A bicycle odometer (which measures distance travele8f 3- hbhpgf5*o09w j*xfhgvod) is attached near the wheel hubfp*03oof*x 8hgvwh-f 5 g h9bj and is 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 veb1aq mq94jj jnk)r5g8locity. 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 componenjq1r kn8gbj4qjm) 59a t of linear acceleration change?

Could a nonrigid body be described bypmp nd7vo2(ur .57qz8 ig;drx a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a yssiij :5*2aggreater 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 toxb u (5aobj8:xu:uv.p do a sit-up with your hands behind your head than when your arms are stretched out in front of you? A diagram :8ab jp :(5uuux b.oxvmay help you to answer this.

A 21-speed bicycle has sev8jo1r1kod q/5pe4gmd*cv2a f en sprockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear *5a2j1/c vmdqr 4gopfod ke18 sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legs with fl8t ;k j,jvhh00(cy/kbwd g+.- kluuzd esh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynau w0ujkk;b.d k0j 8z(-l+yvhh /cdgt,mics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35x 6s7 -jj +yqlm(jdka9so1et9) carry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? If the netpw /xt7)sz+yz.d0 iaf torque on a system d0/.pi)7f + tazyzwsxis zero, is the net force zero?

Two inclines have the same height b 18 kfbj80 5aqpaulz1cut 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 greazbc q1aufl k8 1ja0p58ter? Explain.

Two solid spheres simultaneously start rolling (mf r,df.4 gq. l8*kqvwfrom rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Whick lf ..4vmdw*gfqq, r8h has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. The4ri2lg5 zm1yuy start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greategr2imz4yl1u 5r rotational KE?

We claim that momentum and angular momentum are conserved. Yet most bh229g7pnig:;sb nzeepg b/8 w * ika;moving or rotating objibn*p gw2/bse97g hebg ikzn;:a;2 p8ects eventually slow down and stop. Explain.

If there were a great mi xn7c; 9a6 w 0uawqz;0zrp*kntgration of people toward the Earth’s equator, how would this affect the lengzaxw6 9;7pqz* 0cawn uktn0r;th of the day?

Can the diver of Fig. 8–29 do a somersault witn o) qw-x y*a.g,szz/4lq (po+ kihd5dhout having any initial rotation when sdn.4ldspi 5o xqz)yo/ q+akh -(w,g*z he leaves the board?

The moment of inertia of a rotating solid disk about an axis throuz (c 39tbzx)slgh its center of mzcb9t3zlx s)( ass 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 h(9pf v:j4obauolding a 2-kg mass in each outstretc (pfoj b:vu94ahed hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the samel2hr csh +v(m8hin/z /qn (ch* mass. However, one is hollow and the other is solid. Describe an experiment to determine whichm( hnhzqi8n+/ sc*vch( /r2hl is which.

The angular velocity ouh;64t :-kbp ycl0uc gf a wheel rotating on a horizontal axle points west. In what direction is the k gc-l;6c4:ypb 0huut linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge o o)0 8kebhjm2nf a large freely rotating turntable. What happens if you walk toward the centerbme0 ok)j h2n8?

A shortstop may leap into tdqnrh6o ; cwxc-1x72hw) pv-m he air to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If )mx; p6hhncw w1qo dv-cx-2r 7you 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 cons1x dr j-(by3qwervation of angular momentum, discuss why a helicopter must have more bxdq jr3(-yw1 than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles ihd(d kwqwz5xm(or s )i0 )39ken 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 oa*zor2 bk,q9k f an amazing coincidence. Calculate, using the information inside the Front Cover, the angular d2*rz oqb9ka,k iameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beam divergesthiw.-,wkk) vs8x h,p at an anwk pvi.8 ,w t)k-hsx,hgle $\theta$ (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 motora4d(hl 80umhs is turned off during operation, the blades slow to rest in 3.0 s. What is the angu0dusmh(8a l 4hlar acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ballka;:yktw ;whab d73 (n on a level floor 3.5 m to another child. If the ball makes 15.0 revolutions, what is its diam;k 7bknwt d;:yh3w a(aeter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 k/ n xmg;6z,bjv n9l srj+fns32m. How many revolutions do the wheels mlb m2+ nsgjvnx fr93,sz n;6j/ake?    $rev$

  (a) A grinding wheel 0.35 m in diametey/k.v9qec. h zr rotates at 2500 rpm. Calculate its angularhvq9/zye ..ck velocity 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 revolution in 4.0 s (Fig. 8–3azgzv,tsp64i *- cn9st0k1a p8). (a) Whatisz46vza, *atkg1 s0p nt-9 cp is the 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 velocity of jbeq1far) uv;p lxr,u ;kas .) v1n8(wthe 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 a pointlwph24u ,ui8,qtv47fxug 4if
(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+cwessxtpv/ma n7i/:) 0fq u 1 2svdz. particle 7.0 cm from the axis of ro s:ft1cv+ u iea m)wp/nqv/.sz2d70sxtation is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter whe3xbi5th;v6*lnb0 r li el accelerates uniformly about its center from 130 rpm to 280 rpm * bh;nvix56ir b30lt lin 4.0 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 radiusd1d,fzl r12(oz 4uuoq $R_1$ 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 spnt5 6nl9kz ,dnacecraft put themselves into a slow rotation to distribute the Sun’s energy6nk, nl9nd5tz 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 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 rpm in 220dexdruvw:)/k ne *jl2 7 nok3) s. Through how many revolutions did it turn in this tine e ux*)7:n k32 /lwddkor)vjme?    $rev$

