. If you suddenly drop the masses, wi g/7zs)y;d u( uvetyj:ll your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look idenhs(os/9)m/o f ;dp qihtical and have the same mass. However, one is hollow and the other issd fqhi;o 9)(ms/ /poh solid. Describe an experiment to determine which is which.

The angular velocity of a qirtmt41 ameh8h*6c5wheel rotating on a horizontal axle points west. In what direction is the linear velocity mhm6tct re8i h54 1a*qof 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 the edge of a large freely rotat4ngt: t0n5axim( xde+ing turntable. What happens if you n+i : 5g(exmtx4nda0t walk toward the center?

A shortstop may leap into 7fx5b zl/68 hj aw:/:h ibjbspthe air to catch a ball 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 hips and legs rotate in the opposite direcfzj8 h/lb: bp5j hw6b7 a/x:sition (Fig. 8–36). Explain.

On the basis of the law of conservation ovd3 w- 1wn*snwgtkfh(6 pw 8,kf angular momentum, discuss why a helicopter must have more thwwkw-fk, *sdn683g ( vnhwt1 pan one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians-qck x-/olziycgzd(s 5m-v tj,e;t (2: (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 1tjy2si7 nch -of an amazing coincidence. Calculate, using the information inside the Front Covei2ycstj1-7 hnr, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed aefpw2 lu2)f xc+w2b p3z3ll )jt the Moon, 380,000 km from Earth. The beam diverges at an awue+2)fw p) lz 2xc3l3plbf2jngle $\theta$ (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 atxl w25s p,,jrj a rate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angulaw,ps2j rl ,5xjr acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. Iy p*65j dd0lka mkik;jqaef935( sqo*jvo2, b f the ball makes 15.0 reb5s20laqykoovi (5,9jm3dae**kfk;pd j j q 6volutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutij 0sbnahqr,: 8ons do the wheels ma q:n8,r haj0sbke?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Cakoi; fb4jls: 2(3us5 e6srg celculate its angular velocit:cr; lg6 4jsebkfes3is (2ou5y in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete rk0fl0 13)m(bmzqwy qej/ lvk,evolution in 4.0 s (Fig. 8–38). (a) What is the linear smq00/l1lfkmzb ,wy3 kvq ej)( peed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earthnks m-tku/t x/94giz: (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a point kmsrh5xlv o16 .c*fvvz2pkm+ vz2u8 /
(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 the5k ,5mh6)hhyi5c. r2kcj kmqg axis of rotation is to experience an accelehkmqr5j h.g h c,k2y5)i6k5 cmration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from9;kas;o beng - 130 rpm to 280 rpm in 4.0 s. Det9sg ;keb- ;oanermine
(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 radius3*ffole5w 3 wb $R_1$ 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 spacecraftvbckk;: wag (1l; cf2t put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute durinw :c;t ( bfac1vgkl2;kg 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 15,000 r34 fqm2hij 8s7 soav1jpm in 220 s. Through how many revolutions did it f217j4vo83jsqm ahsi turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Cabjkmjg s d-kwt4)xo: 3m4;c8vlculate
(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 highq) msqfj(;qz :speed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions befmqq;:(z fq)js ore 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 360 rpm in 6.5 2 yp 04 egrlbsasn8/j,s. How far will a point on the edge of the 0nabej 8 l4rg2/ss y,pwheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 8501vo1v7qb7h q rrev/min It turns 1500 revolutions before it qb7vvrh q71o 1comes 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 rd *n)wlj1y rm5kok1 hiyjzh*-q3qa /5 evolutions as the car reduces its speed uniformly from 95km/hjh /-*)5 onmlqiq 1yah3kk5 jyrwz1* d 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 its speed uniformly *v53 51,tfbpn nd ypaxfrom 95km/h to 455y xvp3p1*f, 5nadn tbkm/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 pfq j9/ y) f5.waj1fodduts all her weight on each pedal when climbing a hill. The pedals rotate in a circle of radius 17 cm. odfy wfa1 9.f)5 qj/dj
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force jz ,6 xq;4(7obphu*cbn txrq7b;p kx0of 55 N on the end of a door 74 cm wide. What is the magnitude of the torque if the force isp 6;;0qpk*c7 h, (jb4 ozxbxxtrqnub7 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 ajbv n7b0.s xy,bout the axle of the wheel shown in Fig. 8–39. Assume that70s bvx bn,j.y a friction 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 r; 9-vzi zb1,/ac vxm tg;pu*sbb m-*gwod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net tzgzm x,t;u vmvai-s /pg9** ;1cb wbb-orque on this system.

