Motion in a Circle. Content 1. Kinematics of uniform circular motion 2. Centripetal acceleration 3. Centripetal force.

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JJ 014 H PHYSICS (9646) Motion in a Cicle Motion in a Cicle Content 1. Kinematics of unifom cicula motion.

Centipetal foce Leaning Outcomes Candidates should be able to: (a) expess angula displacement in adians.

(d) descibe qualitatiely motion in a cued path due to a pependicula foce, and undestand the centipetal

1 JJ 014 H PHYSICS (9646) Motion in a Cicle Motion in a Cicle Content 1. Kinematics of unifom cicula motion. Centipetal acceleation 3. Centipetal foce Leaning Outcomes Candidates should be able to: (a) expess angula displacement in adians. (b) undestand and use the concept of angula elocity to sole poblems. (c) ecall and use = ω to sole poblems. (d) descibe qualitatiely motion in a cued path due to a pependicula foce, and undestand the centipetal acceleation in the case of unifom motion in a cicle. (e) ecall and use centipetal acceleation a = ω, a = (f) ecall and use centipetal foce F = mω, F = m to sole poblems. to sole poblems. Physics Dept Page 1 of 13

2 JJ 014 H PHYSICS (9646) Motion in a Cicle Intoduction Thee ae many types of motion in the wold: Linea motion, pojectile motion, cicula motion and simple hamonic motion, to name a few, that we would study. Besides taelling in a staight line as studied in Kinematics ealie, we ae inteested in soling poblems that inoled motion in a cued path. Fo example, a ca going aound a bend, a satellite obiting Eath, an electon obiting a nucleus, a olle-coaste ide. In this chapte, we will study the motion of a body moing ound a cicula path with a unifom (i.e. constant) speed which is descibed as a unifom cicula motion. a) Expess angula displacement in adians The diagam shows how angle θ is defined: θ = ac length adius of ac = s whee s is the cued distance along the ac of adius θ is measued in adians. Hence one adian (1 ad) is the angle when the ac length is the same as the adius of the ac. The adian is the angle subtended by an ac length equal to the adius of the cicle. Fo one complete cicle: ac length, s = cicumfeence of cicle = π ac length cicumfeence of cicle π Fo one eolution, θ = = = = π ad adius of ac adius To descibe the angle tuned duing motion in a cicle, we use the tem angula displacement. Angula displacement is the angle tuned about the cente of the cicle. If a paticle moes fom A to B, the change in angula displacement in this case is Δθ = (θ - θ 1 ) ad. Angula displacement is measued in adians. Example 1 The lase on a CD playe is 5.0 cm fom the cental hole of the disc. What length of the disc is scanned by the lase when the disc tuns though an angle of 0.45 adians? ac length s θ = = Length scanned, s = θ = (0.45)(5) =.3 cm adius of ac Physics Dept Page of 13

JJ 014 H PHYSICS (9646) Motion in a Cicle b) Undestand and use the concept of angula elocity to sole poblems If something is spinning, it is not possible to state a unique elocity with which it is

3 JJ 014 H PHYSICS (9646) Motion in a Cicle b) Undestand and use the concept of angula elocity to sole poblems If something is spinning, it is not possible to state a unique elocity with which it is taelling, as diffeent pats of it ae taelling with diffeent elocities (due to its changing diections). To oecome this poblem, the tem angula elocity is used. Angula elocity ω is the ate of change of angula displacement. Symbol : ω (small lette omega ) SI Unit of angula elocity : adians pe second (ad s -1 ) Fo constant angula elocity, θ π ω = = t T The peiod T of cicula motion is the time taken fo one eolution. Since π adians is tuned in one eolution : angle tuned in one eolution time fo one eolution = angula elocity o π T = ω c) Recall and use = ω to sole poblems Let a paticle tael aound a cicle with unifom angula elocity, ω. Let the length of the ac taelled be s. Since s = θ ds d( θ ) dθ Linea elocity, = = = = ω dt dt dt = ω Example The minute hand of a clock is moing at constant angula elocity ω. Which point on the hand, A, B o C is moing the fastest? As all points A, B and C lie on the same minute hand, they must all be moing with the same angula elocity ω. Fom = ω, points futhe fom the cente (lage ) will hae geate speeds. Hence, point C is moing the fastest. Physics Dept Page 3 of 13

