PHY 171 Practice Test 3 Solutions Fall 2013

Transcription

1 PHY 171 Practice et 3 Solution Fall 013 Q1: [4] In a rare eparatene, And a peculiar quietne, hing One and hing wo Lie at ret, relative to the ground And their wacky hairdo. If hing One freeze in Oxford, Ohio, Wherea hing wo toat in Key Wet, Florida, Which hing ove with a larger angular peed errily adorned in hi crion thneed? a) hing One. b) hing wo. c) Both hing have the ae angular peed. d) Inufficient inforation. Solution: c he angular peed i the ae for any point in the rotating rigid body. he two hing are at different ditance fro the axi of rotation of the Earth (larger in Key Wet than in Oxford), but their linear peed are alo different (alo larger in Key Wet by a proportional aount) o, fro ω = v/r, their angular peed are equal. arry he next four quetion refer to the following ituation: arry pin a arry-goround of a = 100 kg and radiu = 1.5. he yte tart fro ret a the child applie a force of contant agnitude F = 15 N but changing direction, alway tangent to the edge of the erry-go round. he friction i negligible. Q: [4] Conidering the erry-go-round a a thin cylinder, what i the agnitude of it angular acceleration? a) rad/ b) 0.15 rad/ c) 0.0 rad/ d) None of the above. Solution: c he angular acceleration reult fro Newton nd law applied to the rotation under the torque correponding to the force F: F F I 0.0 I 1 rad Q3: [4] Conidering that arry ove cloe to the edge of the erry-go-round, what i the angular velocity in rotation per inute, rp of the erry-go-round after arry run 15 eter in a circle? a).4 rp b) 4.0 rp c) 9.5 rp d) 19 rp Solution: d Becaue arry ove a the point on the edge of the erry-go-round, the correponding angular diplaceent i given by the arc L = 15 that he travel, by Δθ = L/ = 10 rad. Furtherore, the tangent force i contant in agnitude and it ake a contant angle with repect to it ar (that i, the radiu), o it torque i contant. Hence, we can ue the kineatic of uniforly accelerated rotation: 0 rad rad 1 rot 60 ec ec π rad 1 in rp. Criven! arry ain t no dizzy bairn! 1

2 PHY 171 Practice et 3 Solution Fall 013 Q4: [4] What i ary centripetal acceleration after he travel 15 eter in a circle? a) arry i a lady, ir! She ain t got no darn centerpeetal aeleration! b) 6.0 / c) 0.15 / d) Inufficient inforation, becaue arry peed cannot be found. Becaue arry ove like the point on the edge of the erry-go-round, her acceleration i the ae a that of the repective point. hence, uing the calculation fro the previou quetion, ac rad Q5: [4] What i the iniu aount of energy that arry pent a he traveled 15 eter in a circle? a) 5 J b) 150 J c) 15 J d) Zero, ince the force i centripetal force i perpendicular on the travelled path. Solution: a he quetion can be anwered uing either of variou equivalent calculation; for intance, 1. Note that arry applie a force that i tangent to the path that he travel and the iniu energy i the work done by thi force. For any hort enough egent of her path, the force i parallel with the egent, o the net correponding work i W FL 5 J.. Alternately, one can calculate the work done by the torque of the force F, in which cae the diplaceent i angular: W F L FL 5 J. 3. oreover, one ay note that by the Work Energy heore the work i ued to increae the rotational kinetic energy of the erry-go-round, o the energy i KE I 4 5 J 1 1 rot. Q6: [4] A particle ove in a circle of radiu r with a uniforly varying angular peed ω = ct, where c i a contant and t i the tie. herefore, the tie dependent agnitude of it total linear acceleration i given by 4 a) a rc t b) 6 a rct 4 c t c) a rc d) a rc c t 4 1 Solution: d In general, the acceleration of a point oving on a circular trajectory i pointing inide the circle. It can be conidered a having two coponent: a coponent a t tangent to the trajectory decribing how the agnitude of the velocity change, and a radial (centripetal) coponent a r decribing how the direction of the velocity i changing. herefore, the net linear acceleration i: r t a a a r r. In our cae the acceleration i contant, α = ω/t = c. Hence a r rc r c t r c rc c t

3 PHY 171 Practice et 3 Solution Fall 013 Q7: [4] A particle of a i connected to one end of a pring of force contant k, and relaxed length L 0. he particle i rotated horizontally uch that the pring tretche to a length L, a in the figure. Which of the expreion below ot likely repreent the peed v of the particle? a) v kl k L k v b) v LL L0 Lg in F e θ c) k v LL L0 k d) v LL L0 Solution: c he role of centripetal force i played by the elatic force in the pring, o v k k L L v L L L L 0 0 Q8: [4] Earth a planet of a = kg ake one revolution per day. What hould be the radiu of the Earth in order for object to be apparently weightle (zero noral) at the equator? a) b) c) d) N g he period of rotation of the Earth i 1 day 4 h 60 in 1 h 60 1 in he object are kept on the circular trajectory of Earth radiu by a centripetal force g N v g N 4 4. ecall that the gravitational acceleration depend on the radiu of the planet: g = G/. herefore, ince apparent weightlene ean that N = 0, the correponding radiu i given by g G G G ω Q9: [4] What i the gravitational acceleration at a ditance fro the Earth urface equal to half the radiu of the Earth? a) 9.8 / b) 4.4 / c) 0.18 / d) 4.9 / r At a ditance r fro the urface of the Earth, the gravitational acceleration i g g G G r r / 4.4 / r 1 r 1 r 3