  The bolts on the cylinder head of an engine require tightening to a tor2zp r9 h;gox1gs-4j f jzk)9 j9vgwxh(que of 38 92 k jjsh9fjw4o -rph ;xv9gg(z)z x1g$m \cdot N$ 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 whcz5 gi+7y ;es6pcyhh2o17u ksuj4o c/en the axis+4/cc 2iy7;6 kc7e yjhos h go1uuz5ps of rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of in6z,mt 1t9+eegco.3 cbrk, hxrertia of a bicycle wheel 66.7 cm in diameter. The rim and tire ,o.cemb6xtrer3+ 9 tz,c kh1ghave 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 the end of a thin, light rod isnvm/*z3psm2 +i p8n bd rotated in a horizontal circle of radius 1.m8+nd3vip*zmp n/ s2 b2 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 rotatmv4u)llw-o v6 x 3r5zving at constant angular speed (Fig. 8–x z)- 65 wlvuov4vmr3l42). The friction force between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia o9b0wfrcu m10zf the array of point objects shown in Fig. 8–43 abo09rb1c z0 muwfut
(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 two8), bomobu. fl oxygen atoms 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 cylindrical satellite spinning a)t ve b:aur3 ya:7eu/nt the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite hasuu3b/ y r) vt:a:ea7en 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 unifo),5 un .bb agxkux,:n3vl ms9fg t6qz*rm cylinder with a radius of 8.50 cm and a mass of 0.580 kq,gxs,u mg 5 3xf:b)lb*nu9z6 vt.ank g. 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, 1 q0yp i3nt/lp-rypx0 accelerating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merr - ;dby+lhm+wny-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, neglect ny+mbd; -+lhwing frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rot s7m3kw+0wg sdkk 9j1, pdu1etating at 10,300 rpm is shut off and is eventually brought uniformly to rest by aks p,7 kwwu9e0m1kd3gd1jst+ 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 3.6v;+li*koh7 ex1:ax fois4ru6;,rpc u3fjb 4q -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 thkzx -m 2r3bqp1rg --fce forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated - p12kfrqm-zc3-br xg 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 lonv x+r 99th+d far9k 7k:bse677qjgq uvg thin rod, as shown in Fig. 8–46 +ba9jhqv7+ xufk:stkr9dr 69 v7e g7q.
(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 two mas:.s pc.z)fpl- nwjw.g ses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest)yckj7islq . * within four full turns (revolutions) and releac7. ) *yksiqjlses 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 momenta/zbil s(ul 5o2n9)ie 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 deve.rql6kg kwj* v eji0/ jkk2;0ulops 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 radius 9.0 cm rolls cv4 +qe x-5* cegxzc/8b w)llswithout slipping down a lane at 3.3 *vec5 bgxl q8cxzlc)-ew+s/4 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth wit2 x t0vvq-jfa2pe*6p jh respect to the Sun as the sum of two te0 f622*axjv ejp-vtqp rms,
(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 as ,1b4lcz85 x aic5atb mass of 1640 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylindb t8icxacls5,514 abz er.    J

