CHAPTER 5: Circular Motion; Gravitation
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1 CHAPER 5: Cicula Motion; Gavitation Solution Guide to WebAssign Pobles 5.1 [1] (a) Find the centipetal acceleation fo Eq a R v ( 1.5 s) s (b) he net hoizontal foce is causing the centipetal otion, and so will be the centipetal foce. F R a R ( 5.0 kg) ( 1.4 s ) 35.5 N 5. [5] he obit adius will be the su of the Eath s adius plus the 400 k obit height. he obital peiod is about 90 inutes. Find the centipetal acceleation fo these data k k 6780 k ( 6 ) a R 4( 4( sec 9.18 s 60 sec 90 in 1 in 5400 sec 1 g 9.80 s ) 0.9 gs Notice how close this is to g, because the shuttle is not vey fa above the suface of the Eath, elative to the adius of the Eath. 5.3 [6] o find the peiod, the otational speed (in ev / in) is ecipocated to have in/ev, and then conveted to sec/ev. Use the peiod to find the speed, and then the centipetal acceleation. a R v 1in 45 ev 60 sec in ( s) 0.16 sec ev 3.6 s 0.16 v ( 0.16 ) sec s 5.4 [7] See the fee-body diaga in the textbook. Since the object is oving in a cicle with a constant speed, the net foce on the object at any point ust point to the cente of the cicle. (a) ake positive to be downwad. Wite Newton s nd law in the downwad diection. F R g + F 1 a R v 4.00 s F 1 ( v g) ( kg) s ) 3.73 N ( 1
2 his is a downwad foce, as expected. (b) ake positive to be upwad. Wite Newton s nd law in the upwad diection. F R F g a v 4.00 s F 1 ( v + g) ( kg) s ) 9.61 N ( his is an upwad foce, as expected. 5.5 [8] he centipetal foce that the tension povides is given by F v. Solve that fo the speed. R v F ( 75 N)( 1.3 ) R 0.45 kg 15 s 5.6 [9] A fee-body diaga fo the ca at one instant of tie is shown. In the diaga, the ca is coing out of the pape at the eade, and the cente of the cicula path is to the ight of the ca, in the plane of the pape. If the ca has its axiu speed, it would be on the vege of slipping, and the foce of static fiction would be at its axiu value. he vetical foces (gavity and noal foce) ae of the sae agnitude, because the ca is not acceleating vetically. We assue that the foce of fiction is the foce causing the cicula otion. F R F f v µ s F N µ s g v µ s g ( 0.80) ( 77 ) ( 9.8 s ) 5 s Notice that the esult is independent of the ca s ass. 5.7 [10] In the fee-body diaga, the ca is coing out of the pape at the eade, and the cente of the cicula path is to the ight of the ca, in the plane of the pape. he vetical foces (gavity and noal foce) ae of the sae agnitude, because the ca is not acceleating vetically. We assue that the foce of fiction is the foce causing the cicula otion. If the ca has its axiu speed, it would be on the vege of slipping, and the foce of static fiction would be at its axiu value.
3 F R F f v µ s F N µ s g µ s v g ( 1 s + *( 95 k h) 3.6 k h - ), s 0.84 Notice that the esult is independent of the ca s ass. 5.8 [13] At the top of a cicle, a fee-body diaga fo the passenges would be as shown, assuing the passenges ae upside down. hen the ca s noal foce would be pushing DOWN on the passenges, as shown in the diaga. We assue no safety devices ae pesent. Choose the positive diection to be down, and wite Newton s nd law fo the passenges. F F N + g a v F N ( v g) We see fo this expession that fo a high speed, the noal foce is positive, eaning the passenges ae in contact with the ca. But as the speed deceases, the noal foce also deceases. If the noal foce becoes 0, the passenges ae no longe in contact with the ca they ae in fee fall. he liiting condition is v in g 0 v in g ( 9.8 s )( 7.4 ) 8.5 s 5.9 [16] (a) At the botto of the otion, a fee-body diaga of the bucket would be as shown. Since the bucket is oving in a cicle, thee ust be a net foce on it towads the cente of the cicle, and a centipetal acceleation. Wite Newton s nd law fo the bucket, with up as the positive diection. F R F g a v v ( F g) ( 1.10 ) 5.0 N.00 kg.00 kg 9.80 s 1.73 ( 1.7 s (b) A fee-body diaga of the bucket at the top of the otion is shown. Since the bucket is oving in a cicle, thee ust be a net foce on it towads the cente of the cicle, and a centipetal acceleation. Wite Newton s nd law fo the bucket, with down as the positive diection. 3