4 PHY 171 Practice et 3 Solution Fall 013 Q10: [4] ecall that a black hole a a tartup idea i an object collaped until light cannot ecape fro it. Suppoe that planet Earth were to be queezed into a black hole. How any tie aller would be the axiu radiu of thi black hole (called Schwarzchild radiu, r S ) copared to the noral radiu r E of the planet? You ay need the peed of light in vacuu of about /, and the noral ecape peed of planet Earth of about /. a) Nonene: Earth cannot be ade into a black hole, no atter how uch it i collaped. It jut doen t have enough a. b) tie aller c) tie aller d) tie aller Solution: d Indeed, Earth doen t have enough a to collape pontaneouly; thi happen only i uperaive tar. However, the quetion doen t ak about a pontaneou collape: it ugget a cenario where the planet i forced to hrink by oe echani. o copare the radii, one can divide the forula for the ecape peed c for the black hole to that for the noral planet. Becaue the a of the black hole tay the ae a the a E of Earth (it i jut collaped into a uch aller volue), we ee that the radiu of noral earth i larger than the radiu of the reulting black hole a given by the iple calculation: c G r r r c v r r v E S E E tie. G E re S S hat i, Earth ha to be hrunk to about 8.8- radiu to becoe a black hole. Q11: [4] A DVD with oent of inertia of about kg rotate with unifor angular acceleration 00 rad/. Starting fro ret it reache it noinal angular peed in about.4 econd. What i the iniu energy pent in order to bring the dik to it noinal angular peed? a).7 J b) J c) J d) 3.1 J Solution: d Since the dik tart fro ret, the noinal angular peed i given by t 480 rad. Hence, ince the energy it need i at leat equal to it rotational kinetic energy, the iniu energy i 1 K I 3.1 J. Q1: [4] A 70-c tick with a 1.4 kg i ounted on a pivot a in the figure. What i the agnitude of the net torque acting on the tick with repect to the pivot? a) 1.6 N r b) 3.6 N c) 0.69 N d) 0 g Solution: c he weight of the tick act in it center of a at a ditance r = 5 c fro the pivot. herefore gr 0.69 N Q13: [4] wo ae, 1 = 1.0 kg and =.5 kg, are held together by a ale rod with length L = 1.5. he yte rotate about it center of a. What i it oent of inertia? a) 1.1 kg b) 1.6 kg 1 c).1 kg d) Inufficient inforation ince it depend on the angular velocity. x c 4

5 PHY 171 Practice et 3 Solution Fall 013 In order to calculate the poition of the center of a, chooe an x-axi with origin in the poition of a 1. In thi cae, the poition i given by 1 0 L L xc Subequently, we can calculate the total oent of inertia with repect to the center of a uing: I I1 I 1 xc L xc 1.6 kg Q14: [4] A heavy phere and a light phere have the ae radiu and the ae kinetic energy. Which ha the greatet angular oentu? a) he light phere b) he heavy phere c) he angular oenta are equal d) It depend on the angular potential energy he rotational kinetic energy i related to the angular oentu by KE L I L KE I. rot rot he oent of inertia i a eaure of rotational inertia. If the a i ditributed the ae, the heavier i the object the greater i the repective oent of inertia. Hence, ince the kinetic energy i the ae, the heavier phere ha a larger L. Q15: [4] Scrat it on top of hi acorn which i tuck in the center of an ice dik with radiu =.0 and a = 0 kg, rotating with angular peed ω = 11 rad/. Scrat a i 1 = 1.0 kg and acorn a i = 30 g. Scrat leave hi acorn in the center, and walk radially to the edge of the dik. Conidering Scrat and hi acorn a pointlike ae, what i the angular peed of the yte when Scrat reache the edge? a) 11 rad/ b) 10 rad/ c) 9.97 rad/ d).7 rad/ he angular oentu of the yte bug-dik i conerved. Since the acorn i left in the center it doen t have a oent of inertia, o we get I L L I I 10 rad 1 dik i f dik 1 dik Idik ω 5