JJ 014 H PHYSICS (9646) Motion in a Cicle Example 3 A tain is taelling on a tack which is pat of a cicle of adius 600 m at a constant speed of 50 m s -1. What is its angula elocity? = ω 50 ω = = = 0.

6 ad h -1 ω T 4 h Howee, we typically state ω in ad s -1. π π Hence ω = = = 7.

4 JJ 014 H PHYSICS (9646) Motion in a Cicle Example 3 A tain is taelling on a tack which is pat of a cicle of adius 600 m at a constant speed of 50 m s -1. What is its angula elocity? = ω 50 ω = = = ad s Example 4 What is the angula elocity of the Eath about its axis? Angula elocity of the Eath s otation about its axis π π π T = ω = = = 0.6 ad h -1 ω T 4 h Howee, we typically state ω in ad s -1. π π Hence ω = = = 7.3 x 10-5 ad s -1 T s d) Descibe qualitatiely, motion in a cued path due to a pependicula foce, and undestand the centipetal acceleation in the case of unifom motion in a cicle Unifom cicula motion is the motion of an object moing in cicula path at constant speed. Accoding to Newton s 1 st Law, eey object continues in its state of est o unifom motion in a staight line unless a esultant extenal foce acts on it to change that state. If an object is taelling with constant angula elocity in a cicle, the diection of motion is continually changing (i.e. not taelling in a staight line). This implies that : the elocity is changing but the speed is constant. thee must be a esultant foce acting on the object, but diected towads the cente of the cicle. Physics Dept Page 4 of 13

5 JJ 014 H PHYSICS (9646) Motion in a Cicle Analysis of Unifom Cicula Motion C Conside an object moing in unifom cicula motion at constant angula elocity, ω, fom A to B. If no esultant foce acts on the object, by Newton s 1 st law, the object will tael fom A to C at constant elocity,. Howee, since the object taels along the cued path fom A to B, this implies that a esultant foce F must hae acted on the object to change its elocity. Instead of aiing at C, the object aies at B. This means that the object must hae been pulled towads the cente of the cicle "by this esultant foce" F. ω O F B F A Points to note: 1. Since the object is moing at constant speed, this means that thee is no component of acceleation in the diection of motion, implying thee is no component of esultant foce acting along the tangential path of the object. [Conesely, if the speed is changing (not constant), thee will be a component of esultant foce acting tangentially to the cicle. This component causes a change in speed.]. Howee, since the elocity keeps changing (i.e. only the diection of motion is changing), this must imply that thee must be a esultant foce acting on the object. The only diection that this esultant foce can act is towads the cente of the cicle. This esultant foce is called centipetal foce. Centipetal foce is the esultant foce acting on an object in unifom cicula motion and is diected towads the cente of the cicle. 3. Since the object expeiences a esultant foce towads the cente of the cicle, by Newton s nd law, thee must be an acceleation towads the cente of the cicle. This acceleation is called centipetal acceleation, which is a ecto with units m s -. In unifom cicula motion, the speed of the object does not change, but the diection is changing (i.e. elocity is changing). Hence, the centipetal acceleation does not change the magnitude of the elocity, but only changes its diection. 4. The centipetal foce is NOT an additional foce acting on the object. It is the esultant of all the foces acting on the object and acts towads the cente of the cicle. It can be a combination of foces like tension, fiction, gaitational foce, electic foce o nuclea foce. It is called centipetal foce because of what it is doing - moing object in a cicula path. It does not appea in any fee body diagams as a fundamental foce, athe it is a esultant foce. 5. The centipetal foce is always pependicula to the diection of motion. Hence, this foce does not do any wok on the object. Physics Dept Page 5 of 13

