Physics for Scientists and Engineers. Chapter 8 Dynamics II: Motion in a Plane

Transcription

1 Physics fo Scientists and Enginees Chapte 8 Dynamics II: Motion in a Plane Sping, 008 Ho Jung Paik

2 Dynamics in Two Dimensions Daw a fee-body diagam Use Newton s second law in component fom: F ma, F x x y may Sole fo the acceleation If the acceleation is constant, use the twodimensional kinematic equations: 5-Ma-08 Paik p.

3 Pojectile Motion In the absence of ai esistance, a pojectile has only the gaitational foce acting on it: Fom Newton s second law, a x ( F ) ( F ) G m x 0, a g The etical motion is fee fall, while the hoizontal motion is one of constant elocity y G m y 5-Ma-08 Paik p. 3

4 Pola s tz Coodinates Usually the adial unit ecto points outwad, so a ˆ The textbook uses an inwad pointing adial unit ecto, so a + ˆ Make sue you ae consistent in the signs of the adial ectos 5-Ma-08 Paik p. 4

5 Cicula Motion Kinematics 5-Ma-08 Paik p. 5

6 Cicula Motion Dynamics 5-Ma-08 Paik p. 6

7 Unifom Cicula Motion A foce causing a centipetal acceleation acts towad the cente of the cicle F ma, F m It causes a change in the diection of elocity ecto If the foce anishes, the object would moe in a staight line tangent to the cicle 5-Ma-08 Paik p. 7

8 Example 1: Amusement Pak Ride An amusement pak ide consists of a lage etical cylinde that spins about its axis fast enough such that any peson inside is held up against the wall when the floo dops away. Expeiment shows that the angula otation ate has to be lage enough, o the peson falls. What is the necessay otation ate? The coefficient of static fiction between peson and wall is μ s, and the adius of the cylinde is R. 5-Ma-08 Paik p. 8

9 Example 1, cont The fiction holds the peson up! Radial : Vetical: Thus f s,max F F y s ma c ma Fo stability,, y n m R, f mg ma m μsn mg R gr ω μ R s a y 0 μ s g Rμ s y mg 5-Ma-08 Paik p. 9

10 Example : Conical Pendulum The object is in equilibium in the etical diection and undegoes unifom cicula motion in the hoizontal diection. Find the tangential speed. : y : F F y T T y mg m 0 T m Since T T sinθ, Lsinθ T m T y T cosθ, Lsin θ sinθ Lg sinθ tanθ 5-Ma-08 Paik p. 10

11 Flat Hoizontal Cue Static fiction supplies the centipetal foce Maximum speed at which the ca can negotiate the cue is max : F ma, f s m y : F 0, n mg 0 f s s y μ n μ mg, s m μ mg s μ g Note, this does not depend on the mass of the ca 5-Ma-08 Paik p. 11 μ g s s

12 Banked Cue A component of the nomal foce supplies the centipetal foce With zeo fiction, the ca can negotiate the cue with speed g tanθ m : F nx, y : Fy n nsinθ, n ncosθ mg x n m x y ( mg / cosθ )sinθ m mg g tanθ 5-Ma-08 Paik p. 1 n y 0

13 Banked Cue, cont With zeo fiction, thee is only one speed that a ca can tun in a cicle on a banked cue If the ca went slowe than (g tanθ) 1/, it would slide to a smalle adius, i.e. downhill If it went at a highe speed, it would slide to a lage adius In a eal situation, fiction would come into play peenting this sliding The necessay fictional foce would, howee, be less than if the oad wee flat 5-Ma-08 Paik p. 13

14 Example 3: Flat Cued Road The coneing pefomance of an automobile is ealuated on a skidpad, whee the maximum speed that a ca can maintain aound a cicula path on a dy, flat suface is measued. Then the centipetal acceleation, also called the lateal acceleation, is calculated as a multiple of the fee-fall acceleation g. A Dodge Vipe GTS can negotiate a skidpad of adius 61.0 m at 86.5 km/h (53.7 mph). Calculate its maximum lateal acceleation. a [( 86.5 km/h)( 1h / 3600 s)( 1000 m/1km) ] 0.966g 61.0 m 1g 9.80 m/s 5-Ma-08 Paik p. 14

15 Example 4: Egg Cate on Tuck A cate of eggs is located in the middle of the flat bed of a pickup tuck as the tuck negotiates an unbanked cue in the oad. The cue may be egaded as an ac of a cicle of adius 35.0 m. If the coefficient of static fiction between the cate and tuck is 0.600, how fast can the tuck moe without the cate sliding? The static fiction supplies the centipetal foce: f s m /. Since thee is no etical acceleation, n - mg 0. Theefoe, f s m / µ s n µ s mg, (µ s g) 1/. [(0.600)(35.0 m)(9.80 m/s )] 1/ 14.3 m/s (3mph) 5-Ma-08 Paik p. 15

