A bicycle odometer (which meash;y019 q1as3 xp pqmmj0x7ibn ures distance traveled) is attached near the wheel hub and isxjb3 pqmyaxm7 qin90 1ph0s1; designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular ve*ag-alapdmd+ sl (fg9e n2+ r9locity. 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 aag-felg+( an a*9+s9pm rdl d2cceleration change?

Could a nonrigid body be described by a single value of the angulgcwb b0swh016qdm eu )1 k78star velocity $\omega$ Explain.

Can a small force ever exert a grea m/9wx gi1kcb7 :lludev; /-ehter torque than a larger force? Explain.

If a force $\vec{F}$ acts on an object such that its lever arm is zero, does it have any effect on the object’s motion? Explain.

Why is it more difficult to do a sit-up with your hands behind yourt3o:f4olb drfzxrq,u 7;/h 4k head than when yourf4x4ulbf, 3qrotk/:zr7h ;od 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 ; ouyiuip/krg ae2 mie 59h ;w2:-l 6et9ord,zwheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rearu5;or;9/ g:6luahpee,9k rtd-e2m o iy2iz iw sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legs 8 -1cqevlxx(kyt qp84with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dy4q8y1q p-c(v ek l8txxnamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, r8g v)gxy1t:dmkua)5 narrow beam?

If the net force on a)0gbkm v4/ws ;/ h k,j-toz knhzp5h0i system is zero, is the net torque also zero? If the net torque on a system is zero, is the net f0hkkv zkimo)sn 04h - pb/,g5htwj;z/orce zero?

Two inclines have the same height but make/4bw3i,ttv ez5ztgc x nd7y 85 different angles with the horizontal. Thwn ttzz7 xc8 55tv3b/ g,ydei4e same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an i-12. vx6z yo3b5 lpcawhrnlp9ncline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which hv59 rlaob x c.p6plh32yzwn-1as the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius a)bs gs7) h;kfwnt;,odnd the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater to;, fwd;t ssb7)ho)kgntal kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conser.- 02 prrqbc3brrozc wykg7t3:9 8szived. Yet most moving or rotating objects eb98s k2qor0p:-g crr37 rt y.c wiz3zbventually slow down and stop. Explain.

If there were a great migration of people toward the Earth’ evz b53jhq7w9s equator, how would this affect the hbz3v5 j9e7qw length of the day?

Can the diver of Fig. 8–29 do a somersault withouc1*(h1rctllhf.4,c ;i gf b tat having any initial rotation when she leaves thfgrb4ct;1 lcfia1 *,htc h.(le board?

The moment of inertia of a rotating s:p(6f- i rjtcvolid disk about an axis through its center ofr v j-6:ptci(f 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 holrx0- tlt + aav*8d 2rzrggy3i7ding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? r8 rai- 3lat72*tz gxy g0v+rdExplain.

Two spheres look idenp3 pw7fo ;r;wvtical and have the same mass. However, one is hollow and the other is solid. Describe an experim;o3 7;pp rwwvfent to determine which is which.

The angular velocity of a c.1 :tgkh:lu1kben+n wheel rotating on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the whe l1:n e:.kktb1ch+gu nel? 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 of2gj(*y* mph6bx5 hfwo9r7dwpel+ + fo a large freely rotating turntable. What happens if you walk t jfg(e+b hhw5wy*2orx ml*6fdp 7o9p+ oward the center?

A shortstop may leap into the air to1 0;9on zcxiyp .p.usa(r0sba catch a ball and throw it quickly. As he throws the ball, the upper part of his body r.np0 0zxayaucob (1s.s; p9riotates. 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 conservation of angular momentum, discuss why a hel ynzu 54h8zl5iicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can o5u zliy8n45z hperate to keep the helicopter stable.

