Chapter 7 Rotational Motion and the Law of Gravity
|
|
- Tabitha Bradford
- 5 years ago
- Views:
Transcription
1 Chapte 7 Rotational Motion and the Law of Gaity What is a Rigid Body? Rotational Kinematics Angula Velocity ω and Acceleation α Unifom Rotational Motion: Kinematics Unifom Cicula Motion: Kinematics and Dynamics Applications Newton s Law of Uniesal Gaitational Attaction Planetay motion
2 Rigid Body and its Motions Until now we consideed only the motion of point-like objects. Objects with extended size can be consideed as a collection of many point-like paticles. When these paticles do not moe with espect to each othe, the system is called a igid body: it cannot be defomed. The geneal motion of a igid body can be split into two types: Tanslational (Linea) Motion Rotational (Angula) Motion we need an angula fomalism Object and teminology:,,, a, F, p...,,,,, L... Linea elocity Rigid body A paticle of the body Tanslation elocity Cente o axis of otation
3 Angula Kinematics Angula Position θ In puely otational motion, all points on the object moe in cicles aound the axis of otation ( O ), with each point descibed by a ecto position Def: The angle θ made by the position ecto with espect to an abitay axis (say x) is called angula position θ O y θ x Ex: see the two points on the adjacent bicycle wheel: notice that, as long they ae not on the same adius, the points on the igid body will hae diffeent angula positions In ou appoach to otations, angles will be measued in adians: 1 adian is the angle at the cente of a cicle subtending an ac equal to the adius of the cicle When the angle at the cente is expessed in adians, the length of the ac subtended is gien by: l Ex: The cicumfeence π of a complete otation subtends an angle of π Conention: angles measued counteclockwise ae positie angle measued clockwise ae negatie + l = θ θ = 1 ad l = θ θ
4 Angula Kinematics Angula Displacement and Velocity How can we use angula positions to descibe otations? Notice that een though the angula positions of diffeent points of a wheel ae in geneal diffeent, when the wheel otates, all points otate though the same angle Def: The change in angula position of all the points on a otating igid body is called angula displacement: 1 Δθ θ θ1 Abitay adius x Then, if we want to efe to how fast the angula position changes we hae to define fist the aeage angula elocity as the angula displacement diided by time: 1 Theefoe, like in the linea case, the instantaneous angula elocity is gien by: lim t 0 t 1 t t t Ex: If the bicycle wheel makes two complete otations eey second, we say that it has a constant angula speed of (π/1 s) = 4π ad/s SI ad s
5 Angula Kinematics Angula Acceleation. Angula ecto diections Hence we can define the aeage angula acceleation as the ate at which the angula elocity changes with time: 1 t t t 1 lim t 0 t SI ad s Hence, the instantaneous angula acceleation: The diection of angula elocity is gien by a ight hand ule 0 0 Ex: If the bicycle wheel spins each second though an angle lage and lage by π, we say that each complete otation its aeage angula elocity is π, so its instantaneous angula elocity inceases by π ad/s, so it has a constant angula acceleation π ad/s Although it is not as intuitie as in the tanslational case, the angula elocity and acceleation ae ectos, pependicula on the cicle of otation: The diection of angula acceleation is paallel o antipaallel with the angula elocity depending on whethe ω inceases o deceases
6 Relating Linea and Angula Kinematics So, the angula displacement, elocity and acceleation chaacteize the entie igid body: all points hae the same Δθ, ω and α Howee, the linea distance, elocity and acceleation of points at aious adii fom the axis of otation ae diffeent: each point has a diffeent Δl, and a The linea kinematics of each point on a igid body can be elated to the oeall angula chaacteistics based on the elationship l = θ Fo instance, conside a wheel otating with constant angula speed ω. A point at distance fom the cente of otation will otate with constant linea speed taeling an ac Δl in a time Δt: theefoe, we obtain l t t So, as long as they ae not at the same distance fom the cente of otation, the diffeent points on a igid body hae diffeent linea speeds, inceasing fom zeo in the cente of otation to a maximum alue on the oute im of the otating igid body 3 = ω 3 = ω Δl = Δθ 1 = ω 1 Δθ 3 1 ω ω x
7 How about Acceleation? The acceleation is a bit moe complex since in geneal the ecto linea acceleation of a paticle in cicula motion is not tangent to the tajectoy. Howee, it can be consideed as haing two components one tangent to the tajectoy (paallel o anti-paallel with the elocity) and one pependicula on the elocity: a t : tangent, descibes how the magnitude of the linea elocity aies a : adial (o centipetal), descibes how the diection of the elocity aies The component at of a paticle at distance fom the axis of otation can be easily elated to the angula acceleation α of the igid body: a a t a t t t a x The component a (also called centipetal since it always point towad the cente of otation) makes necessay a sepaate discussion a few slides futhe ω
8 Execise : Components of acceleation A paticle moes as shown in the figue. Between points B and D, the path is a staight line. Let s figue out the net acceleation ectos in points A, C and E along the path epesented below, fo each of the following cases: a) the paticle moes with steadily constant speed b) the paticle moes with steadily inceasing speed c) the paticle moes with steadily deceasing speed
9 Angula Kinematics Unifomly acceleated otation The linea-angula elationship poides an easy way to descibe cicula motion with constant angula acceleation α, by simply noticing that each point on the otating igid body acceleates unifomly along the espectie ac of cicle Assume that the motion stats at t 0 = 0 when the otation is chaacteized by θ 0, ω 0 0 ω 0 t 0 = 0 At time t ω Δl x Δθ x Then, if the angula acceleation α is constant, at a late instant t, The linea motion of one paticle of the igid body at distance fom the cente of otation l t a t 1 0 a t 0 a l l t t t t if each linea quantity is diided by, we obtain t t t t t t The otational motion of the entie igid body (alid fo any of its pats)