6 JJ 014 H PHYSICS (9646) Motion in a Cicle As thee is no displacement in the diection of the esultant foce (esultant foce is always pependicula to the elocity), thee is no wok done on the object moing in unifom cicula motion. Hence thee is no change in the kinetic enegy of the object. Example 5 [N93/I/6] Which of the following statements is coect fo a paticle moing in a hoizontal cicle with constant angula elocity? A the linea momentum is constant but the kinetic enegy aies B the kinetic enegy is constant but the linea momentum aies C both kinetic enegy and linea momentum ae constant D neithe the linea momentum no the kinetic enegy is constant E the speed and the linea elocity ae both constant. Ans A Remaks about statement Incoect. Linea momentum is a ecto and since the diection is changing, the linea momentum aies (i.e. it is NOT constant). Kinetic enegy is a scala. As the object is moing at constant angula elocity, the kinetic enegy is constant. B C Coect statement. Incoect. KE is constant, but linea momentum (ecto) is always changing due to the changing diection of elocity. D Incoect. KE is constant. E Incoect. Speed is constant, but elocity (ecto) is changing. e) Recall and use centipetal acceleation a = ω, a = to sole poblems. Centipetal acceleation is gien by [the poof of this equation is not necessay]. Centipetal acceleation, a = = ω 1. The diection of the centipetal acceleation at any instant is in a staight line towads the cente of the cicle (cicula path).. Fo cicula motions at constant speed, the smalle the cuatue (adius ) & lage the speed, the lage the centipetal acceleation, a. Physics Dept Page 6 of 13

7 JJ 014 H PHYSICS (9646) Motion in a Cicle Example 6 The maximum speed of the blades on otay lawn-mowes is esticted to aoid dange fom flying stones. If the ate of otation of the blades in one paticula model is 3500 eolutions pe minute and the blade has a adius of 0.3 m, find a) the angula elocity of the blade. b) the linea elocity of the tip of the blade. c) the centipetal acceleation of the tip of the blade. a) Blade moes 3500 eolutions pe minute. Hence, time fo ONE eolution = b) = ω c) a = o a = ω f) Recall and use centipetal foce F = mω, F = m to sole poblems. Fom Newton s nd Law, if the mass emains constant, then esultant foce, F = ma. In cicula motion, centipetal acceleation, a = = ω Hence, esultant foce towads the cente of cicle (centipetal foce) is gien by: Centipetal foce, F = m = mω Pocedue to sole cicula motion questions: Step 1: Identify the object and the cente of the cicula motion Step : Daw the fee body diagam of the object in cicula motion Step 3: Resole all foces and find the esultant foce pointing towads the cente of the cicula motion Step 4: equate this esultant foce poides the centipetal foce. Physics Dept Page 7 of 13

8 JJ 014 H PHYSICS (9646) Motion in a Cicle Example 7 Fo each object, daw all the eleant foces, and wite down the expession fo the centipetal foce (fee body diagams). a) of mass m tied to a sting, descibing a cicula path in a etical plane o T mg Object at bottom of cicle Resultant foce to the cente, F = T - mg m T mg = Object at top of cicle Object at side of cicle Note: Thee ae two types of cicula motion possible in this etical cicle - unifom speed and non-unifom speed. The speeds of objects in a etical plane ae geneally not constant (and difficult to maintain at constant speed), hence is not unifom cicula motion. This implies the centipetal acceleation cannot be the same magnitude thoughout the motion. b) A stunt die diing his ca inside a etical cicula tack. Physics Dept Page 8 of 13