  A sphere of radius 20.0 cm z/7 gl)td g+lgh)ot a /-+mr esq 9edrrg99f;iand mass 1.80 kg starts from rest and rolls without slio9lh 9e+e)z+tr /-/ dm;qdgtir f) gg7 lsgra9pping 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 starts t mt-)2 cfbxt vjzzi6p*w+q,.s-aio-d9 bgun9o fall and its lower end does not slip. What will be the speed of the upp9m z.sw b+ -f*9-it vuj6qa2nt c,zbpxod i)-ger 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 rotatl0u) ,ldrf:k nrfv0x 3bbq-,wing on the end of a thin string in a circle of radius 1.10 mrr:x uf)b00d-3qvkwf ,lbn l, at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grindikj 9kq8hc6n 4j5.wdf-k 6o zqeng wheel of radius 18 cm when rotating at 1ok5 j.68z-j 4cqenk fdkwh6q9500 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 isxt:w/0 cxum6a rotating at a rate of 1.3rev/s If he raises his arms to xmxa:w 0t6u/ca 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 f l8cye d1y58fshown in Fig. 8–29) can 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 1.5 s when in the tuck positio f58elyc18yfdn, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial rate3+u 7luiii:k s of 1.0 rev every 2.0 s to a f7:l iu3iis +kuinal 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 center 0o+ f8/ymeclk ot:5evr03s k zat a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and di 5cy8f/kvr:+z omk0s elt3e 0oameter 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 / boz *,x alfdx-tfp6h:dv9v+t2 h(zgspinning 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 the Ear8 fc+dc**kr dwl)psu6th
(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 droppq1z(ecy.)7fny iw/rlu sw+4f ed onto an identical disk rotating at angwir1s nlu+()z4q.fefyw/y7c ular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns pfnsen3ru w* 7o7pom2rq+/o8at 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 platfoywvgc .xgo/,:rm of radius 3.0 m a:/ cgyvxgw.o, nd 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 velocs i8mjq:c rf+bh41t/zity of 0.8 r q cfjs/+zt1b4:8mhi$rad/s$ 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 whjx+64y5zi ytwql sw4 3fui;;hite 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 rotation rate be?(roun ;3tz5x 4 ys6w q4yfh;ii+wjuld 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 exc6qvab,piy.ns 9(,y-oetr 9 my*s xln/ ess 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 7d u7vqko* n3cnkdds) 3m5k2 d$ 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 platfo 3snikri:6 /s6oxmm: crm, initially at rest, that can rotate freely without friction. The moment of :iirm6ns6 ox/ km3s:c inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of iv.8 bfl)aoptf*dv (-a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of ine.)v- * flibt8adovpf( rtia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground wit8idf)jb(8 v ngh the end of the rope lying on the top edge of the spool. A person grabs tjnd8 fg i)v8b(he 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 Eartf4 8o r2mjyb1a,fr(q oh. Determine the ratiorob8qm(r fya12,jf4 o 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 Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rateek4**/n kvhq0v 4twow 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 shape3nxu u:3mb :v( 90:yljatjimh of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angul jb: 3:m(h ljnyx90v3i:mtauuar acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindricolnc,o7 br* gyp-(ma1b5o4yp al disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of massn4b,ary1(lco p m opg*-o5yb7 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 o*vkqk2j )uhz1f 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 withh)5kclkv0/zyw i1: n)nr q pm, mass 8.0 times as great, were rotating5l /h,p cvnqk:wi)zy mnr)01k 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 mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use energy stored io9e2jtoa+k12 bw6q a ma8+sxth t*mq+n a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flyw+ 9qam1t8wxmkqas oae 6+o+ t2tjb* 2hheel “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-qsxm9 h (7udh7b rv.v9quhj4 an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal srg8lx7oj 1lc,h(fz ; yurface 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 y2 96y 6ngxqqiL 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. Assume that the force of gravity acts at the center of mass of the rod, as shiq6q 2 ng6xyy9own. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the floor, andzulu-33- a5 c6nl k(+eamfyde we want to exert a hoc5ea(zu3k3ldle - u f6 +ymna-rizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on /ugpkrx79 wp/a flat road is making a turn with a radius The forces acting on the cyclis/kupgpr7xw9 /t 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 rocr wo -jav3wtgj/4 wc2)8ld3w ck6gn8hk into a sling of length 1.5 m and begins whirling the rvj klg/4anwd6)8-jwhg8 wtcc 2w33r oock 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 the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder ddb/iyjb;u ) 1and 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 againstduy id b);/bj1 her body.   

  You are designing a clutch assembly which consists of two cylindrical plt .xwx)p(e,d0 .pnm.k + ,d.eokx yugnates, of mass .,0xuxgnxpe.dm +,.pko)k d ey(nw . t$M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 lzgs h+7al*2 nFig. 8–58. What is the minimum value of the vertical height h t2+h*nl zgal7s hat 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, bu hh04l ermx7;5-uauemt do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 iai.zz frk9by(;p-oj*7quwt t b6m13 revolutions as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. - t 7tarpm.ik9*b;b zjq1o63uwz fi(y(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