10 Obseation useful in poblems: Rolling Motion (Without Slipping) In figue (a), if the wheel of adius is olling without slipping, the point P on the im is at est with espect to the floo when it touches it, while the cente C moes with elocity to the ight In figue (b), the same wheel is seen fom a efeence fame whee C is at est. Now point P is moing with elocity. Since P is a point at distance fom the cente of otation, we hae: Theefoe, when a wheel, o a sphee, o a cylinde olls, its tanslational speed (that is, the speed cm of its cente of mass) is elated to its angula speed ω by cm a) Wheel seen by someone on the gound: ω C P b) Wheel seen by someone on the bike: ω C Caution: een though this has the same fom as the elation between linea speed of a point and the angula speed, it is not the same thing. P
11 Poblems: 1. Unifomly acceleated otation: An automobile engine slows down fom 4500 pm to 500 pm in.5 s. Calculate a) its angula acceleation (assumed constant) b) the total numbe of eolutions the engine makes in this time pm ad 60 s pm ad 60 s s 150 ad s 83 ad. Rotational and linea motion: You ae to design a otating cylindical axle to lift buckets of cement fom the gound to a ooftop. The buckets will be attached to a hook on the fee end of a cable that waps aound the im of the axle; as the axle tuns, the buckets will ise. a) What should the diamete of the axle be in ode to aise the buckets at a steady.00 cm/s when it is tuning at 7.5 pm? b) If instead the axle must gie the buckets an upwad acceleation of m/s, what should be the angula acceleation of the axle be? pm ad s ad s 60 s 4 s s
12 Centipetal Acceleation So, the adial (o centipetal) acceleation descibes how the diection of the elocity changes: that is, if the object moes in a staight line, a = 0 To see how a is elated to the speed, we ll conside a paticle moing in a cicle of adius with α = 0, that is, a constant speed (caution! the elocity is not constant) Assume that the paticle taels an ac-distance Δl subtending an angula displacement Δθ in a time Δt Then, looking at the coesponding change in elocity of the eoling paticle and using the definition of acceleation, we find that the centipetal acceleation is gien by This expession is alid fo any paticle moing along a cued tajectoy: if the cuatue of the path can be fitted locally by a cicle of adius and the instantaneous speed is, the expession aboe gies the adial component of the acceleation in the espectie point a 1 1 ~Δl Δθ Two simila tiangles fom, such that: l l a l t t a a a a path path
13 T Some useful quantities and a summay We can intoduce a new set of paametes descibing peiodic motion: 1. Fequency f : the numbe of eolutions pe time. Peiod T: time equied to complete a eolution Fequency and peiod ae the inese of each othe: Summaizing list of coespondences between linea and otational quantities: Linea Type Rotational Relation x displacement θ x = θ elocity ω = ω a t acceleation α a t = α a acceleation - a = ω f 1 T Using the idea of peiod we can see once moe how the elationship between the linea elocity and the angula elocity makes sense and anothe elationship fo a : a f 1 s Hetz, Hz SI T SI s
14 Unifom Cicula Motion Kinematics The unifom cicula motion is the motion of a paticle in a cicle of constant adius at constant speed, such that α is zeo Being always tangent to the cicula path the ecto instantaneous elocity changes diection, albeit its magnitude stays constant Hence, the tangential acceleation a t is zeo, and the net acceleation is gien only by the centipetal acceleation a pointing eeywhee pependicula on the elocity Comments: Physical situation: a paticle moing in a cicle: The magnitude of the centipetal acceleation is lage if the speed is lage The magnitude of the centipetal acceleation is lage if the adius of otation is small path a Constant speed Ex: if a ca takes a tun at high speed, it will hae a lage centipetal acceleation than when taking it slowly Ex: if a ca takes a tun shap tun (small adius), it will hae a lage centipetal acceleation than when taking a wide tun a
15 By Newton s nd Law, since in the cicula motion the acceleation is necessaily not zeo (since the elocity must change in diection), we see that fo an object to be in unifom cicula motion thee must be a net foce acting on it. We aleady know the acceleation, so can immediately wite the foce: F ma m This centipetal foce is not a new foce: any net foce pointing adially inwad the cicula tajectoy (pependicula on the elocity) can play the ole of centipetal foce, since it has as a esult a change in diection of elocity Comments: Unifom Cicula Motion Dynamics A common misconception is to assume that an object on a cued tajectoy is thown out of it by an outwad centifugal foce. We now see that the foce must be actually inwad The objects taking tuns ae appaently pushed outwad by thei inetia, while the centipetal foce keeps it on the tajectoy If the centipetal foce anishes, the object flies off tangent to the cicle, not outwad as if a centifugal foce wee pesent path F
16 Execise 1: How Angelina Succumbed to Bad Physics In the moie Wanted, bullets ae cued by skilled tattooed assassins. Fo instance, Angelina kills heself by fiing a bullet in a cicle passing though the skulls of some bald dudes befoe hitting he. Say that the bullet has a mass m = 4. g and a muzzle speed of 600 m/s. Also, say that the dudes aanged themseles coneniently in a cicle with adius = 5.0 m. a) How big should be the centipetal foce keeping the bullet on the cicula tajectoy? Meditate about the possible oigin of such a foce and how ealistic is such a scenaio b) How fast should Angelina toss the gun to the sweaty guy in the middle fo the scene to make sense?