  Express the following angles in radians: (a) 30 y h7dmk6*r vxd cv5(d6$^{\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. Calculatext,urkdd7zs h;bb93s tb/ )s(, using the information inside the Front Cover, s);rbbdds/ (k zbh97 ,ut t3sxthe angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directl4;q r,zvlmr0 ed at the Moon, 380,000 km from Earth. The beam diverges at an anglevr;qm r0l,4zl $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. When the motor ik 0 9u )w08zql zhq:mn-5uli)muae+mys turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as ml+z 8muih 5m90ze) qyua)k nulw0q-:the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to anothex99nmsem pgq+/* v /xsr child. If the ball makes 15.0 revolutions, what is its diamen mx9*gmvsp/xq9 + se/ter?    m

  A bicycle with tires 68 cm ie682 pgotru4q/ b uor-0ar.uk n diameter travels 8.0 km. How many revolutions do the wheels make? r 8opu0e rgq2k/4.uo6urba t-    $rev$

  (a) A grinding wheel 0.35 m ined1jk up1vya3e)e 6)l diameter rotates at 2500 rpm. Calculate its angular velo1y l36vujeee a)1d)pkcity 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-riw3t)7v7kcb:2blkr 5 t 9lu wiound makes one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear rl3ktuilikt527: bvw7w c9 b )speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the E;o7h,hz kczhn)3 ;zlgarth (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 6b4.4 r1tjk+e h5fzjrtpxg e0
(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 zgf (gk -9xbp-the axis of rotation is to experiencek9zpfx-g bg-( an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelh)j+ow *fowvxz xe kmwm*( o*ik/ 6b*0erates uniformly about its center from 130 rpm to 280 rpm in 4.0 s. Determin/(xo**m h o vke0zf6j*kiw*w)mw+xobe
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radius g hnjv+x4(z4g z.by/)cp 7dez$R_1$ 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 putmx-c (9phr es. 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 during a 12-min time interval. The spacecraft can be thought of as a cylindem hr-pc9x.e(sr 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 rpm in 220 s. Through hy))g z8ugslu* ow )z8guuy*)sg lmany revolutions did it turn in this time?    $rev$