9 JJ 014 H PHYSICS (9646) Motion in a Cicle c) A conical pendulum (mass m) descibing a cicle in a hoizontal plane. Notice that the hoizontal component of the tension will now poide the centipetal foce while the etical component balances the weight of the pendulum. Thei ecto sum is not zeo since thee must be a esultant foce as the centipetal foce. Foces towads the cente of cicle: m Tsin θ = Foces in the etical: T cos θ = mg Example 8 The diagam epesents a cyclist making a left tun on a ough suface at constant speed,, as iewed fom behind. The total mass of the bicycle and ide is m and thei combined cente of gaity is at G. If R is the esultant foce of the nomal eaction and the fictional foce, which ecto diagam epesents the diections of the foces acting on the bicycle and its ide? Example 9 A satellite of mass 800 kg is obiting the Moon in a cicula path of adius 1760 km. The pull of the moon on the satellite is 1300 N. Find the speed of the satellite and its peiod of otation. In this example, the only foce acting on the satellite is the foce by the moon on the satellite, which is aleady diected towads the cente of the cicle (i.e. moon). 3 m ( ) F = 1300 = = = = 1690 m s -1 m 800 Peiod of otation, T = π π π = = ω = 6540 s Physics Dept Page 9 of 13

JJ 014 H PHYSICS (9646) Motion in a Cicle Banking of oads The pat of the oad that is cued is usually tilted at an angle (i.e. banked). Banking is beneficial to the safety of a ca moing oe a cued oad.

If the tilt angle is just ight, the nomal foce will be able to poide the entie centipetal foce. This will allow the ca to negotiate the cue een if thee is no fiction between its tyes and the oad.

Find the appopiate banking angle fo a 900 kg ca taelling at 0.5 m s -1 in a tun of adius 85.

10 JJ 014 H PHYSICS (9646) Motion in a Cicle Banking of oads The pat of the oad that is cued is usually tilted at an angle (i.e. banked). Banking is beneficial to the safety of a ca moing oe a cued oad. This is because on a banked cue, the nomal foce exeted by the oad on the ca contibutes to the centipetal foce equied fo the ca to tun. If the tilt angle is just ight, the nomal foce will be able to poide the entie centipetal foce. This will allow the ca to negotiate the cue een if thee is no fiction between its tyes and the oad. Example 10 If a oadway is banked at the pope angle, a ca can ound a cone without any assistance fom fiction between the tyes and the oad. Find the appopiate banking angle fo a 900 kg ca taelling at 0.5 m s -1 in a tun of adius 85.0 m Resoling foces etically: Resoling foces hoizontally: tan θ = Example 11 (Aiplane taking a bank) An aicaft is taelling at a constant speed of 180 m s -1 in a hoizontal cicle of adius 0 km. (a) Find the centipetal acceleation of the aicaft. (b) Daw the fee body diagam of the aicaft. O (c) Calculate the angle of the bank, θ, of the aicaft. θ (a) Centipetal acceleation, a = 180 = 3 0x10 = 1.6 m s - Physics Dept Page 10 of 13

11 JJ 014 H PHYSICS (9646) Motion in a Cicle (b) L O θ mg (c) Since the plane is in equilibium in the etical diection, Resultant foce (etically), F y = ma y = 0 L cos θ = mg (1) Since plane is pefoming cicula motion, thee must be a esultant foce towads the cente of the cicle (i.e. centipetal foce), m Centipetal foce, F = Equations () / (1): tan θ = g θ = tan 1 m Lsin θ = () = tan g ( 9.81)( 0 10 ) = 9.4 Example 1 (The otating spaceman) A spaceman in taining is otated in a seat at the end of a hoizontal otating am of length 5 m. If he can withstand acceleations up to 9g, what is the maximum numbe of eolutions pe second pemissible? Calculate the peiod of the otating am in maintaining this no. of eolutions. Using a = ω 9g = 5ω 9g ω = = 4. ad s -1 5 Maximum numbe of eolutions in 1 second = π π Peiod of otation, T = = = 1.5 s ω 4. ω 4. = = 0.67 π π Physics Dept Page 11 of 13