17 Execise : Ball eoling as held by a sting A ball of mass m is connected by a sting of length and moed into a cicula tajectoy with constant speed. Let s estimate the foce a peson must exet on the sting to make the ball eole in a hoizontal cicle. T a) What is the natue of the centipetal foce exeted on the ball? b) Based on Newton s nd Law, what is this foce in tems of gien quantities? Poblem 3. Tension as a centipetal foce: Now let s assume that the ball fom the execise aboe is swung in a etical cicle, still with a constant speed. a) Detemine the tension in an abitay point of the cicle whee the adius makes an angle θ with espect to the hoizontal b) Use the esult fom pat (a) to detemine the tension in the sting when the sting is hoizontal, on top of the cicle, and at the bottom of the cicle.
18 Poblems: 4. Nomal as a centipetal foce: A small emote-contol ca with mass m = 1.60 kg moes at a constant speed of = 1.0 m/s in a etical cicle inside a hollow metal cylinde that has a adius of = 5.00 m. What is the magnitude of the nomal foce exeted on the ca by the walls of the cylinde at a) an abitay point on the loop b) point A (bottom of the etical cicle) c) point B (top of the etical cicle) 5. Conical pendulum: A bob of mass m is suspended fom a fixed point with a massless sting of length L (i.e., it is a pendulum). What tangential speed must the bob hae so that it moes in a hoizontal cicle with the sting always making an angle θ fom the etical?
19 Nonunifom Cicula Motion Elementay obseations If the elocity of a eoling paticle changes in magnitude as well, the objects is said to be in a nonunifom cicula motion As any eoling object, the object has a centipetal acceleation: a Howee, in this case, the paticle also has a tangential acceleation: at Physical situation: a paticle moing in a cicle: path a a Changing speed a t Hence, at a cetain moment when the instantaneous angula speed is ω, the instantaneous net acceleation of the paticle has a magnitude gien by: a a a 4 t Accodingly, the net foce acting on the object will contibute to the change in elocity diection with a centipetal component, and to the change in speed with a tangential component
20 Newton s Law of Uniesal Gaitation The idea We leaned that Eath exets a foce on any mass the weight. We studied how the weight contibutes to the motion of the object, and we quantified its stength though its effect when it acts alone: the gaitational acceleation g Howee, what is this foce? Fo instance, is it acted only by Eath? Quiz: Based on what law can we infe that the gaitational foce is not acted only by Eath? The fomulation of a coheent theoy of gaity (consistent with his mechanics) is anothe example of Newton s many fundamental contibutions to Physics He ealized that gaity acts between any masses: it is just logical to assume that the downwad foce that pulls onto an apple o a peson must be hae the same natue as the attaction exeted by Eath on the Moon in ode to keep it on its obit Moeoe, by studying the motion of the planets, he deduced that the magnitude of the gaitational attaction must ay inesely popotional to the squae of the distance between the inteacting masses weight
21 Eey two paticles in the uniese attact each othe with gaitational foces diectly popotional to the poduct of thei masses m and M, and inesely popotional to the squae of the distance between them. Comments: Newton s Uniesal Law of Gaitation Quantitatiely Fg G mm Gaitational constant G Nm kg 11 The foce of attaction is along the line connecting the centes of gaity of the objects (which in the cases we ae inteested in coincides with thei centes of mass o symmety) Ex: Two paticles: m F g F g M Hence, the distance between objects is the distance between thei centes of gaity Note the pesence of Newton s 3 d law in this law: mass m attacts mass M and ice-esa with action-eaction foces Two sphees: m Two squae plates: M The gaitational constant G is ey small, meaning that the gaitational attaction will be hadly obseable between small masses m M
22 Newton s Uniesal Law of Gaitation Detemining G The fist detemination of the alue of the gaitational constant G is sometimes attibuted (wongly) to Si Heny Caendish Caendish actually tied to find expeimentally the density of Eath. His data was only much late used to calculate G Caendish s appaatus was based on the obseation of the gaitational attaction between elatiely small objects using a tosion balance If the foce of attaction is known, G can be calculated easily Moden esion of Caendish s expeimental aangement:
23 Poblems: 6. Pinciple of supeposition: Thee identical paticles of mass m ae positioned at thee cones of a isosceles ight tiangle of equal sides a, as in the figue. Calculate the total gaitational attaction on mass 1. m 1 a a m a 3 m
24 Gaitational Acceleation In a moe geneic context Until now, we assumed the local gaitational acceleation constant. Howee, wee see that this is just an appoximation sing the weight does depend on the distance to the cente of attaction (Eath): h m mg h mg m g h G mme me me 1 g h G G h 1h E E E E On the suface of the Eath h 0 so m E mme me mg ~ 9.8 m s h0 G g G E E Theefoe, the gaitational acceleation at altitude h aboe the suface of the Eath can be witten: g h g 9.8 m s 1h 1h E So, the acceleation due to gaity aies oe the Eath s suface due to altitude, local geology, and the shape of the Eath, which is not quite spheical. E
25 Applications of the Law of Gaity Satellites The motion of a satellite can be associated with a pojectile tajectoy missing the suface of the Eath due to the high speed. If the speed is too small, the satellite falls on the suface (paths 1-) In ode to put a satellite on the obit (paths 3-5), it must hae a minimum speed called satellite speed, which can be estimated assuming a cicula obital tajectoy of adius. Then mm E 1 mg ma G m Howee, if the initial speed is too geat, the satellite flies out into space and becomes a spacecaft as it flies along open (paths 6-7) The minimum speed necessay fo the object to escape the Eath s gaity is called escape speed and can be estimated by noticing that ey fa away fom eath the gaitational potential enegy anishes. Theefoe mm m E E 0 0 G 1 KEinitial PE PE E m G E 1 m G E m E Launching towe E E
