General Physics (PHY 2130)

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1 Geneal Physics (PHY 130) Lectue 11 Rotational kinematics and unifom cicula motion Angula displacement Angula speed and acceleation

2 Lightning Review Last lectue: 1. Foces; laws of motion field foces and contact foces foce of fiction and gavitational foce tension Review Poblem: Calculate gavitational attaction between two students (say, 70 kg and 90 kg) that ae 1 mete apat.

3 Example: Question: Calculate gavitational attaction between two students (say, 70 kg and 90 kg) that ae 1 mete apat F m1m 11 N m 70kg 90kg 7 G N kg ( 1m) Extemely small Compae: weight of the 70 kg (154 lb) peson F mg 686 N

4 Angula Displacement Recall fo linea motion: displacement, velocity, acceleation Δ Δ f i, v, Δt Need simila concepts fo objects moving in cicle (CD, mey-goound, etc.) As befoe: need a fixed efeence system (line) use pola coodinate system a Δv Δt Angles measued CW ae negative and angles measued CCW ae positive.

5 Angula Displacement Evey point on the object undegoes cicula motion about the point O Angles geneally need to be measued in adians Note: θ s length of ac adius ad π π θ [ ad] θ [degees] 180

6 Example A wheel has a adius of 4.1 m. How fa (path length) does a point on athe cicumfeence tavel if the wheel is otated though angles 30, 30 ad and 30 ev espectively?

7 Angula Displacement The angula displacement is defined as the angle the object otates though duing some time inteval Δθ θ f θ i Evey point on the disc undegoes the same angula displacement in any given time inteval

8 Angula Velocity The aveage angula velocity (speed), ω, of a otating igid object is the atio of the angula displacement to the time inteval ω θ t f f θ t i i Δθ Δt

9 Angula Speed The instantaneous angula velocity (speed) is defined as the limit of the aveage speed as the time inteval appoaches zeo ω Δθ lim Δ t 0 Δt Units of angula speed ae adians/sec (ad/s) Angula speed will be positive if θ is inceasing (counteclockwise) negative if θ is deceasing (clockwise)

10 Angula Acceleation What if object is initially at est and then begins to otate? The aveage angula acceleation, α, of an object is defined as the atio of the change in the angula speed to the time it takes fo the object to undego the change: α ω t f f ω t i i Δω Δt Units ae ad/s² Similaly, instant. angula accel.: α Δω lim Δ t 0 Δt

11 Notes about angula kinematics: When a igid object otates about a fixed axis, evey potion of the object has the same angula speed and the same angula acceleation i.e. θ, ω, and α ae not dependent upon, distance fom hub o axis of otation

12 Examples: 1. Bicycle wheel tuns 40 evolutions/min. What is its angula velocity in adians/second? ev 1min π ads ω 40 8π adians sec min 60sec 1ev 5.1adians sec. If wheel slows down unifomly to est in 5 seconds, what is the angula acceleation? ω f ω i α Δt 0 5 ad 5sec sec 5ad sec

13 Examples: Given: 1. Angula velocity: 40 ev/min. Time t 5 s 3. How many evolution does it tun in those 5 sec? Find: 1. θ? Recall that fo linea motion we had: x v t + Pehaps something simila fo angula quantities? 1 θ ω0t + αt 1 5 ad sec 1ev θ ( ev) 6.5ad 10 evolutions π ( 5sec) + ( 5ad sec)( 5sec) 0 1 at 6.5ad

14 Analogies Between Linea and Rotational Motion Rotational Motion About a Fixed Axis with Constant Acceleation Linea Motion with Constant Acceleation ω ω + αt i v vi + at 1 Δθ ω it + α t Δ x v i t + 1 at ω i ω + αδθ v v + aδx i

15 Relationship Between Angula and Linea Quantities Displacements Δθ Δs Speeds Acceleations Δθ 1 Δt o ω 1 Δs Δt v a α

16 Relationship Between Angula and Linea Quantities Displacements Speeds s θ v ω Acceleations a α Evey point on the otating object has the same angula motion Evey point on the otating object does not have the same linea motion