  An automobile engine slows down 5ma:.dr1 r.gwkv5vkdfrom 4500 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 the stresses of flying highspeed jets in a whzl,.hayk9ow.3 +xqwbgwk f 7 yg2m2z -irling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before -oqf7g.+.m zw9 gzlh23kyx, ky bw aw2reaching 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 36 avy9hie67z(. f*roxh 0 rpm in 6.5 s. How far will a point on t.ez9ao x6rh (i*vf7hyhe edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 1500 e u9cc-ib my3 (2;baxjrevolutions beforebcjyxa -3;ie2 (b9m cu 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 ma,cqh*3ol5 m (s*;fvah.a5a fad c/uajke 65 revolutions as the car reduces its speed uniformly from a* 3so;vcfal daj5 hu qhf,ma(a5/ *.c95km/h 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 revolutio. dh-c(tg 67hgrdczjfd qen dp2)ul -rc99h 68ns as the car reduces its speed uniformly from 95km/h to 45km/h The 9.h8lcdh 6z g rdddt6 e )gpjc hcfu2q(-79-rntires 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 p u rqd)(ssb .st4gi*;all her weight on each pedal when climbing a hill. gps;s4b *rsd (q t.)iuThe pedals rotate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end jdm8q/7h r6-re13vdqvtgr sxys1,x0w :6 ryxof a door 74 cm wide. What is the magnitude of the torque if the force is exerq/r :x6dqt3 rv1-vxheg1rxm w8j7,0y 6dssryted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig. 8–3sx2v ;o*azedqj 3 4zr xi/9oe+9. Assume that a fricz er;x qadzo9so/+j3e 4*2vxition 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 wh0lucnfi-(h h:j/ sa-q; xdc (o6 5sbbkich pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the mfc-(b 0su-q5cjko(6 h;saxb hndli/: agnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tighten9g0-*-r rk2kn;epq wniab yf3mkb() ting to a torque of 38fi-tnkky *rm3ebk-)a09g b; rpn( q2 w $m \cdot N$ 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 i ufkuji3q 847k5itfm0(5 5 altf9pi gmnertia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is through its tg jp85(0kfika tqulu7 39i5m5fi 4mfcenter.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameteras 7cy0rro0v 3. The rim and tire have a combined mass 3rcv7aosr0y0 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 an2 jwlv-r3k 50f jbtc7 thin, light rod is rotated in a horizontal circle of radius t0kf-j vcj n7 2wrb35l1.2 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 op+7qnw +dmq;f gz+x 7bowl on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The frictnqgpd+w7xm qz + o+7;fion 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 of th 2lzu+c4g6.5r w ,d ldfa1evqle array of point objects shown in Fig. 8–43 ab.a,lzgwfdecl4 l1q rd52u6 +vout
(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 consistw8. h;q0uwxvl bwx)ees73 i+f2un vw 9s of two 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 at the correct-s3rue+yp or9jh-lwki5c88o rate, enginr35rulk8wh jyo+e-9i8-s c poeers 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 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.dapr,3 *9ktkqc l(v r050 cm and a mass of 0.580 kg.pqk kt*l ,vdar9r3c0( 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, acceleratis(-d ynawx*9ow)i)pfng it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangent fyoktuqw xm4((;./u2w1yj, mz4rh fx ially on a small hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neghk(2rfotqf 4.j 1u zy,(4 ; /wuwmmyxxlecting 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 broughgy- j0jf94gmga(0 im ped-tf:fc .w,ut uniformly to rest bm (wm,:a yjff. edjcu-p4t00g f g9g-iy 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 vsza*e +- dzacvl 0n,jx .-nn1accelerates 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 8sotk5f;;wzy,7eb o+zr21b pa1 da o raction of the forearm, which rotates about the elbop ozt7bfdr1, a1wb 2 oao5yk;s8rz;e +w joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thinsw y)wym kk3,u(kc2 3 vc3wn2h rod, as shown in Fig. 8–46c)3wh ( y32vkycw wnk mu2ks3,.
(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 co:; ;7au7y)+bmkii w ark:/3nja jjqjbnsists 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 accelera5)kia bg9) bw:cp.uy vtes the hammer from rest within four full turns (revolutions) and releases it at 5gp.9)c b a i:v)ykubwa 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 mokk(ewb-7k h;zrx 9n3 oment 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 torque of 280x*wr*k,g jz:6jlq h.jf9 h0yd $m \cdot N$ 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 without sliq ns*g1b lib4 w0th. erke/ kw0,hof53pping down a lane awlqrohk*eh0g15.b4/b 3s, e in0fwk tt 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Eagi;02 twyukl6mcj 2,xrth with respect to the Sun as the sum of two ter02i;c ygtxwl 2, ujmk6ms,
(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. Hod1liw/vm yl- 8w much net work is required to accelerate it from rest to a rotation rate/l8 dw- lv1iym 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 starts from rest and rolzqbo yje*9 ;s;qbki10ls without slipping down a1 j 9q;es*iz0kyob ;qb 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 to 1muub om7 vg9;fall and its lvm1 u7gmbo u;9ower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating oah/umvy7a 3(*x xkng 7n the end of a thin string in a circle of radius 1.10 m x7(3*avk y hg /x7manuat an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angu1qr-emh ; y:eelar momentum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm wh 1:me;-qrhey een rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotating atqsj+rp15sc)b nhee 1as +u6p7 a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases t+p br+cas1qsje1e7s5 pu n 6h)o 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce her moment ofxj:xl3 tuw (u/ inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If shex3xu lt: /uwj( makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial rate of f ie;3hvh0ncjdo*a 1*1.0 rev every 2.0 s to a final rate of e30 j ofv1;d*h*c hnai3 $rev/s$ 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 aro z2l/cdbxq ))hund a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?zxdl )b2c/ q)h    $rev/s$