26 Applications of the Law of Gaity Keple s Laws 1. The path of each planet about the Sun is an ellipse, with the Sun at one focus. Comment: This behaio is due to the specific 1/ dependence of the foce on the distance.. The position of each planet about the Sun sweeps out equal aeas in equal times. Comment: This behaio is explained by the conseation of angula momentum (we ll lean about it in the next chapte) 3. The peiods of the planets ae popotional to 3/ powes of the semi-majo axis lengths of thei obits. Comment: This is explained by Newton s nd Law applied to the obiting body. Fo any two planets, T T a a 1 1 3
Chapters 5-8. Dynamics: Applying Newton s Laws
Chaptes 5-8 Dynamics: Applying Newton s Laws Systems of Inteacting Objects The Fee Body Diagam Technique Examples: Masses Inteacting ia Nomal Foces Masses Inteacting ia Tensions in Ropes. Ideal Pulleys
More informationChapter 7-8 Rotational Motion
Chapte 7-8 Rotational Motion What is a Rigid Body? Rotational Kinematics Angula Velocity ω and Acceleation α Unifom Rotational Motion: Kinematics Unifom Cicula Motion: Kinematics and Dynamics The Toque,
More informationr cos, and y r sin with the origin of coordinate system located at
Lectue 3-3 Kinematics of Rotation Duing ou peious lectues we hae consideed diffeent examples of motion in one and seeal dimensions. But in each case the moing object was consideed as a paticle-like object,
More informationDYNAMICS OF UNIFORM CIRCULAR MOTION
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
More informationUnit 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
More informationShree Datta Coaching Classes, Contact No Circular Motion
Shee Datta Coaching Classes, Contact No. 93698036 Pof. Deepak Jawale Cicula Motion Definition : The motion of the paticle along the cicumfeence of a cicle is called as cicula motion. Eg. i) Motion of eath
More informationCircular Motion. x-y coordinate systems. Other coordinates... PHY circular-motion - J. Hedberg
Cicula Motion PHY 207 - cicula-motion - J. Hedbeg - 2017 x-y coodinate systems Fo many situations, an x-y coodinate system is a geat idea. Hee is a map on Manhattan. The steets ae laid out in a ectangula
More informationChapter 13 Gravitation
Chapte 13 Gavitation In this chapte we will exploe the following topics: -Newton s law of gavitation, which descibes the attactive foce between two point masses and its application to extended objects
More informationChap13. Universal Gravitation
Chap13. Uniesal Gaitation Leel : AP Physics Instucto : Kim 13.1 Newton s Law of Uniesal Gaitation - Fomula fo Newton s Law of Gaitation F g = G m 1m 2 2 F21 m1 F12 12 m2 - m 1, m 2 is the mass of the object,
More informationrt () is constant. We know how to find the length of the radius vector by r( t) r( t) r( t)
Cicula Motion Fom ancient times cicula tajectoies hae occupied a special place in ou model of the Uniese. Although these obits hae been eplaced by the moe geneal elliptical geomety, cicula motion is still
More informationUnit 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
More informationω = θ θ o = θ θ = s r v = rω
Unifom Cicula Motion Unifom cicula motion is the motion of an object taveling at a constant(unifom) speed in a cicula path. Fist we must define the angula displacement and angula velocity The angula displacement
More informationUniform Circular Motion
Unifom Cicula Motion Intoduction Ealie we defined acceleation as being the change in velocity with time: a = v t Until now we have only talked about changes in the magnitude of the acceleation: the speeding
More informationAP Physics 1 - Circular Motion and Gravitation Practice Test (Multiple Choice Section) Answer Section
AP Physics 1 - Cicula Motion and Gaitation Pactice est (Multiple Choice Section) Answe Section MULIPLE CHOICE 1. B he centipetal foce must be fiction since, lacking any fiction, the coin would slip off.
More informationMotion in a Plane Uniform Circular Motion
Lectue 11 Chapte 8 Physics I Motion in a Plane Unifom Cicula Motion Couse website: http://faculty.uml.edu/andiy_danylo/teaching/physicsi PHYS.1410 Lectue 11 Danylo Depatment of Physics and Applied Physics
More informationChap 5. Circular Motion: Gravitation
Chap 5. Cicula Motion: Gavitation Sec. 5.1 - Unifom Cicula Motion A body moves in unifom cicula motion, if the magnitude of the velocity vecto is constant and the diection changes at evey point and is
More informationPhysics C Rotational Motion Name: ANSWER KEY_ AP Review Packet
Linea and angula analogs Linea Rotation x position x displacement v velocity a T tangential acceleation Vectos in otational motion Use the ight hand ule to detemine diection of the vecto! Don t foget centipetal
More informationPhysics 4A Chapter 8: Dynamics II Motion in a Plane
Physics 4A Chapte 8: Dynamics II Motion in a Plane Conceptual Questions and Example Poblems fom Chapte 8 Conceptual Question 8.5 The figue below shows two balls of equal mass moving in vetical cicles.
More informationThe study of the motion of a body along a general curve. the unit vector normal to the curve. Clearly, these unit vectors change with time, u ˆ
Section. Cuilinea Motion he study of the motion of a body along a geneal cue. We define u ˆ û the unit ecto at the body, tangential to the cue the unit ecto nomal to the cue Clealy, these unit ectos change
More informationOSCILLATIONS AND GRAVITATION
1. SIMPLE HARMONIC MOTION Simple hamonic motion is any motion that is equivalent to a single component of unifom cicula motion. In this situation the velocity is always geatest in the middle of the motion,
More informationSections and Chapter 10
Cicula and Rotational Motion Sections 5.-5.5 and Chapte 10 Basic Definitions Unifom Cicula Motion Unifom cicula motion efes to the motion of a paticle in a cicula path at constant speed. The instantaneous
More informationAnswers to test yourself questions
Answes to test youself questions opic. Cicula motion π π a he angula speed is just ω 5. 7 ad s. he linea speed is ω 5. 7 3. 5 7. 7 m s.. 4 b he fequency is f. 8 s.. 4 3 a f. 45 ( 3. 5). m s. 3 a he aeage
More information4. Two and Three Dimensional Motion
4. Two and Thee Dimensional Motion 1 Descibe motion using position, displacement, elocity, and acceleation ectos Position ecto: ecto fom oigin to location of the object. = x i ˆ + y ˆ j + z k ˆ Displacement:
More information10. Force is inversely proportional to distance between the centers squared. R 4 = F 16 E 11.
