DYNAMICS OF UNIFORM CIRCULAR MOTION
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1 Chapte 5 Dynamics of Unifom Cicula Motion Chapte 5 DYNAMICS OF UNIFOM CICULA MOTION PEVIEW An object which is moing in a cicula path with a constant speed is said to be in unifom cicula motion. Fo an object to moe in a cicula path, thee must be a foce exeted on the object which is diected towad the cente of the cicula path called the centipetal foce. This centipetal foce gies ise to centipetal acceleation. This centipetal foce can be poided by tension in sting, fiction acting between ties and the oad, o the gaitational foce holding a satellite in obit aound a planet o sta. QUICK EFEENCE Impotant Tems centipetal acceleation the acceleation of an object moing in cicula motion which is diected towad the cente of the cicula path centipetal foce the cental foce causing an object to moe in a cicula path peiod the time fo one complete cycle o eolution satellite motion the motion of a satellite obiting unde the influence of gaity in a cicula o elliptical path aound a lage mass tangential elocity the elocity tangent to the path of an object moing in a cued path unifom cicula motion motion in a cicula path of constant adius at a constant speed 70
2 Chapte 5 Dynamics of Unifom Cicula Motion Equations and Symbols p = T ac = m Fc = = GM E whee = elocity o speed of an object in cicula motion o obiting satellite = adius of cicula motion o obit T = peiod of eolution a c = centipetal acceleation F c = centipetal foce m = mass G = gaitational constant = 6.67 x N m / kg M E = mass of the eath = 5.98 x 10 4 kg DISCUSSION OF SELECTED SECTIONS Unifom Cicula Motion, Centipetal Acceleation and Centipetal Foce The law of inetia states that if an object is moing it will continue moing in a staight line at a constant elocity until a net foce causes it to speed up, slow down, o change diection. If a foce on an object is constantly applied towad a cental point, the object will be pulled away fom its staight-line path, and will follow a cicula path. As long as the sting is pulled towad this cental point, the ball will continue moing in a cicle at a constant speed. An object moing in a cicle at a constant speed is said to be in unifom cicula motion (UCM). Notice that een though the speed is constant, the elocity ecto is not constant, since it is always changing diection due to the cental foce that the sting applies to the ball, which we will call the centipetal foce F c. Centipetal means cente-seeking. The thee ectos associated with unifom cicula motion ae elocity, centipetal foce F c, and centipetal acceleation a c. These ectos ae dawn in the diagam of a ball on the end of a sting shown below. F c a c 71
3 Chapte 5 Dynamics of Unifom Cicula Motion Notice that the elocity ecto is tangent to the path of the ball, and points in the diection the ball would moe if the sting wee to beak at that instant. The centipetal foce and acceleation ae both pointing towad the cente. Although the centipetal foce does not change the speed of the ball, it acceleates the ball towad the cente of the cicle, constantly changing its diection. The time it takes fo the ball to complete one eolution is called the peiod T. Since peiod is a time, we will measue it in seconds, minutes, hous, o een yeas. On the othe hand, fequency f is the numbe of eolutions the ball makes pe unit time. Units fo fequency would include eolutions eolution s,, o any time unit diided into eolutions o cycles. Anothe second hou name fo e/s is hetz. The elationships between peiod, fequency, speed, centipetal foce, and centipetal acceleation ae listed in the Equations and Symbols list aboe. Some othe examples of centipetal foce is the gaitational foce keeping a satellite in obit, and fiction between a ca s ties and the oad which causes the ca to tun in a cicle. 5.5 Satellites in Cicula Obits As a satellite obits the eath, it is pulled towad the eath with a gaitational foce which is acting as a centipetal foce. The inetia of the satellite causes it to tend to follow a staight-line path, but the centipetal gaitational foce pulls it towad the cente of the obit. m F G = F C M 7