  An automobile engine s7j8ggf 0m+qhz lows down from 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 theboup3dr, 6k*2hldwg ct3+l f(8 dhg,qr9rw7a stresses of flying highspeed jets in a whirling “human centrifuhd6u, rbgq373o9 r*c+rawl 8g lp(d kdfwth2,ge,” which takes 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 diameter8+rfj41 theeg,q.0lafd+fao accelerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in this thrjaq4+f ee .,agfo0 f8+l1d time?    m

  A cooling fan is turned off when it is running at 850rev/min It turnq/m6 j2 gxzier:h)t9is 1500 revolutions before it comes to a stop. j) e6tgh2 /qrmiizx:9
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as thw)-erw, 6jgq pe car reduces its speed uniformly from 95km/h to 45km/h The tires have a 6j wgep,) q-wrdiameter 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 revaah t8fcm(v2 (olutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.80 m.2a( c (fatm8hv
(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 alloc+ y:q-+i6nmfrl5zo yk)h+q jy-c( *dia6 mr her weight on each pedal when climbing a hill. The pedals rm*y l+oy-jqyc( 5fo6d+mairncz -kqr6:h)+iotate 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 7v)jm3 (a+xc2:md oi:l76b sdpx ihvkon the end of a door 74 cm wide. What is the magnitude of the7px 3h(2 mdlmc7o6sak iiv:vx)db+:j torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig. 8–39wb qjp(qy)m1t)vgb76c k wi )5. Assume that a friction tt 5ky7pjqcivw )qm()b1)g w6borque 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 whim* qh 938hwz6w1hw*ygu7jz gd ch pivots as shown in Fig. 8–40. Ini*ugyd6h mzz7wq83 hgj1* hw9wtially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engineei l0g+ t1vv8h require tightening to a torque of 38lv1 g8+hvie0 t $m \cdot N$ 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 sphers 0jqdz:lid/ +e of radius 0.648 m when the axis of rotation is through its centerld: idzsq0/+j.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. The kej/91v5 g(uy5jt eyw rim and tire have a combjuy5yj twg(ke 5 9/ve1ined mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on t3tig(pxkdu58m1 m jx ,he end of a thin, light rod is rotated in a horizontal circle of radius 1.2 m1(m3xd mip 5 8kuxt,jg. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating at constant an7-9 4pcc yalub6h psn4gular speed (Fin la44-hcps9cuy7b p6g. 8–42). 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 of the array bar+ce.u*4(oh g2jv b of point objects shown in Fig. 8–43 about a4e ujob rg.(*+2chbv
(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 c(04vh9mn g1;awh mluhonsists 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 cyv)cn+ m2t61xkh9p qbhmg- h*oucm.6i lindrical satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if mghi 1669p q *kco)nh2.mcmxhvt -u+bthe 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 8.50 cm and a mass of d7u- 0l ro*r*a yty:hv0.580-7vrd o lt: u*aryhy*0 kg. 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,b.auijqq 7 zo3xulst+u 3 y62 bat, 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 s gfyz(ym(g/b5 )/ih 6hoj.byhand-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. Cso)5gyjigmh(fbzhy6b/ / .(yalculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is evy6te d2. cc 7o+v;kfaventually brought uniformly to res cv6f ;.e+d2cvaky 7ott by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a 3.6-kg baj2b5yfp erm /8non6j/ o* jkd1ll 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 th rnbzmmdq36 4ve,li 1 izn,rn r+c9xb62v,9vhe action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is nl m+ex3rn,4, d6cz zv 9r9qmni61ivbrv,2hbaccelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as))uamoo 3zs6u1oy)vt shown in Fig. 8–46.u)ystu6 oa v3o1om )z)
(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 consi;s ;j7sia3wuosts of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest within f) czg;vppreg.cz/( +lour full turns (revolution eg.cgrc/+p)( lzvz; ps) and 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 of s-2 /pqbb4ky ai4t8wdu cu2b0$3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque of orjr 7r2 kbzao c3(f1-(-fze e280 $m \cdot N$ 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 slipping dovq-k ko1re79ar7 hcc )wn a larckc ov-q)k 177hre9ane at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth wit5vducl aj6n560vv 9kl h respect to the Sun as the sum of two termc vj0dal5v k5vl966u ns,
(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 v1i- hd*p jyk9kg and a radius of 7.50 m. How much net work is required to accelerate it*jh91 pyvki -d from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and masaa3h78jf jhb(y1/8z - lnqymos 1.80 kg starts from rest and rolls without slipping down a 30. hjja8ly bh/(ma y 8fq3-o1z7n0 $^{\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 star od.e:(q,9w4atcptikts to fall and its (aot .pt:e,w kq9 i4dclower 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 wxe9 ./vmzhi2b0bz*ywfng1 1.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an anguvwe0*g1 z 9n/f1xmzbh.i w 2yblar speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg unifo*0jskfdl2/ u xrm cylindrical grinding wheel of radius 18 cm when rotating at 1500 rp0fu l/sxdj2 *km?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotating at a ratemf dk41a *mrhu *hm ze 6 ng:jckutn5cf6;4v;1 of 1.3rn;*c6hz1jk1t6c:nmu4*rgva4ehf 5 dmkum; f ev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce her mometpvs 0-wk eh :5.k7dmvnt of inertia by a factor of ak 0.tv shew75 dk-:vmpbout 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial ratei( rdth r60qhc*3 h6ay of 1.0 revrha3r q tc(06hydi6h* every 2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis through its center at a frf o or+7vr 2m0j)z/hlxequency of 1.5rev/+ )h vxzmfjro0o r/2l7s 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?    $rev/s$