4 F R F + g a v v ( F + g) If the tension is to be zeo, then v ( 0 + g ) g ( 1.10 ) ( 9.80 s ) 3.8 s he bucket ust ove faste than 3.3 /s in ode fo the ope not to go slack [18] Conside the fee-body diaga fo a peson in the Roto-ide. F N is the noal foce of contact between the ide and the wall, and F f is the static fictional foce between the back of the ide and the wall. Wite Newton s nd law fo the vetical foces, noting that thee is no vetical acceleation. f f F y F g 0 F g If we assue that the static fiction foce is a axiu, then F µ F g F g µ. f s N N s But the noal foce ust be the foce causing the centipetal otion it is the only foce pointing to 4 the cente of otation. hus F R FN v. Using v, we have F N. Equate the two expessions fo the noal foce and solve fo the coefficient of fiction. Note that since thee ae 0.5 ev pe sec, the peiod is.0 sec. F N 4 s g µ s g µ s s Any lage value of the coefficient of fiction would ean that the noal foce could be salle to achieve the sae fictional foce, and so the peiod could be longe o the cylinde salle. hee is no foce pushing outwad on the ides. Rathe, the wall pushes against the ides, so by Newton s 3 d law the ides push against the wall. his gives the sensation of being pessed into the wall [0] A fee-body diaga fo the ball is shown. he tension in the 4
5 suspending cod ust not only hold the ball up, but also povide the centipetal foce needed to ake the ball ove in a cicle. Wite Newton s nd law fo the vetical diection, noting that the ball is not acceleating vetically. g F y F sin g 0 F sin he foce oving the ball in a cicle is the hoizontal potion of the tension. Wite Newton s nd law fo that adial otion. FR F cos ar v Substitute the expession fo the tension fo the fist equation into the second equation, and solve fo the angle. Also substitute in the fact that fo a otating object, v. Finally we ecognize that if the sting is of length L, then the adius of the cicle is L cos. F cos g v cos 4 4 L cos sin sin g g ( sin 1 4 L 4 L 9.80 s ) s sin º 9.80 s he tension is then given by F g kg sin sin 5.94º 14. N 5.1 [8] he spacecaft is thee ties as fa fo the Eath s cente as when at the suface of the eath. heefoe, since the foce as gavity deceases as the squae of the distance, the foce of gavity on the spacecaft will be one-ninth of its weight at the Eath s suface. F G 1 g 9 Eaths suface 9 ( 1350 kg) 9.80 s N his could also have been found using Newton s law of Univesal Gavitation [30] he foce of gavity on an object at the suface of a planet is given by Newton s law of Univesal Gavitation, using the ass and adius of the planet. If that is the only foce on an object, then the acceleation of a feely-falling object is acceleation due to gavity. 5
6 F G G M Moon g Moon Moon ( kg) ( ) g Moon G M Moon N kg Moon 1.6 s 5.14 [31] he acceleation due to gavity at any location on o above the suface of a planet is given by g GM, whee is the distance fo the cente of the planet to the location in planet question. Planet g planet M Planet M Eath 1 M 1 G G G g 1.5 R 1.5 ( 1.5 R ) Eath Eath Eath Eath 9.8 s s 5.15 [35] In geneal, the acceleation due to gavity of the Eath is given by g GM Eath, whee is the distance fo the cente of the Eath to the location in question. So fo the location in question, g 1 g 10 suface G M Eath 1 G M Eath R Eath 10R Eath R Eath [38] he distance fo the Eath s cente is REath + 50 k Calculate the acceleation due to gavity at that location. g G M Eath G M Eath N kg 1g s 9.80 s ( 0.94 gs kg ( ) s his is only about a 7.5 eduction fo the value of g at the suface of the Eath [43] he speed of a satellite in a cicula obit aound a body is given by v GM body, whee is the distance fo the satellite to the cente of the body. So fo this satellite, 6
7 v G M body G s ( kg) ( ) M Eath R Eath N kg 5.18 [57] Use Keple s 3 d law fo objects obiting the Sun. Icaus Eath 3 Icaus Eath Icaus Eath Icaus Eath / ( d ( ) 365 d /3 1.6 ( [58] Use Keple s 3 d law fo objects obiting the Sun. ( Neptune Eath ) Neptune Eath Neptune Eath 3 Neptune Eath 3/ 1 yea 4.5 ( 10 9 k 1.5 ( 10 8 k 3/ 1.6 ( 10 yeas 5.0 [66] A fee-body diaga of azan at the botto of his swing is shown. he upwad tension foce is ceated by his pulling down on the vine. Wite Newton s nd law in the vetical diection. Since he is oving in a cicle, his acceleation will be centipetal, and points upwad when he is at the botto. F F g a v v ( F g) he axiu speed will be obtained with the axiu tension. v ax ( F ax g) ( 1400 N ( 80 kg) ( 9.8 s )) kg 6.5 s 5.1 [16.1] he agnitude of the Coulob foce is F kq 1 Q / ( N /C )(.50 C)(.50 C)/(3.0 ) N
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