NSWRS - P Physics Multiple hoice Pactice Gavitation Solution nswe 1. m mv Obital speed is found fom setting which gives v whee M is the object being obited. Notice that satellite mass does not affect obital
More information06 - ROTATIONAL MOTION Page 1 ( Answers at the end of all questions )
06 - ROTATIONAL MOTION Page ) A body A of mass M while falling vetically downwads unde gavity beaks into two pats, a body B of mass ( / ) M and a body C of mass ( / ) M. The cente of mass of bodies B and
More informationCh04: Motion in two and three dimensions (2D and 3D)
Ch4: Motion in two and thee dimensions (D and 3D) Displacement, elocity and acceleation ectos Pojectile motion Cicula motion Relatie motion 4.: Position and displacement Position of an object in D o 3D
More informationPROJECTILE MOTION. At any given point in the motion, the velocity vector is always a tangent to the path.
PROJECTILE MOTION A pojectile is any object that has been thown though the ai. A foce must necessaily set the object in motion initially but, while it is moing though the ai, no foce othe than gaity acts
More informationObjective Notes Summary
Objective Notes Summay An object moving in unifom cicula motion has constant speed but not constant velocity because the diection is changing. The velocity vecto in tangent to the cicle, the acceleation
More informationRecap. Centripetal acceleration: v r. a = m/s 2 (towards center of curvature)
a = c v 2 Recap Centipetal acceleation: m/s 2 (towads cente of cuvatue) A centipetal foce F c is equied to keep a body in cicula motion: This foce poduces centipetal acceleation that continuously changes
More informationPhysics 107 TUTORIAL ASSIGNMENT #8
Physics 07 TUTORIAL ASSIGNMENT #8 Cutnell & Johnson, 7 th edition Chapte 8: Poblems 5,, 3, 39, 76 Chapte 9: Poblems 9, 0, 4, 5, 6 Chapte 8 5 Inteactive Solution 8.5 povides a model fo solving this type
More informationPhysics 1114: Unit 5 Hand-out Homework (Answers)
Physics 1114: Unit 5 Hand-out Homewok (Answes) Poblem set 1 1. The flywheel on an expeimental bus is otating at 420 RPM (evolutions pe minute). To find (a) the angula velocity in ad/s (adians/second),
More informationPhysics 231 Lecture 17
Physics 31 Lectue 17 Main points of today s lectue: Centipetal acceleation: a c = a c t Rotational motion definitions: Δω Δω α =, α = limδ t 0 Δt Δt Δ s= Δ θ;t = ω;at = α Rotational kinematics equations:
More informationExtra notes for circular motion: Circular motion : v keeps changing, maybe both speed and
Exta notes fo cicula motion: Cicula motion : v keeps changing, maybe both speed and diection ae changing. At least v diection is changing. Hence a 0. Acceleation NEEDED to stay on cicula obit: a cp v /,
More information3.2 Centripetal Acceleration
unifom cicula motion the motion of an object with onstant speed along a cicula path of constant adius 3.2 Centipetal Acceleation The hamme thow is a tack-and-field event in which an athlete thows a hamme
More information3.3 Centripetal Force
3.3 Centipetal Foce Think of a time when ou wee a passenge in a ca going aound a shap cue at high speed (Figue 1). If the ca wee going fast enough, ou might feel the side of the ca doo pushing on ou side.
More informationChapter. s r. check whether your calculator is in all other parts of the body. When a rigid body rotates through a given angle, all
conveted to adians. Also, be sue to vanced to a new position (Fig. 7.2b). In this inteval, the line OP has moved check whethe you calculato is in all othe pats of the body. When a igid body otates though
More informationCircular motion. Objectives. Physics terms. Assessment. Equations 5/22/14. Describe the accelerated motion of objects moving in circles.
Cicula motion Objectives Descibe the acceleated motion of objects moving in cicles. Use equations to analyze the acceleated motion of objects moving in cicles.. Descibe in you own wods what this equation
More informationAH Mechanics Checklist (Unit 2) AH Mechanics Checklist (Unit 2) Circular Motion
AH Mechanics Checklist (Unit ) AH Mechanics Checklist (Unit ) Cicula Motion No. kill Done 1 Know that cicula motion efes to motion in a cicle of constant adius Know that cicula motion is conveniently descibed
More informationKinematics of rigid bodies
Kinematics of igid bodies elations between time and the positions, elocities, and acceleations of the paticles foming a igid body. (1) Rectilinea tanslation paallel staight paths Cuilinea tanslation (3)
More informationNEETIIT.COM. Angular Displacement. Page - 1
- Download ou andoid App. 1. ANGULA DISPLACEMENT Intoduction : Angle subtended by position ecto of a paticle moing along any abitay path w..t. some fixed point is called angula displacement. (a) Paticle
More information- 5 - TEST 1R. This is the repeat version of TEST 1, which was held during Session.