  (a) What is the angular momenton8z:f 6:qcw xum of a figure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momento5oaq,9(geup5cob:6cv: - exk. lmhmum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$

(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical 7o9te aihb/ar z;d8b* .ghv y9disk of moment of inertia I is dropped onto an identical disk 9v ;9. hadb8ob hzi/gaer7t*yrotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turnsw7)j fk9xo4t6 7cs tin at 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at th):0ca/l2ci,eu ndu2g9; cqh2 n5ipota ,de f xe center of a rotating merry-go-round platform of radius 3.0 m and mhexqocip2d5a0n:anlueuc ft9 ,gd,/;c2 )i2oment 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 w)8-takt5c m0g dt(kl.raw i4jith an angular velocity of 0.8 )rd (8ja. wck-0 m4a5gtkttli$rad/s$ 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 iyl 0r-v145yq. qjj ftznto a white dwarf, losing about half its mass in t14j.r 0-y fqtvql j5zyhe 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 excess of 120 xmqcl;7v8zx yvy f7s (81y ;wu$km/h$ 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 xmisv;svdk5 48: eib9$ 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 ax aul. w0a+c7ewi;h6y*7vdty( dq ej,1ypn2i t rest, that can rotate freely without friction. The momen(cane*yp uti wdy 1ld, 6w. 7yhxi;q27+evj0at 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 diamete (7e a:9y/jtvu, hm jzz0vlqi1r merry-go-round turntable that is mouy0/ujm9,1 lq(h:zj vt7vizeanted 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 with thv moh-70 ha6d8jdoql) e end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the sq6h8jl dv07h )o-aomd pool’s center of mass move?

  The Moon orbits the Earth such that the samersm+6 pl.tts5m 7 ,rro side always faces the Earth. Determine the ratio of the Moon’s spin ang7 p+ 6motrlsm.r,5r tsular 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 ray9v0 g9aegvhp 01 h /cs7q7iofte of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of ram cz8 +mtem,s* n39jthdv8gu:7e fok*dius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the mot*nj, cmmdte*89z+g moue8 shk73 f:vtor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks8.ez/hab51 nrtib)e l , each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrice.a1ilz )t/85r nebbh al 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 wheo9(m,pqa +g7j vdu5m eel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as eb6,wynu:z.w*-sqdeho wf 4/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, losing three-quarters of its mass in the process, what would its rotation speed be? Assume t ehy ow6b/ews:. , fd4w-zu*qnhat 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 lm6l+3pml8 u8z;j ljjfow-pollution automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical fly;uj38j +mf pjlzl m6l8wheel 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 illustratfi*:vlnita8, es 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 rolli/nxh u:.qopw7th9e s uu28zm1ng on a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot flfa*(8j dyj1t reely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. tf*1 yl(a8jjd 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 l: l85lzwobc/bm,toqkf+ 05 hvertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a slcol8fowq/0 ,:zlm+k 5tb5hbtep against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v28 gh,:xh9 )7grf mqq9.tqqk a empmi,=4.2m/s on a flat road is making a turn with a radius The forqh9tmg:f pe9,,r7. x h)8amkmqq g 2qices acting on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5 m and begins whirl+6+ ;;xv ew ib/mb7bzxbq j5jzing 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 axbij6be;q /v;+7 jwm zz+b xb5chieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s+:bnu y dff1;u5 /fyhsx,p+gj body as a solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly agu,1uy; d fjn gy:hsx++/ff5pbainst her body.   

  You are designing a clutch assembly which consists of tb4hkcefao,/ /-m /n.*ifofxn wo cylindrical plates, of maia ,c xom. hb-*en/4fno//f kfss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 radiu1iyq2e o*. /kg)fwtg rs r rolls along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest*/)kwqretfgyo.1i 2 g point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assume a n9jpanh ;88y$r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutionnh4e..+mk2 tdarza(u lk .l lx3t* z)ps as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m.k+z.3pa lrk.mu.ld( h a*z2ttn 4)xle (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