A bicycle odometer (which measures distance travebwy,mbs q* xbvcu o1 21;yx;.harj17eled) is attached near the wheel hub and is designed for 27-inch wheels. What hap;yhbv . bm1o r1b2x*w sqeu,ca ;71xjypens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. D9wvg bn)x b+6 wa)i3xhoes a point on the rim have radial and/or tangential accwx+3 )an ibwg)9h x6bveleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described bwslkxo1 b41cclgr535 y a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larlu9u)n/c+ew fger 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 your head thancjf luz8sx+/0 when your arms are stretched out in front of you? A diagram may help you to anxjczf/+slu80 swer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the 69mx8th gwr3f 5 u977lmeyh2bv* xtk spedal cranks. In which gear is it harder to pedal, 5shgvl97*mh2xy u xbm8r39k 7ttfwe6 a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slendeby-qonfw u:4z:lnfqv8;yjrc .l8kbj 9+ ;; vlr lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explajbk frnl8ub.;o wf:vq q+: n z;y- l894;cyjvlin why this distribution of mass is advantageous.

Why do tightrope walkers (F 4t xeo 8i,n-k;mfxm+cig. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is fmt9 oet*+ a7ethe net torque also zero? If the net torque on a system is zero, is*fm9teeat7+ o the net force zero?

Two inclines have the same height b h e0ah jwq1su9(eld67ut 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? Exh u j(a0796desh1qwel plain.

Two solid spheres simultaneously start rolling (from rest) down an incline.,g5vv sapzbw8s)ooy gi4m:- 2 One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total-8v,yi vz o5g mpg: )ws2oa4sb kinetic energy at the bottom?

A sphere and a cylinder have a (z4y2ie6zzy 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 boty4 6z2e( izzaytom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet mo )o*d-4h e6qf h0 9d3,igip4cyi tbzbrst moving or rotating objects eventually slow down and stop. Expla 3f4ch0,bz4)q*6r teg odidii -y9pbh in.

If there were a great migration of people toward the Earth’s equator4gtvf91wws/ a 1c,s jq, how would this aff ft gw9j1ss,c 1qva4w/ect the length of the day?

Can the diver of Fig. 8–29 do a somersault without hamhxu 3,kr.+snt segx/+3i9jk ving any initial rotation when xnh grj/s.9mkk ,te3i+xu +3sshe leaves the board?

The moment of inertia of a rotating solid di(3m:d yjinhz .sk about an axis through its center of ma. ni :3dzymh(jss 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 hb.rlolne bw) .*r7 j e1eh84pqolding a 2-kg mass in each outstretched hand. If you suddenly dropr*p j4l.e)rhwo8. e7nl1bbq e the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hollout/yu n h)q)xo8siy /7w and the other is solid. Describe an experime /ui87)uy yhtxq )o/nsnt to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle pointseddrljyxn: 6(+gs 1j*u.:u *ftdz5cb west. In what direction is the lineajgs(te l* 1fbd *nd.z+65yrd :x:uu jcr 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 of a large fnno p g- t(h,n:znht5:reely rotating turntable. What happens if you walk toward the centen:(o5tnh, tz-:n hn pgr?

A shortstop may leap into the air to catch a ball and throw it 5v.t a2r*u5xhi)m7dbd 7l isnquickly. 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 direction (Fig. 8–36).xbid75dmv *t7suh5.in l a 2)r Explain.

