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 Motion: Section 4.6
In addition to tanslation, objects can otate Thee is otation eveywhee you look in the univese, fom the nuclei of atoms to spial galaxies Rotational Motion Spialing Galaxies Need to develop a vocabulay fo descibing otational motion
In ode to descibe otation, we need to define How to measue angles? You know degees, but in the scientific wold RADIANS ae moe popula. Let s intoduce them.
Angula Position in pola coodinates Conside a pue otational motion: an object moves aound a fixed axis. y Instead of using x and y catesian coodinates, we will define object s position with: s aclength x s θ in adians!, Its definition as a atio of two length makes it a pue numbe without dimensions. So, the adian is dimensionless and thee is no need to mention it in calculations (Thus the unit of angle (adians) is eally just a name to emind us that we ae dealing with an angle). If angle is given in adians, we can get an aclength spanning angle θ Use Radians to get an ac length s 60 s s 3
Examples: angles in adians y s ac length x 2 / 2 4 Apply s / 2 2 adians! y s 2 s 2 2 ad x 360 2 ad 1 ad 360 / 2 57.3
Now we need to intoduce otational kinematic quantities like we did fo tanslational motion. Angula displacement, Angula velocity Angula acceleation fo otational kinematic equations
Angula displacement and velocity Angula displacement: 2 1 The aveage angula velocity is defined as the total angula displacement divided by time: t The instantaneous angula velocity: d lim t 0 t dt ; ; ; Angula velocity is the ate at which paticle s angula position is changing. Fo both points, θ and t ae the same so is the same fo all points of a otating object. That is why we can say that Eath s angula velocity is 7.2x10-5 ad/sec without connecting to any point on the Eath. All points have the same. So is like an intinsic popety of a solid otating object.
Sign of Angula Velocity When is the Angula velocity positive/negative? As shown in the figue, ω can be positive o negative, and this follows fom ou definition of θ. (Definition of θ: An angle θ is measued (convention) fom the positive x-axis in a counteclockwise diection.) Example: Fo a clock hand, the angula velocity is negative
ConcepTest Bonnie sits on the oute im of a mey-go-ound, and Klyde sits midway between the cente and the im. The mey-go-ound makes one complete evolution evey 2 seconds. Klyde s angula velocity is: Bonnie and Klyde A) same as Bonnie s B) twice Bonnie s C) half of Bonnie s D) one-quate of Bonnie s E) fou times Bonnie s t 1ev 2sec 2 ad ad 2sec sec is the same fo both abbits The angula velocity of any point on a solid object otating about a fixed axis is the same. Both Bonnie and Klyde go aound one evolution (2 adians) evey 2 sec. Klyde Bonnie
A paticle moves with unifom cicula motion if its angula velocity is constant. Now, if a otation is not unifom (angula velocity is not constant), we can intoduce angula acceleation const Angula acceleation
Angula Acceleation The angula acceleation is the ate at which the angula velocity changes with time: 2 t 2 1 t 1 Aveage angula acceleation: 2 1 t t t 2 1 Units ad/s 2 Instantaneous angula acceleation: lim t 0 t d dt The units of angula acceleation ae ad/s 2 Since is the same fo all points of a otating object, angula acceleation also will be the same fo all points. Thus, and α ae popeties of a otating object
The Sign of Angula Acceleation, α Initial Initial Initial Final Final Final Positive and lage f i f i 0 0 t t t t f Positive and smalle So, α is positive if ω is inceasing and is counte-clockwise. i Positive and smalle f Positive and lage α is negative if ω is deceasing and ω is counteclockwise. i α is positive if ω is deceasing and ω is clockwise. Initial Final α is negative if ω is inceasing and ω is clockwise.