- 5 - TEST 1R This is the epeat vesion of TEST 1, which was held duing Session. This epeat test should be attempted by those students who missed Test 1, o who wish to impove thei mak in Test 1. IF YOU
More informationPHYSICS 220. Lecture 08. Textbook Sections Lecture 8 Purdue University, Physics 220 1
PHYSICS 0 Lectue 08 Cicula Motion Textbook Sections 5.3 5.5 Lectue 8 Pudue Univesity, Physics 0 1 Oveview Last Lectue Cicula Motion θ angula position adians ω angula velocity adians/second α angula acceleation
More informationCircular Motion. Mr. Velazquez AP/Honors Physics
Cicula Motion M. Velazquez AP/Honos Physics Objects in Cicula Motion Accoding to Newton s Laws, if no foce acts on an object, it will move with constant speed in a constant diection. Theefoe, if an object
More informationDiscover the answer to this question in this chapter.
In a oto ide such as the one shown in the figue, what is the maximum peiod of otation that the oto ide can hae so that people do not slip down the wall if the coefficient of fiction between the wall and
More informationΣF = r r v. Question 213. Checkpoints Chapter 6 CIRCULAR MOTION
Unit 3 Physics 16 6. Cicula Motion Page 1 of 9 Checkpoints Chapte 6 CIRCULAR MOTION Question 13 Question 8 In unifom cicula motion, thee is a net foce acting adially inwads. This net foce causes the elocity
More informationChapter 5 Force and Motion
Chapte 5 Foce and Motion In chaptes 2 and 4 we have studied kinematics i.e. descibed the motion of objects using paametes such as the position vecto, velocity and acceleation without any insights as to
More informationChapter 8. Accelerated Circular Motion
Chapte 8 Acceleated Cicula Motion 8.1 Rotational Motion and Angula Displacement A new unit, adians, is eally useful fo angles. Radian measue θ(adians) = s = θ s (ac length) (adius) (s in same units as
More informationDescribing Circular motion
Unifom Cicula Motion Descibing Cicula motion In ode to undestand cicula motion, we fist need to discuss how to subtact vectos. The easiest way to explain subtacting vectos is to descibe it as adding a
More informationMotion in a Circle. Content 1. Kinematics of uniform circular motion 2. Centripetal acceleration 3. Centripetal force.
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
More informationPhysics 181. Assignment 4
Physics 181 Assignment 4 Solutions 1. A sphee has within it a gavitational field given by g = g, whee g is constant and is the position vecto of the field point elative to the cente of the sphee. This
More informationROTATORY MOTION HORIZONTAL AND VERTICAL CIRCULAR MOTION
ROTATORY MOTION HORIZONTAL AND VERTICAL CIRCULAR MOTION POINTS TO REMEMBER 1. Tanslatoy motion: Evey point in the body follows the path of its peceding one with same velocity including the cente of mass..
More informationChapter 12. Kinetics of Particles: Newton s Second Law
Chapte 1. Kinetics of Paticles: Newton s Second Law Intoduction Newton s Second Law of Motion Linea Momentum of a Paticle Systems of Units Equations of Motion Dynamic Equilibium Angula Momentum of a Paticle
More information2013 Checkpoints Chapter 6 CIRCULAR MOTION
013 Checkpoints Chapte 6 CIRCULAR MOTIO Question 09 In unifom cicula motion, thee is a net foce acting adially inwads. This net foce causes the elocity to change (in diection). Since the speed is constant,
More informationChapter 5 Force and Motion
Chapte 5 Foce and Motion In Chaptes 2 and 4 we have studied kinematics, i.e., we descibed the motion of objects using paametes such as the position vecto, velocity, and acceleation without any insights
More informationUniform Circular Motion
Unifom Cicula Motion constant speed Pick a point in the objects motion... What diection is the velocity? HINT Think about what diection the object would tavel if the sting wee cut Unifom Cicula Motion
More informationDEVIL PHYSICS THE BADDEST CLASS ON CAMPUS IB PHYSICS
DEVIL PHYSICS THE BADDEST CLASS ON CAMPUS IB PHYSICS TSOKOS LESSON 6- THE LAW OF GRAVITATION Essential Idea: The Newtonian idea of gavitational foce acting between two spheical bodies and the laws of mechanics
More informationDiscover the answer to this question in this chapter.
In a oto ide such as the one shown in the figue, what is the maximum peiod of otation that the oto ide can hae so that people do not slip down the wall if the coefficient of fiction between the wall and
More informationChapter 4. Newton s Laws of Motion
Chapte 4 Newton s Laws of Motion 4.1 Foces and Inteactions A foce is a push o a pull. It is that which causes an object to acceleate. The unit of foce in the metic system is the Newton. Foce is a vecto
More informationCIRCULAR MOTION. Particle moving in an arbitrary path. Particle moving in straight line
1 CIRCULAR MOTION 1. ANGULAR DISPLACEMENT Intoduction: Angle subtended by position vecto of a paticle moving along any abitay path w..t. some fixed point is called angula displacement. (a) Paticle moving
More informationChapter 5. Uniform Circular Motion. a c =v 2 /r
Chapte 5 Unifom Cicula Motion a c =v 2 / Unifom cicula motion: Motion in a cicula path with constant speed s v 1) Speed and peiod Peiod, T: time fo one evolution Speed is elated to peiod: Path fo one evolution:
More informationm1 m2 M 2 = M -1 L 3 T -2
GAVITATION Newton s Univesal law of gavitation. Evey paticle of matte in this univese attacts evey othe paticle with a foce which vaies diectly as the poduct of thei masses and invesely as the squae of
More informationRotational Motion. Lecture 6. Chapter 4. Physics I. Course website:
Lectue 6 Chapte 4 Physics I Rotational Motion Couse website: http://faculty.uml.edu/andiy_danylov/teaching/physicsi Today we ae going to discuss: Chapte 4: Unifom Cicula Motion: Section 4.4 Nonunifom Cicula