On the basis of the law of conservation of angular momentum, di pwiae,h1ticrpn 6.0veuhf/*-w vn79scuss why a helicopter must ha n.6eww10i*ui c-np fa h,pvev/79rth ve more than 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: (a), vv1c f1q zttric-3p 1uvm.s)ere.6 h 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidence. Calculate, usif(s*(:k cpjj(tju g- wcso2b(ng the information inside the Frontw((os2j :kcjjb-u( *ftpcs (g Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the M: 9: ijtu lov2.xpf-xqyavhu :fta5 ic:/:s)goon, 380,000 km from Earth. The beam diverges at an angla2i: 9:fxptqtuyo/v5cg .v jh:) :u:a-s ixfle $\theta$ (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 of3rk hg,yj4v6a 6500 rpm. When the motor is turned off during operation, the blades slow to resa 4hy 6gkjrv3,t in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If thax )0p xb4uzpr i( 1(ww75gqhbe ball makes 15.0 revolutions, what is its dqbiw gh xa w0 z5p7(bp1xr)(4uiameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutio cq6uoksv8((smg 0pjv 2a1/v hns do the wheels mh v( gao0 2mqv v81/kcss(p6ujake?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate itcdqs0w pp/j:6s angul6jqdc/ :wsp p0ar 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 ln 54b2j;jrr n4nwhno2qfp uq+ u.8s)in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from the c;2jlqr ubqn)p o4rj+5. n4uhf2sw nn8enter?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity o ppb94vc/ml1,vy1 vhih juc (j4u5r-o f the Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a p3 xh: ulmd ;sxexj89b*t)c k.foint
(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 ths 4s8si awp *nennnfw*r 7+e-lx9yb *sr 8/6ide axis of rotatn*rsba*i88ix ssen+6 * l/fd7ernpyw w9-n4sion is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uni 4so)6rp)t -uv:gupz/ls7r v uformly about its center from 130 rpm to 280u7ps-r)tugvovsr /) l up:6 4z rpm in 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 radiush m.vul7j 08lt $R_1$ 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 fi .ziv yf:;wq7w1a* pti3qo.Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, theqfwyiftw3q.i ia *vp7 :;zo1.y 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-4ikmmq mxc+8 to 15,000 rpm in 220 s. Through how many revolutiockim -xmq+48m ns did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200.;jkyu j9uy, vsq00ku0er-dfd 2h(wb 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 q3v 0l 0kvgyqyyk;o o 3ec-7w/whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its finagwy;3 vqk voy-/kl0 ecqy73o0l 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 37jyzdwc/0 i /m60 rpm in 6.5 s. How far will a point on the edge of the wheel have70cmyjw/zid/ traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 1y9no a+ g q(9sgnw)x ggv858ij500 revolutions befyq9g 8x 9g ai 8+jonns5gwv()gore it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed uniform m1.jnp lfe 5,-w aruh mol(u1(t5+qmely from 95km/h to 41 lowl+tme(q,u(ra f1m-pn m5jhe5. u5km/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 revolutiot2i on sf-1js 5let9 94ulevu8ns as the car reduces its speed uniformly from 95km/h to 45k t8i4lulfevse9 92t nsoju1 5-m/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each pedal when climbing a vw -f0cqrqd-0hcf-d0- 0qrq vwill. The 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 N4. dp6ljgz 6c.lfqu1q on the end of a door 74 cm wide. What is the magnitude1c gud.lq jlp 6fz.6q4 of the torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig. 8–39. Assume+gm(au zpv*gg* djnoq 3w vhjn1j-*4( that a frictijqj -jagp+v** h(vn g3dun z*o1m(4wgon 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 ztk46 **1ulz)ypm/gmirn 3 ws which pivots as shown in Fig. 8–40. Initially the rod is held y pul/i)mgs64*n w z1rmkz*t3in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylindera3/n++ywewb c head of an engine require tightening to a torque of 38 a w e+wnc/3b+y$m \cdot N$ 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 uq mix/. wkp-/a 10.8-kg sphere of radius 0.648 m when the axis of rotatixq i.p/u/-wk mon is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of b)/mue*ik:hm a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can bmk h*:)iemu/ be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotajs/ciwv t w2b0;(p5izted in a horizontal circle 2w 5pbz0j( ;/is tivcwof radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a pot5 xz dq;27d* gdlwfa3dter’s wheel rotating at constant angular speed (3l7gdz*q;2wa 5df xddFig. 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 of point objects shown in Fig. 8–4mzs+c 9gt7df3 3 +9g d7ft s3cmzabout
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen atoms whose total massafsc nr/ xdtdz ue*- 9/a32fs, 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 ri or+ gtf n-c:6rt+6jzate, 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 ifj66rttczf+ no :ig- r+ the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uwx)ns:dp9 lqby(gs3 *niform cylinder with a radius of 8.50 cm and a mass of 0.580 kg. Calcq w )p3sbglxn*sy( :9dulate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerating it from rest to 3u,sb8m l, f5xh $rev/s$ 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 merry-go-(zel0 *djyc-qf*d ry *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 requir*cqf l*yddj 0re* (z-yed 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 eventual8zl/ y tcg0il9ly brought uniformly to rest by a frictio giz0l9y/ 8tlcnal 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 33 lsa);pqgk ldo1e,a1iux /fa 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 ac1lho(w7 0ykkc tion of the forearm, which rotates abouyw7ok0h l (1ckt the elbow 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 ca+ uevj ma+a25fn be considered a long thin rod, as shown in Fig. 8–46a+efjv+ ma25u .