12 JJ 014 H PHYSICS (9646) Motion in a Cicle Things I hae leant today / Things I found inteesting / Questions I still hae Refeence Textbooks: 1. Compehensie Physics fo A leel (3 d edition), K.F. Chan, Chales Chew & S.H. Chan. Fedeal Study Aids.. Physics ( nd edition), Robet Hutchings. Bath Adanced Science. 3. Calculations fo A-leel Physics (3 d edition), T.L. Lowe & J.F. Rounce. Stanley Thones (Publishes) Ltd. 4. Physics ( nd edition), James S. Walke Exta Reading What is the maximum G foce a human can suie? That depends on the duation theeof. If subjected to constant acceleation, it would be doubtful if someone could withstand 9g fo moe than a few minutes. Afte a few minutes at that ate, one faints fom the lack of blood to the bain, and pesumably afte that, some damage will occu, like heat failue, bain hemohage and so on. But fo ey shot duations, ey high acceleations can be suppoted, although some damage can esult. Colonel John Paul Stapp of the US Ai Foce did seeal expeiments, stapping himself to a ocket sled, and detemined that 3g was an acceleation someone could walk away fom, which then became the acceleation used in the design of fighte jet seat. Stapp subjected himself to 45g and aboe, but since he did that so often, ended up with damage to his eyes. The Bitish Fomula One ace Daid Puley cashed in 1977, his ca going fom 173 km/h (107 mph) to 0 in only 66 cm (two feet) (which means he hit a wall and the ca stuctue compessed to deceleate him). He boke many bones, but suied. This deceleation of 178g is belieed to be the highest ee suied by a human being. [Except fom Physics Dept Page 1 of 13

JJ 014 H PHYSICS (9646) Motion in a Cicle Summay 1) A body taelling at constant speed in a cicula path has acceleation towads the cente of its path.

It is pependicula to the diection of motion (tangential to the cicula path), and hence, no wok (as defined) is done by the centipetal foce.

find the esultant foce pointing towads the cente of the cicula motion Step 4: This esultant foce poides the centipetal foce. Examples of Applications: 1.

13 JJ 014 H PHYSICS (9646) Motion in a Cicle Summay 1) A body taelling at constant speed in a cicula path has acceleation towads the cente of its path. This acceleation is called a centipetal acceleation. ) The centipetal foce is the esultant foce acting towads the cente of the cicula path. It is pependicula to the diection of motion (tangential to the cicula path), and hence, no wok (as defined) is done by the centipetal foce. 3) Pocedue to sole cicula motion questions: Step 1: Identify the object and the cente of the cicula motion Step : Daw the fee body diagam of the object in cicula motion Step 3: Resole all foces and find the esultant foce pointing towads the cente of the cicula motion Step 4: This esultant foce poides the centipetal foce. Examples of Applications: 1. Washing Machine In washing machine, the dums poide the necessay centipetal foce (eaction of the dum acting on the clothes) to keep the clothes in cicula motion. In the dums thee ae holes but they cannot poide the centipetal foce. When the dum is otated with high speed, wate paticles acquie linea speed and the speed is high so that they moe in a spial path. The adius of the cicula path is inceasing coesponding to the speed. Thus, it is the lack of necessay centipetal foce that causes the wate paticle to moe away fom the cente though the holes along a tangential path and not along the adius. Physics Dept Page 13 of 13

Unit 6 Practice Test. Which vector diagram correctly shows the change in velocity Δv of the mass during this time? (1) (1) A. Energy KE.

Unit 6 Practice Test. Which vector diagram correctly shows the change in velocity Δv of the mass during this time? (1) (1) A. Energy KE. Unit 6 actice Test 1. Which one of the following gaphs best epesents the aiation of the kinetic enegy, KE, and of the gaitational potential enegy, GE, of an obiting satellite with its distance fom the