More information2/26/2014. Magnetism. Chapter 20 Topics. Magnets and Magnetic Fields. Magnets and Magnetic Fields. Magnets and Magnetic Fields
Magnets and Magnetic ields Magnetism Howee, if you cut a magnet in half, you don t get a noth pole and a south pole you get two smalle magnets. ectue otes Chapte 20 Topics Magnets and Magnetic ields Magnets
More informationPhys 201A. Homework 5 Solutions
Phys 201A Homewok 5 Solutions 3. In each of the thee cases, you can find the changes in the velocity vectos by adding the second vecto to the additive invese of the fist and dawing the esultant, and by
More informationSection 26 The Laws of Rotational Motion
Physics 24A Class Notes Section 26 The Laws of otational Motion What do objects do and why do they do it? They otate and we have established the quantities needed to descibe this motion. We now need to
More informationF g. = G mm. m 1. = 7.0 kg m 2. = 5.5 kg r = 0.60 m G = N m 2 kg 2 = = N
Chapte answes Heinemann Physics 4e Section. Woked example: Ty youself.. GRAVITATIONAL ATTRACTION BETWEEN SMALL OBJECTS Two bowling balls ae sitting next to each othe on a shelf so that the centes of the
More informationHoizontal Cicula Motion 1. A paticle of mass m is tied to a light sting and otated with a speed v along a cicula path of adius. If T is tension in the sting and mg is gavitational foce on the paticle then,
More informationLecture 13 EXAM 2. Today s Topics: Rotational motion Moment of inertia. Tuesday March 8, :15 PM 9:45 PM
Lectue 13 Rotational motion Moment of inetia EXAM uesday Mach 8, 16 8:15 PM 9:45 PM oday s opics: Rotational Motion and Angula Displacement Angula Velocity and Acceleation Rotational Kinematics Angula
More informationRotational Motion: Statics and Dynamics
Physics 07 Lectue 17 Goals: Lectue 17 Chapte 1 Define cente of mass Analyze olling motion Intoduce and analyze toque Undestand the equilibium dynamics of an extended object in esponse to foces Employ consevation
More information= 4 3 π( m) 3 (5480 kg m 3 ) = kg.
CHAPTER 11 THE GRAVITATIONAL FIELD Newton s Law of Gavitation m 1 m A foce of attaction occus between two masses given by Newton s Law of Gavitation Inetial mass and gavitational mass Gavitational potential
More informationPhysics 111 Lecture 5 Circular Motion
Physics 111 Lectue 5 Cicula Motion D. Ali ÖVGÜN EMU Physics Depatment www.aovgun.com Multiple Objects q A block of mass m1 on a ough, hoizontal suface is connected to a ball of mass m by a lightweight
More informationCircular Motion & Torque Test Review. The period is the amount of time it takes for an object to travel around a circular path once.
Honos Physics Fall, 2016 Cicula Motion & Toque Test Review Name: M. Leonad Instuctions: Complete the following woksheet. SHOW ALL OF YOUR WORK ON A SEPARATE SHEET OF PAPER. 1. Detemine whethe each statement
More informationLab #9: The Kinematics & Dynamics of. Circular Motion & Rotational Motion
Reading Assignment: Lab #9: The Kinematics & Dynamics of Cicula Motion & Rotational Motion Chapte 6 Section 4 Chapte 11 Section 1 though Section 5 Intoduction: When discussing motion, it is impotant to
More informationPhysics 201, Lecture 6
Physics 201, Lectue 6 Today s Topics q Unifom Cicula Motion (Section 4.4, 4.5) n Cicula Motion n Centipetal Acceleation n Tangential and Centipetal Acceleation q Relatie Motion and Refeence Fame (Sec.
More informationGravitation. AP/Honors Physics 1 Mr. Velazquez
Gavitation AP/Honos Physics 1 M. Velazquez Newton s Law of Gavitation Newton was the fist to make the connection between objects falling on Eath and the motion of the planets To illustate this connection
More informationISSUED BY K V - DOWNLOADED FROM CIRCULAR MOTION
K.V. Silcha CIRCULAR MOTION Cicula Motion When a body moves such that it always emains at a fixed distance fom a fixed point then its motion is said to be cicula motion. The fixed distance is called the
More informationRotational Motion. Every quantity that we have studied with translational motion has a rotational counterpart
Rotational Motion & Angula Momentum Rotational Motion Evey quantity that we have studied with tanslational motion has a otational countepat TRANSLATIONAL ROTATIONAL Displacement x Angula Position Velocity
More informationChapter 13: Gravitation
v m m F G Chapte 13: Gavitation The foce that makes an apple fall is the same foce that holds moon in obit. Newton s law of gavitation: Evey paticle attacts any othe paticle with a gavitation foce given
More informationESTIMATION MODELS USING MATHEMATICAL CONCEPTS AND NEWTON S LAWS FOR CONIC SECTION TRAJECTORIES ON EARTH S SURFACE
Fundamental Jounal of Mathematical Physics Vol. 3 Issue 1 13 Pages 33-44 Published online at http://www.fdint.com/ ESTIMATION MODELS USING MATHEMATICAL CONCEPTS AND NEWTON S LAWS FOR CONIC SECTION TRAJECTORIES
More informationBetween any two masses, there exists a mutual attractive force.
YEAR 12 PHYSICS: GRAVITATION PAST EXAM QUESTIONS Name: QUESTION 1 (1995 EXAM) (a) State Newton s Univesal Law of Gavitation in wods Between any two masses, thee exists a mutual attactive foce. This foce
More informationPhysics 111. Lecture 14 (Walker: Ch. 6.5) Circular Motion Centripetal Acceleration Centripetal Force February 27, 2009
Physics 111 Lectue 14 (Walke: Ch. 6.5) Cicula Motion Centipetal Acceleation Centipetal Foce Febuay 7, 009 Midtem Exam 1 on Wed. Mach 4 (Chaptes 1-6) Lectue 14 1/8 Connected Objects If thee is a pulley,
More informationPhysics 2A Chapter 10 - Moment of Inertia Fall 2018
Physics Chapte 0 - oment of netia Fall 08 The moment of inetia of a otating object is a measue of its otational inetia in the same way that the mass of an object is a measue of its inetia fo linea motion.