(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 con0.qweob c,rq,sists of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammwhdjd5 wlx5/*er from rest within four full turns (revolutions) and releases it at a s l 5jx*/5wddhwpeed 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. .1 c/qsjch hq-di+x kac4xu5 $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque of 2t ny9e1al:-bc5 p3hx-z5 lldu80 $m \cdot N$ 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 withor 1bn(a nzas8dn,u(l5ut slipping down a lane 1alnnrsd a nzbu8 (,5(at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the sum of /5aswju y- h/wt a-wy/w5uj/shwo terms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg andyeolhgm g9- 88frm,.05g8 gfoy g w.vv a radius of 7.50 m. How much net work is required to acceleratehgr5 8o8e,f0gy.8f - mv9yg.logvmgw it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 k4zy v- t /(tm8:f(pdvp nl 2mg/uf5jrcm and mass 1.80 kg starts from rest and rolls without slipping dovmyd g nur- pp52/ j8fttv: /z4(flm(kwn 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 bal.8lov tj fk(oq:5 rqf1anced vertically on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the .q t8lfo5q:1k (fovrj pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular m7ej /og qf;q*yomentum of a 0.210-kg ball rotating on the end of a thin str/fj;geyq*7 qo ing in a circle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kgsvy46gq/ 5e;iw d+gwpt tas(yf3l+ 7p uniform cylindrical grinding wheel of radius p+tw +p(a;e7i wy46l /v dq35tfssggy18 cm when 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 rotatinmouvmuimn p* 4n3 g lu.y;g*u td596k(g at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decre6l4ym i* k(5.gn d;g n 3utvoumpu9u*mases 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 moment oub;2 hsy6wcl)2tsi2y0 7skc afmf9 5tf 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 posit29f2c cfhs i25s; ky7m6wu0bs a) ytltion, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation ratl:uiopi t 9h;5e from an initial rate of 1.0 hlt 5;o9uiip:rev 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 verti 2k7hj:gpt-q j eh)0kf/gw8 dqcal 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 tjp7:jdt2/ hk0h -qeg gkqw )f8he wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentu4 5pl azj-p/rtm 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 momec2;pp/qdj t5 mt/1dm w2l;qwyntum 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 d5)y fxdx.2d taisk of moment of inertia I is dropped onto an identical disk rotating af)x d ytx2.5dat angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at.inn+f4c fovn3d9z v8mz ip4, 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 cen,bfi3d5xr sh91q z*s wter of a rotating merry-go-round platform of radiuxfi *w39 h s5rsdqz,1bs 3.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angular velocity w4r ,ruye n5(oof 0.8 5 erro(y,wnu4 $rad/s$ 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 1rl:0dq gl0+ litygv8r4n . enwhite dwarf, losing about half its mass in the process, and windingrgl0t0 8yn4e 1. :q drlinv+lg 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 ex9 pmmw*m cc99ocess 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 pu0rya2rb*e t,q/ b,g$ 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 resp+i k511 a lgdzrp+k/gpna;c(t, that can rotate freely without friction. Thep+zp5/1;l1 kg cidp ag(+ ankr moment 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 t:e )cnr:s0io ehe edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia of 1700 )e eno0: :ircs$kg \cdot m^2$ 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 the end of the rope lfwu 59:ukvx 8wying on the top edge of the spool. A person grabs the end of the rope and walks a distance L,wx kw9:85vuf u 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 sugpbij aot4imbh )8/rv.a1 7/hch that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital 8h 7vo/1ir4/bbjg pami.t )haangular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate5fgb fj o0kq*( 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 shapey pto +o,qq zbi6n0)v2/co fu/ of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleratq/o ,z/u 02to+ypnfcq o6 bi)vion. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each rj ;2n0zx+e*rv68f *(iuz1um fjid uiof mass 0.050 kg and diameter 0.075 m, joined by a (concentric)frvf+i*ijj;u8d 1 e( *0urunxm 2i6zz thin solid cylindrical 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 tpp78u fg6bi4s;asq2twvq1 8hhe 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 w8nqhpj9b 5mqc okfm 0- ux(gf i()1(arith mass 8.0 times as great, were rotatingumm (fkpo15fc8ar9 qixgn)j0-qh b (( 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 automobzwp 3qa nfxv25d 187d ll6+yxgile 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 flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheelyad5nzx dqp8 gxl 6f1w v3+72l “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 illustrate (h:ast/ xo8tus 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 oy ijd0 g2awami*n e g6u+qi(1*n 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 freely (i.e., we ignore fric9nr(lhvozvbhe1 2fe:w62xc.tw 3 -ma tion) 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 shown. [Hint: See Fig. 8–r2awx3c ( w2 mo:n9.lbeh ve6t-1zvh f21g.]

A wheel of mass M has radius R. It is standing vertically on the flkvtr 5v4 ;8b szkj /c4tmdp;lbwiih 88y.*3 zxoor, and we want to exert a horizontal force F at its axle so that it will climb a step against which bs8p8ctzvk 445 3; iljk;dt8*wiy b.x/ mrzhvit rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road ix4/jqc3 f6az.r bn*hds making a turn with a radius Thehj azb/3fcqr.*n d6 x4 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 rock into a slingi 4xtk /kkiqpw c2-+c5 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 2ii kq/x+pkwk45c-c tto achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a wljrfi22 sm y0:a+v *2c3dnqtsolid 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 ar2fj3s 2ida0*wv2 c yqlntm :+rms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists obw djmym )lgm5-)p1 8of two cylindrical plates, of mass y)m1 p) dl womm5bjg-8$M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 rou tg ;l9wngrg2y/eod7; gh track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the tr7;y o2r;lnwd 9g g/etgack? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, bp 0lerlk9-ra* ut do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 5* ; o +16 e.le) xjnbxhrqyymur,8mfkrevolutions as the car reduces its speed uniformly from 90ojey;xqh *ur 6n k)fy5m +mr.b8lx1,ekm/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s