4 Chapte 5 Dynamics of Unifom Cicula Motion If a satellite of mass m moes in a cicula obit aound a planet of mass M, we can set the centipetal foce equal to the gaitational foce and sole fo the speed of the satellite obiting at a paticula distance : F c m = = F G GmM = GM Example 1 Venus obits the sun at a adius of 1.08 x m with an obital peiod of 8 days. Find the mass of the sun in kg. Solution: Fo consistency, let s conet days to seconds: æ 4 h öæ 3600 s ö 7 ( 8 days) ç ç = 1.97 x10 s è day øè h ø = GM GM 4p = T 4p M = GT p = T 3 4p = æ - ç6.67x10 è 11 3 ( 1.08x10 m) Nm ö 7 ( 1.97x10 s) 11 kg ø = 1.9 x10 30 kg The table of constants in the font of you textbook lists the mass of the sun as 1.99 x kg. 73
5 Chapte 5 Dynamics of Unifom Cicula Motion 5.7 Vetical Cicula Motion Conside a ball attached to the end of a sting and swung in a etical cicle. Because of the gaitational foce acting on the ball, it will incease its speed slightly as it falls fom point A to point B, and deceases its speed slightly as it ises fom point B to point A. This also implies that the tension in the sting inceases as the ball falls and deceases as the ball ises. A B In this case, the centipetal foce is not meely the tension foce in the sting. Since centipetal foce is always the net foce acting on an object following a cicula path, the centipetal foce in this case is a combination of the tension foce in the sting and the weight of the ball. Notice that the tension foce and weight ectos at point A (top) ae in the same diection and thei magnitudes can be added to find the centipetal foce at point A, and the tension foce and weight ectos at point B (bottom) ae in the opposite diection and thei magnitudes can be subtacted to find the centipetal foce at point B. This implies that the tension foce must be geate at the bottom of the cicle than at the top. Ty woking the fee esponse question below, and check you answe fo an illustation of the foces inoled in a etical cicle. 74
6 Chapte 5 Dynamics of Unifom Cicula Motion CHAPTE 5 EVIEW QUESTIONS Fo each of the multiple choice questions below, choose the best answe. Unless othewise noted, use g = 10 m/s and neglect ai esistance. 1. A ball on the end of a sting is swung in a hoizontal cicle, otating clockwise as shown. A, When the ball is at a paticula point in the cicle, the diection of the elocity, centipetal foce, and centipetal acceleation ectos espectiely ae shown below: Which of the following best epesents the position of the ball as it otates clockwise? Top View (A) (B) (C) (D) (E) F a. If the sting wee suddenly cut when the ball is at the position shown in answe (E) in the peious question, the subsequent motion of the ball would be (A) to moe to the ight. (B) to moe to the left. (C) to moe to the top of the page. (D) to moe to down and to the ight (E) to moe up and to the left. Questions 3-5: A 40 kg child sits on the edge of a canial ide at a adius of 4 m. The ide makes 3 eolutions in 6 s. 3. The peiod of eolution fo this ide is (A) ½ e/s (B) ½ s (C) e/s (D) s (E) 4 s 4. The speed of the child is most nealy (A) 4 m/s (B) 1 m/s (C) 4 m/s (D) 10 m/s (E) 360 m/s 5. The foce which is holding the child on the ide is most nealy (A) 30 N (B) 160 N (C) 30 N (D) 1440 N (E) 880 N 75
7 Chapte 5 Dynamics of Unifom Cicula Motion Questions 6 7: The figue below shows the top iew of a ca going aound a hoizontal cicula tack at a constant speed in the counteclockwise diection. Assume the fictional foce between the ties and the oad is at its maximum alue. 7. The ca has a mass m and a speed as it moes aound the tack of adius. Which of the following expessions can be used to find the alue of the coefficient of fiction between the ties and the oad? (A) µ = g (B) µ = (C) µ = g g 6. Which of the following ectos epesents the fictional foce acting on the ties of the ca? (A) (B) (C) (D) (D) µ = g m (E) µ = 8. The acceleation due to gaity g at a distance fom the cente of a planet of mass M is 9 m/s. In tems of the obital distance, what would the speed of this satellite hae to be to emain in a cicula obit aound this planet at this distance? (E) (A) = 3 (B) = 3 (C) = 9 (D) = 9 (E) = 3 76