ConcepTest The fan blade is slowing down. What ae the signs of ω and α? The signs of and α A) ω is positive and α is positive. B) ω is positive and α is negative. C) ω is negative and α is positive D) ω is negative and α is negative E) ω is positive and α is zeo 1) is negative (otation CW) 2) is slowing down ( f < i ) Less Negative Moe Negative f i 0 t t f i Fo example 2 ( 5) 3ad / s 3 2 Case 3 (the pevious slide) 2
Now, since we have intoduced all angula quantities, we can wite down Rotational Kinematic Equations Fo motion with constant angula acceleation
Rotational kinematic equations The equations of motion fo tanslational and otational motion (fo constant acceleation) ae identical Tanslational kinematic equations v v o 1 x xo vot at 2 2 v v 2a( x x 2 o at 2 o ) Analogs v x a Rotational kinematic equations o t o o t 1 2 t 2 2 2 2 o o
Fo a otating object we can also intoduce a linea velocity which is called the Tangential velocity Now we need to intoduce a useful expession elating linea velocity and angula velocity
Relation between tangential and angula velocities The tangential velocity component v t is the ate ds/dt at which the paticle moves aound the cicle, whee s is the ac length. v t ds d By definition, linea velocity: v t ds dt = In the 1 st slide, we defined: v t d dt = Remembe it!!!! You will use it often!!! Relation between linea and angula velocities ( in ad/sec) Each point on a otating igid body has the same angula displacement, velocity, and acceleation! The coesponding linea (o tangential) vaiables depend on the adius and the linea velocity is geate fo points fathe fom the axis. The end of the class
ConcepTest Bonnie sits on the oute im of a mey-go-ound, and Klyde sits midway between the cente and the im. The mey-go-ound makes one evolution evey 2 seconds. Who has the lage linea (tangential) velocity? Bonnie and Klyde II A) Klyde B) Bonnie C) both the same D) linea velocity is zeo fo both of them We aleady know that all points of a otating body have the same angula velocity. But thei linea speeds v will be v diffeent because and Bonnie is located fathe out (lage adius R) than Klyde. 1 V 2 t Klyde V Bonnie Klyde Bonnie
Fo a otating object we can also intoduce the Acceleation Tangential acceleation Centipetal acceleation Now we need to intoduce a useful expession elating linea acceleation and angula acceleation
Tangential acceleation The paticle in the figue is moving along a cicle and is speeding up. (Definition) Tangential acceleation is the ate at which the tangential velocity changes, a t = dv t /dt. v tan a t dv dt t = d dt a t a total a t Thee is a tangential acceleation a t, which is always tangent to the cicle. a R Thee is also the centipetal acceleation is a = v t2 /, whee v t is the tangential speed (next slide). Finally, any object that is undegoing cicula motion expeiences two acceleations: centipetal and tangential. Let s get a total acceleation: 2 2 atotal a t a atotal at a
Centipetal acceleation In unifom cicula motion (=const), although the speed is constant, thee is an acceleation because the diection of the velocity vecto is always changing. The acceleation of unifom cicula motion is called centipetal acceleation. The diection of the centipetal acceleation is towad the cente of the cicle. The magnitude of the centipetal acceleation is constant fo unifom cicula motion: vt 2 a (towad cente of cicle) v tan Centipetal acceleation can be ewitten in tem of angula velocity, vt 2 a 2 v tan a R a R (without deivation) v tan a R v tan
ConcepTest Ca on a cuve A ca is taveling aound a cuve at a steady 45 mph. Is the ca acceleating? A) Yes B) No Thee is a Centipetal acceleation
ConcepTest Ca on a cuve A ca is taveling aound a cuve at a steady 45 mph. Which vecto shows the diection of the ca s acceleation? Thee is a Centipetal acceleation pointing towad the cente
ConcepTest Ca on a cuve A ca is slowing down as it dives ove a cicula hill. Which of these is the acceleation vecto at the highest point? Acceleation (slowing down) of changing speed a total a R a v tan tan Acceleation of changing diection v tan
Unifom cicula motion =const A paticle moves with unifom cicula motion if its angula velocity is constant. The time inteval to complete one evolution is called the peiod, T. The peiod T is elated to the speed v: In this case, as the paticle goes aound a cicle one time, its angula displacement is 2 duing one peiod. Then, the angula velocity is elated to the peiod of the motion: d dt t 2 T
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