  (a) What is the angular momentum x18i obqc9r wny)0nbmb 7 /r+)bp 8cqpof a figure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of the Earthp+ ( (9xisxhvc
(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 disr2s c4s8:nstek of moment of inertia I is dropped onto an identical disk rotating at angulsetn82s4 s c:rar speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 7otu,cc :z2o l2.4 $rev/s$ 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 merrybct 9yvl4 k.;*.tjw hrnbn 76t-go-round platform oltry4. 9 *tbn6k7wjbtnvc;h. f radius 3.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is ro v rz96vo.z:3dt9trmr tating freely with an angular velocity of 0zrz3rdt: m6o 9 9vtrv..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 into a whit6(f dvm zvtxmb/: s +g1cavp,p*6pr )pe 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 aw( p+vg/6a 1bpvr:zd6mf), m xp*ptv scay 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 excess ofknafu6s19 9m*y-l fh y 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 z0es-v9c ia ,zpbb58k,/h hye$ 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 frexz7)ha nvqd+ :ely without friction. The azn +7qhv)d :xmoment of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diameter merry-go-round t4h rum-u8w6hzjeps9g ns09almx 1 +5yur+uhmesya nwh480ml 5usprx96gz-j1 9 ntable that is mounted on frictionless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground wi.xulqv4 a( 1yzth 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 aq4yl (.uvz1 xL, 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 7e4gyuzqf:x -Earth. Determine the ratio of th-g:e7 xy 4qufze 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 kkn2a+zuuic cbz7c/7, 2tw:q4 pdy :c 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 u z cvm nv4+w9t(k/xvt59k 1pqp w.-qfthe shape of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-vfqc95ttk+vwk (1x uq.mp9zwnv-4p /s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, eacz fty9wc7 av0-h 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 conservation of energy to c97t-afyzwv0 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 angul,,ta hfn 0:e.qfejh,gar 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 midcc(dd:9f v 2 m) :nn99inwaof our Sun, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 9id(n : ndn w29if:d) 9avcmmc12 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 automobirkyptnk*6 *cmv:45t;nfv 2jk le is for it to use energy stored in a heavy rotating flywheel. Suppose nc4:62yjn t*v*;v m5kp kf trksuch a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrakrream15ph k9ek,8i6)li 8g ctes 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 qeuxy:j 8od8i-if )0won 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 4g0c2znz7 -ij (qpm2fojbq(qL can pivot freely (i.e., we ignore friction) azmfopi 2nj 0qcq(z b-24(7 gjqbout 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 shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the fl+ l difmprp m4o5t t,*k/t)c-joor, and we want to exert a horizontal force F at its axle so that it will climb a step against i+ /mdt-*rt ol4p)t5mpkj c, fwhich it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4 m(p741 q ck6qzhmij)z.2m/s on a flat road is making a turn with a radius Thzm(4 i1mh6p j7qz kc)qe forces acting on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg roc 3v:axr s-r(qtnbp: *lsn 5iv6k into a sling of length 1.5 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 the torque come v:* l6sb3 in(xa sr-vn:pr5t qfrom?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her(/: snk (,ddm9i 7 zimaidv9fk arms as thin rods, making reasonable estimates for the dimensions. Then calcn 7(zfd:maiimiks9dd k ,/v9 (ulate 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 which consists of two cylindrical plates, zli hohq +;k35a .leo;of mhe;kih3.;a ozql +5ol ass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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. )uh7j0n -xu u n6hipft ty9,1q8–58. What isi1u7u 6hj) ntxq,u nf-h ypt90 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 leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, bjs2 zky42 bt1b;ans9a ut do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uniformly frp3nl;0wf v2i. ylapvb, j/0o hom 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration o3j; 2hpvb.l a,lnpvw0f0i oy/f 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