16 Cicula Obits Gaitational foce poides the centipetal acceleation needed fo a cicula obit: An object moes paallel to the suface with speed An object with any othe speed will not follow a cicula obit 5-Ma-08 Paik p. 16

17 Loop-the-Loop At the bottom of the loop, the upwad foce expeienced by the pilot is geate than his weight n bot mg m mg 1 + g At the top of the cicle, the foce exeted on the pilot is less, n n, 1 top mg m ntop mg g bot n top must be positie lest the pilot should fall 5-Ma-08 Paik p. 17

18 Example 5: Wate Pail A pail of wate is otated in a etical cicle of adius 1.00 m. What is the minimum speed of the pail at the top of the cicle if no wate is to spill out? Wite down Newton s nd Law fo the wate, assuming the wate, like the bucket, goes aound in cicula motion. F n n m mg sin θ ma m mg sinθ mg sinθ g n mg sinθ θ 5-Ma-08 Paik p. 18

19 Suppose Example 5, cont 1/ ( ) ( g 9.80 m/s 1.00 m) Then, at some angle θ, n mg sin θ 0. g Fo example, if.63 m/s, n 0 fo θ m/s. Once n 0 is eached, the bucket no longe pushes on the wate to make it go in a cicle with the bucket. At this point, the adial acceleation of the wate becomes a g sinθ and the wate falls away fom the bucket. In ode fo the wate to stay in the bucket at θ 90, (g) 1/ 5-Ma-08 Paik p. 19 1/

20 Non-unifom Cicula Motion The acceleation and foce hae tangential components F poduces the centipetal acceleation F t poduces the tangential acceleation ΣF ΣF + ΣF t 5-Ma-08 Paik p. 0

21 Vetical Cicle Unde Gaity The gaitational foce exets a tangential foce on the object Look at the components of F g Example: pendulum The tension at any point can be found: 5-Ma-08 Paik p. 1

22 Example 6: Pendulum A.00-kg ball is tied to a sting which is m long. The ball swings in a etical cicle unde the influence of gaity. When the ball makes an angle of 0.0 with the etical, its speed is 1.50 m/s. a) What is the tangential acceleation, a θ? b) What is the adial acceleation, a? c) What is the total acceleation? d) What is the tension in the sting? e) What is a θ when the ball passes though the etical? 5-Ma-08 Paik p.

23 Example 6, cont (a) Tangential acceleation a θ F θ mg 9.80 m/s sin ˆ θθ ma (b) Radial acceleation θ (1.50 m/s) a ˆ m (c) Total acceleation a total φ tan -1 a + θ θ, sin ˆ a θ + g m/s 4.50 sinθ ˆ θ ( 4.50 m/s )ˆ -1 ( a / a ) tan ( 3.35 / 4.50) 36.7 θ a 5.61 m/s ˆ θˆ 5-Ma-08 Paik p. 3

24 Example 6, cont (d) Tension in the sting T F m g ma, T cosθ + + mg cosθ m.00 ( 9.80 cos ) 13.7 N (e) Angula acceleation at θ 0 sin ˆ F mg θθ ma, a g θ At θ 0, a θ θ θ 0 m/s sinθθˆ 5-Ma-08 Paik p. 4

25 Fictitious Foces A fictitious foce esults fom an acceleated fame of efeence A foce appeas to act on an object, but you cannot identify the agent fo this foce Although fictitious foces ae not eal foces, they hae eal effects Examples: Centifugal foce: Objects in the ca tuning a cone do slide and you feel pushed to the outside The Coiolis foce is esponsible fo the otation of weathe systems and ocean cuents 5-Ma-08 Paik p. 5

26 Linea Acceleating System Inetial obsee sees F T sinθ ma F x y tanθ T cosθ a / g mg 0 Non-inetial obsee sees fictitious foce θ F x T sin ma 0 F T cosθ mg 0 y tanθ ma / mg acceleation 5-Ma-08 Paik p. 6

27 Rotating System Inetial obsee sees F Non-inetial obsee sees T F m ' centifugal foce T + m centipetal acceleation 0 5-Ma-08 Paik p. 7

28 Centifugal Foce As seen fom the ca As seen fom a helicopte A peson in a hoeing helicopte aboe the ca could descibe the moement of the objects as just going staight while the ca taels in a cued path 5-Ma-08 Paik p. 8

29 Coiolis Foce An appaent foce caused by changing the adial position of an object in a otating fame The esult is the cued path of the object 5-Ma-08 Paik p. 9

30 Ai and Ocean Cuents 5-Ma-08 Paik p. 30