6 PHY 171 Practice et 3 Solution Fall 013 P1: A light cable i wrapped around a cylinder of a = 1.5 kg and radiu = A box of a = 10 kg i connected to the free end of the cable and releaed at level A fro ret. he a ove vertically a ditance y 0 = 0.0 to the level B. Notice that the oent of inertia of a cylinder rotating about it axle i provided on the forula heet. ω α a) [4] Chooe the ground at level B, and write out ybolical expreion in ter of known quantitie and the peed v of the box for: A. echanical energy E A of the yte at level A: E gy A 0 B. echanical energy E B of the yte at level B: E v I v I v B A B y 0 g b) [4] Ue conervation of energy to calculate the peed v of the box at level B. E BEA 0 0 v I gy v gy v y g c) [4] Calculate the intantaneou angular velocity ω of the cylinder when the box i at level B, and indicate it direction on the figure (ue ybol and for direction out and into the page). v 13 rad. Uing the right hand rule for the counterclockwie rotation, we ee that the direction i out of the page:. d) [4] Calculate the angular acceleration of the pulley when the box i at level B and indicate it direction on the figure. he torque acting on the pulley i contant uch that we can calculate the acceleration fro the kineatic of the pulley (or of the a ). While the a travel a ditance y 0 downward, the cable ove together with the ri of the pulley, o the correponding angular diplaceent of the pulley i y (poitive ince the rotation i counterclockwie). 0 herefore, ince the yte tart at ret, we can ue v g 0 1 y 0 61 rad. he angular velocity increae, o the angular acceleration ha the ae direction: out of the page. e) [4] Ue the angular acceleration to calculate the torque rotating the pulley, and then ue the torque to calculate the tenion in the cable. By Newton nd law applied to the rotation of the pulley, the torque i given by 1 I 1.0 N he torque i due to the tenion force acting perpendicular on the radiu. herefore 6.8 N 6

7 PHY 171 Practice et 3 Solution Fall 013 P: A block of a and a all ball of a =.5 kg, are connected to each other by a light tring paed through a hole in a horizontal table. he block i at all tie at ret on the rough urface of the table with coefficient of tatic friction μ = he ball hang under the table and rotate uniforly in a horizontal circle of radiu = 0.5 with the tring aking an angle θ = 35 with the vertical, a hown on the figure. (We ay that a ove a a conical pendulu.) a) [4] On the figure below, ketch the free body force diagra for ae and. Conider the portion of the tring between the a and the hole parallel with the tabletop, a repreented. Label the force eaningfully. θ rough table N f y x θ g g b) [5] Write Newton nd law ybolically for ae and along the direction indicated for each object. he expreion for the a hould contain the angle θ fro the coponent of the tenion. a : F x f 0 a : F y co g 0 F N g 0 F in v y c) [4] Ue one of the equation for a in part (b) to calculate the tenion in the tring. hen ubtitute the tenion in the other equation to calculate the peed v of the ball. he tenion can be calculated fro co g g co 30 N. he peed coe fro the other equation: in v v in 1.3. d) [4] Ue the reult of part (b) and (c) to calculate the friction keeping the a fro oving. What i the iniu a that will prevent the block fro liding on the table? he friction i tatic and i equal to the tenion: f 30 N. coθ On the other hand, the friction cannot be larger than μ n. Hence one can derive the iniu a : f N g in g g 8.0 kg. inθ y e) [3] If the tring i cut at the oent hown in the figure fro part (a), how will the ball ove? Circle one: out of the page to the left downward (freely falling) r r 7

8 PHY 171 Practice et 3 Solution Fall 013 P3: A helicopter rotor blade can be conidered a a long thin rod, a hown in the figure. Each of the three rotor helicopter blade ha length L = 3.75 and ha a a = 160 kg. he rotor i initially at ret, ω 0 = 0, and it tart to rotate and accelerate about a point in the center until it reache an angular peed ω = 0 rad/, after a tie t = 10. a) [3] Calculate the net oent of inertia of the yte conidering the blade a rod, each rotating about one extreity. he net oent of inertia i the u of the individual oent of inertia of the rigid body part. Hence, uing the oent of inertia of rod rotating about an end fro the forula heet, we get 1 I 3I 3 L L 50 kg rod 3 b) [5] Calculate the angular acceleration of the yte. Since the yte rotate with contant angular acceleration and i initially at ret, ω 0 = 0, we get 0 t t.0 rad/ c) [5] Calculate the total angular diplaceent of the yte after a tie t = 10. One way to calculate the angular diplaceent i by uing rad hi i equivalent with about 3 rotation. Alternatively, ince the tie i given one can alo ue the equation 1 1 0t t t 100 rad. d) [3] Calculate the net torque applied to the yte by the engine. By Newton nd law,. I 4500 N e) [4] How uch work wa done on the rotor in tie t? he work i done by whatever force F rotate the rotor uch that it pin through an angular diplaceent Δθ. If the force i applied in a point at a ditance r fro the center of rotation, the point travel an arc Δl = rδθ. Since the force doing the work i contant in agnitude and i tangent to the traveled circle in every point, it work can be derived in ter of the torque aociated with the force: W Fl Fr 450 kj. 8