More informationNewton s Laws, Kepler s Laws, and Planetary Orbits
Newton s Laws, Keple s Laws, and Planetay Obits PROBLEM SET 4 DUE TUESDAY AT START OF LECTURE 28 Septembe 2017 ASTRONOMY 111 FALL 2017 1 Newton s & Keple s laws and planetay obits Unifom cicula motion
More informationChapter 5: Uniform Circular Motion
Chapte 5: Unifom Cicula Motion Motion at constant speed in a cicle Centipetal acceleation Banked cuves Obital motion Weightlessness, atificial gavity Vetical cicula motion Centipetal Foce Acceleation towad
More informationUniform Circular Motion
Unifom Cicula Motion Have you eve idden on the amusement pak ide shown below? As it spins you feel as though you ae being pessed tightly against the wall. The ide then begins to tilt but you emain glued
More information6.4 Period and Frequency for Uniform Circular Motion
6.4 Peiod and Fequency fo Unifom Cicula Motion If the object is constained to move in a cicle and the total tangential foce acting on the total object is zeo, F θ = 0, then (Newton s Second Law), the tangential
More informationAP * PHYSICS B. Circular Motion, Gravity, & Orbits. Teacher Packet
AP * PHYSICS B Cicula Motion, Gavity, & Obits Teache Packet AP* is a tademak of the College Entance Examination Boad. The College Entance Examination Boad was not involved in the poduction of this mateial.
More informationPS113 Chapter 5 Dynamics of Uniform Circular Motion
PS113 Chapte 5 Dynamics of Unifom Cicula Motion 1 Unifom cicula motion Unifom cicula motion is the motion of an object taveling at a constant (unifom) speed on a cicula path. The peiod T is the time equied
More informationPHYSICS NOTES GRAVITATION
GRAVITATION Newton s law of gavitation The law states that evey paticle of matte in the univese attacts evey othe paticle with a foce which is diectly popotional to the poduct of thei masses and invesely
More informationMAGNETIC FIELD INTRODUCTION
MAGNETIC FIELD INTRODUCTION It was found when a magnet suspended fom its cente, it tends to line itself up in a noth-south diection (the compass needle). The noth end is called the Noth Pole (N-pole),
More informationc) (6) Assuming the tires do not skid, what coefficient of static friction between tires and pavement is needed?
Geneal Physics I Exam 2 - Chs. 4,5,6 - Foces, Cicula Motion, Enegy Oct. 10, 2012 Name Rec. Inst. Rec. Time Fo full cedit, make you wok clea to the gade. Show fomulas used, essential steps, and esults with
More informationPhysics 101 Lecture 6 Circular Motion
Physics 101 Lectue 6 Cicula Motion Assist. Pof. D. Ali ÖVGÜN EMU Physics Depatment www.aovgun.com Equilibium, Example 1 q What is the smallest value of the foce F such that the.0-kg block will not slide
More informationCHAPTER 5: Circular Motion; Gravitation
CHAPER 5: Cicula Motion; Gavitation Solution Guide to WebAssign Pobles 5.1 [1] (a) Find the centipetal acceleation fo Eq. 5-1.. a R v ( 1.5 s) 1.10 1.4 s (b) he net hoizontal foce is causing the centipetal
More informationCircular-Rotational Motion Mock Exam. Instructions: (92 points) Answer the following questions. SHOW ALL OF YOUR WORK.
AP Physics C Sping, 2017 Cicula-Rotational Motion Mock Exam Name: Answe Key M. Leonad Instuctions: (92 points) Answe the following questions. SHOW ALL OF YOUR WORK. ( ) 1. A stuntman dives a motocycle
More informationDEVIL PHYSICS THE BADDEST CLASS ON CAMPUS IB PHYSICS
DEVIL PHYSICS THE BADDEST CLASS ON CAMPUS IB PHYSICS LSN 10-: MOTION IN A GRAVITATIONAL FIELD Questions Fom Reading Activity? Gavity Waves? Essential Idea: Simila appoaches can be taken in analyzing electical
More informationTorque, Angular Momentum and Rotational Kinetic Energy
Toque, Angula Moentu and Rotational Kinetic Enegy In ou peious exaples that inoled a wheel, like fo exaple a pulley we wee always caeful to specify that fo the puposes of the poble it would be teated as
More information7.2. Coulomb s Law. The Electric Force
Coulomb s aw Recall that chaged objects attact some objects and epel othes at a distance, without making any contact with those objects Electic foce,, o the foce acting between two chaged objects, is somewhat
More informationLecture 13. Rotational motion Moment of inertia
Lectue 13 Rotational motion Moment of inetia EXAM 2 Tuesday Mach 6, 2018 8:15 PM 9:45 PM Today s Topics: Rotational Motion and Angula Displacement Angula Velocity and Acceleation Rotational Kinematics
More informationName. Date. Period. Engage Examine the pictures on the left. 1. What is going on in these pictures?
AP Physics 1 Lesson 9.a Unifom Cicula Motion Outcomes 1. Define unifom cicula motion. 2. Detemine the tangential velocity of an object moving with unifom cicula motion. 3. Detemine the centipetal acceleation
More informationPhysics 2B Chapter 22 Notes - Magnetic Field Spring 2018
Physics B Chapte Notes - Magnetic Field Sping 018 Magnetic Field fom a Long Staight Cuent-Caying Wie In Chapte 11 we looked at Isaac Newton s Law of Gavitation, which established that a gavitational field
More information