17 ConcepTest A ladybug sits at the oute edge of a mey-go-ound, and a gentleman bug sits halfway between he and the axis of otation. The mey-go-ound makes a complete evolution once each second.the gentleman bug s angula speed is 1. half the ladybug s.. the same as the ladybug s. 3. twice the ladybug s. 4. impossible to detemine

18 ConcepTest A ladybug sits at the oute edge of a mey-go-ound, and a gentleman bug sits halfway between he and the axis of otation. The mey-go-ound makes a complete evolution once each second.the gentleman bug s angula speed is 1. half the ladybug s.. the same as the ladybug s. 3. twice the ladybug s. 4. impossible to detemine Note: both insects have an angula speed of 1 ev/s

19 Peiod and fequency The time it takes to go one time aound a closed path is called the peiod (T). v total distance total time av π T Compaing to v ω: π ω πf T f is called the fequency, the numbe of evolutions (o cycles) pe second. 19

20 Centipetal Acceleation An object taveling in a cicle, even though it moves with a constant speed, will have an acceleation (since velocity changes diection) This acceleation is called centipetal ( cente-seeking ). The acceleation is diected towad the cente of the cicle of motion

21 Centipetal Acceleation and Angula Velocity The angula velocity and the linea velocity ae elated (v ω) The centipetal acceleation can also be elated to the angula velocity Simila tiangles! Δv v Δs Δv v Δs, but a Δv Δt a v Δs Δt Thus: a C v o a C ω

22 Total Acceleation What happens if linea velocity also changes? Two-component acceleation: the centipetal component of the acceleation is due to changing diection the tangential component of the acceleation is due to changing speed Total acceleation can be found fom these components: slowing-down ca a a + a t C

23 Vecto Natue of Angula Quantities As in the linea case, displacement, velocity and acceleation ae vectos: Assign a positive o negative diection A moe complete way is by using the ight hand ule Gasp the axis of otation with you ight hand Wap you finges in the diection of otation You thumb points in the diection of ω

24 Foces Causing Centipetal Acceleation Newton s Second Law says that the centipetal acceleation is accompanied by a foce F ma C m v F stands fo any foce that keeps an object following a cicula path Foce of fiction (level and banked cuves) Tension in a sting Gavity

25 Example1: level cuves Conside a ca diving at 0 m/s (~45 mph) on a level cicula tun of adius 40.0 m. Assume the ca s mass is 1000 kg. 1. What is the magnitude of fictional foce expeienced by ca s ties?. What is the minimum coefficient of fiction in ode fo the ca to safely negotiate the tun?

26 Example1: Given: masses: m1000 kg velocity: v0 m/s adius: 40.0m Find: 1. f?. µ? v f µ mg m N µ 1000 kg 9.8m s 1. Daw a fee body diagam, intoduce coodinate fame and conside vetical and hoizontal pojections F y 0 N mg N mg F x ma f f v ma m. Use definition of fiction foce: 4 N, 1.0 thus ( 0 m s) 1000 kg m Lesson: µ fo ubbe on dy concete is 1.00! ubbe on wet concete is 0.! 4 N diving too fast

27 ConcepQuestion Is it static o kinetic fiction that is esponsible fo the fact that the ca does not slide o skid? 1. Static. Kinetic

28 Example: banked cuves Conside a ca diving at 0 m/s (~45 mph) on a 30 banked cicula cuve of adius 40.0 m. Assume the ca s mass is 1000 kg. 1. What is the magnitude of fictional foce expeienced by ca s ties?. What is the minimum coefficient of fiction in ode fo the ca to safely negotiate the tun? A component of the nomal foce adds to the fictional foce to allow highe speeds tan θ v g

29 Example: Given: masses: m1000 kg velocity: v0 m/s adius: 40.0m angle: α 30 Find: 1. f?. µ? 1. Daw a fee body diagam, intoduce coodinate fame and conside vetical and hoizontal pojections v F x m cos30 f mg sin30 v f m cos30 mg sin N v F y m sin30 N mg cos30 v N m sin30 + mg cos Use definition of fiction foce: 4 N f µ N, thus minimal µ is µ s s fs 3760 N N N s Lesson: by inceasing angle of banking, one deceases minimal µ o fiction with which one can take cuve!

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