8 Chapte 5 Dynamics of Unifom Cicula Motion 9. In geneal, (A) the smalle the obital adius of a satellite, the longe its obital peiod. (B) the lage the obital adius of a satellite, the shote its obital peiod. (C) the lage the obital adius of a satellite, the longe its obital peiod. (D) the smalle the obital adius of a satellite, the smalle its acceleation. (E) the lage the obital adius of a satellite, the geate the gaitational foce acting on it. Fee esponse Question Diections: Show all wok in woking the following question. The question is woth 15 points, and the suggested time fo answeing the question is about 15 minutes. The pats within a question may not hae equal weight. 1. (15 points) A ball on the end of a sting is swung in a etical cicle. The mass m of the ball is 0.5 kg and the adius of the cicle = 0.75 m. The position of the ball is maked eey quate of a eolution in the diagam below. I IV II III 77
9 Chapte 5 Dynamics of Unifom Cicula Motion (a) On the diagam below, daw all of the foces acting on the ball when it is at position IV. Be sue and label each foce. (b) When the ball is at position IV, the tension foce in the sting is twice as geat as the weight of the ball. Detemine the speed of the ball at position IV. (c) On the diagams below, daw and label the foces acting on the ball i. at the top of the cicle (position I) and ii. at the bottom of the cicle (position III). Position I Position III (d) The speed of the ball at position III (the bottom of the cicle) is 5 m/s. Detemine the tension foce in the sting as the ball passes though position III. (e) Suppose the sting beaks just as he ball is at position IV. i. Descibe the subsequent motion of the ball. ii. Use the speed you calculated in pat (b) to detemine the maximum height the ball eaches afte the sting beaks. 78
10 Chapte 5 Dynamics of Unifom Cicula Motion ANSWES AND EXPLANATIONS TO CHAPTE 4 EVIEW QUESTIONS Multiple Choice 1. B The elocity ecto is always tangent to the path and in the diection of motion, and the centipetal foce and acceleation ae towad the cente of the cicle. F a. D At the instant the ball is in the position shown in answe (E) in the peious question, it is moing tangent to the path, down and to the ight. 3. C 6s Peiod T = = s 3e 4. B p p ( 4m) = =» 1 m / s T s 5. D m ( 40 kg )( 1 m / s) F = = = 1440 N 4 m 6. B The fictional foce poides the centipetal foce that causes the ca to moe in a cicle. 7. D m m f = µ FN =, and F N = mg, so µ mg =, and 8. A Fom section 5.6 in the textbook, = g = 9 = 3 µ = g 9. C Keple s 3 d law (section 5.5 in the textbook) states that the squae of the obital peiod is popotional to the cube of the obital adius. Thus, the fathe away a satellite obits, the longe obital peiod it has. 79
11 Chapte 5 Dynamics of Unifom Cicula Motion Fee esponse Question Solution (a) points F T mg (b) points m F T = mg = Soling fo we get ( m/ s )( 0.75m) = 3.9 m s = g = 10 / (c) 4 points Position I Position III i. ii. F T F T mg (The lengths of the ectos ae not necessaily dawn to scale.) mg (d) 3 points Choosing up as the positie diection: SF = ma FT - mg = ma The net foce and the acceleation ae diected towad the cente of the cicle. F F T T æ ö - mg = m ç è ø æ ö = mg + m ç = è ø ( 0.5 kg )( 10 m / s ) + ( 0.5 kg )( 5m / s) 0.75 m = 10.8 N 80
12 Chapte 5 Dynamics of Unifom Cicula Motion (e) points i. At the instant the ball is at position IV, its elocity ecto points upwad, and theefoe when the sting beaks, the ball will ise upwad against gaity. When we descibe the motion of an object, we discuss its elocity (speed and diection), its acceleation, and any path it would follow. In this case, the ball would simply moe staight upwad, acceleating at 10 m/s and deceasing its speed, each a maximum height whee its elocity is instantaneously zeo, then acceleate downwad and incease its speed as it falls to the gound. ii. points The ball is pojected staight upwad a 3.9 m/s, and at maximum height h, the speed of the ball is zeo. = o + gh = 0 o h = = g 10 ( 3.9 m / s) ( m / s ) = 0.76 m 81
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