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: s = 2π Speed: v = s/t = 2π/T
2) Centipetal Acceleation s a c v Diected towad cente (always changing) Magnitude: a = v2 c
Two cas, diving at the same constant speed v, ae taveling though tuns that have diffeent, as shown in the dawing. Which statement is tue about the magnitude of the centipetal acceleation a c of each ca? (a) a c of the ca at A is geate than that of the ca at B, since is geate at A. (b) a c of the ca at A is smalle than that of the ca at B, since is smalle at A. (c) a c of each ca is the same, since both cas ae moving at v. (d) a c of the ca at A is geate than that of the ca at B, since is smalle at A.
3) Centipetal foce s a c v F c Towad cente (paallel to acceleation) since F = ma (always changing) Magnitude: F c = ma c = mv2
Centipetal foce is a eal foce that poduces cicula motion in an inetial fame gavity Tension Fiction, Nomal Centifugal foce is a vitual foce peceived in a otating (non-intetial fame)
Imagine you ae swinging a bucket by the handle aound in a cicle that is nealy level with the gound (a hoizontal cicle). Now imagine thee's a ball in the bucket. What keeps the ball moving in a cicula path? A. contact foce of the bucket on the ball B. contact foce of the ball on the bucket C. gavitational foce on the ball D. the centifugal foce
C&J 5.59 A block is hung by a sting fom the inside oof of a van. When the van goes staight ahead at a speed of 28 m/s, the block hangs vetically down. But when the van maintains this same speed aound an unbanked cuve (adius = 150 m), the block swings towad the outside of the cuve. Then the sting makes an angle θ with the vetical. Find θ.
4) Diving aound cicula cuves a) Unbanked cuve - static fiction povides Fc! f s! F N m! g F c = f s µ s F N = µ s mg F c = mv 2 / v max = µ s g
b) Banked cuve, no fiction y x x : F N sinθ = mv 2 / y : F N cosθ = mg tanθ = v2 g
c) Banked cuve with fiction f s (ii) Net foce:!! F = F + m g!! + f N s Recall ideal angle: f s (i) tanθ = v2 g i) If v 2 > g tan θ, fiction pevents sliding up ii) If v 2 < g tan θ, fiction pevents sliding down
Example: Find maximum velocity fo µ = 0.80, θ = 47º, and = 60 m f s (i) f s (ii) x y F c F N θ θ! F c =! F N + m! g +! f s f s y-motion x-motion mg 0 = F N cosθ mg f s sinθ When v = v max, f s = µf N so mg F N = cosθ µ sinθ F c = F N sinθ + f s cosθ When v = v max, f s = µf N, and using F c = mv2 F N = mv max 2 (sinθ + µ cosθ),
0 = F N cosθ mg f s sinθ When v = v max, f s = µf N so mg F N = cosθ µ sinθ F c = F N sinθ + f s cosθ When v = v max, f s = µf N, and using F c = mv2 F N = mv max 2 (sinθ + µ cosθ), Eliminate F N, and solve fo v max : sinθ + µ cosθ v 2 max = g cosθ µ sinθ v max = 88m/s = 317km/h
5) Satellites in cicula obits a) Speed and adius m F = G mm E 2 Law of gavity = mv 2 / cicula motion ME Radius of obit detemines speed (independent of mass) v = GM E Equal acceleation of all local objects --> weightlessness
(b) Peiod and adius v = 2π T = GM E speed and peiod obit T = 2π3 2 GM E
c) Radius of synchonous obits satellite is stationay above eath s suface Conditions: T = 1 sideeal day above equato 3 2 = T GM E 2π Using T = 24h 4 min = 86160 s gives = 4.22 10 7 m = 6.61R E
It is the yea 2094; and people ae designing a new space station that will be placed in a cicula obit aound the Sun. The obital peiod of the station will be 6.0 yeas. Detemine the atio of the station s obital adius about the Sun to that of the Eath s obital adius about the Sun. Assume that the Eath s obit about the Sun is cicula. a) 2.4 d) 5.2 b) 3.3 e) 6.0 c) 4.0
C&J 5.52 A child is twiling a 0.0120-kg plastic ball on a sting in a hoizontal cicle whose adius is 0.100 m. The ball tavels once aound the cicle in 0.500 s. (a) Detemine the centipetal foce acting on the ball. (b) If the speed is doubled, does the centipetal foce double? If not, by what facto does the centipetal foce incease?
C&J 5.11 A centifuge is a device in which a small containe of mateial is otated at a high speed on a cicula path. Such a device is used in medical laboatoies, fo instance, to cause the moe dense ed blood cells to settle though the less dense blood seum and collect at the bottom of the containe. Suppose the centipetal acceleation of the sample is 6.25 10 3 times as lage as the acceleation due to gavity. How many evolutions pe minute is the sample making, if it is located at a adius of 5.00 cm fom the axis of otation?
Octobe 2008 Midtem exam Q18 A 1200.0 kg ca, taveling at 30.0 m/s, tavels safely (without skidding) aound a fictionless banked cuve of adius 200.0 m. What is the magnitude of the nomal foce on this ca?
6) Centifugal foce and atificial gavity a) Atificial gavity Fc = FN (inside suface) e.g. if = 1700 m, and F c = mg, then mg = mv 2 /, giving v = g = 130 m/s = 468 km/h Centipetal foce acts towad the cente (up, like the nomal foce on eath) but gavity acts down. What has taken the place of gavity, fom the pespective of the cylinde-dwelles? Centifugal foce does not exist (in inetial fames) is a fictitious, pseudo, vitual foce is an inetial foce
An episode of Quiks & Quaks on CBC adio illustates the confusion that can exist about atificial gavity. The question was: What happens when you jump inside a otating cylinde like the one on the movie 2001: A Space Odysey So, what? He ll go staight acoss and his head will smash into the othe side? That s ight. No. That s wong. The simulated gavity does not equie continuous contact with the suface. Fom you pespective (the jumpe s), you will be pulled back to the suface just as you would in a gavitational field, apat fom a small deviation depending on the adius of the cylinde. To undestand this, we will conside a simple inetial foce fist.
b) Inetial foce -- Appaent foce esulting fom an acceleating efeence fame. e.g. An acceleating spaceship (fa fom planets) ma 0 F N = 0? m F = ma a 0 a0 F N ma 0 = 0 Inetial fame F N = ma 0 Inside the ship (zeo acceleation) F = ma F N ma 0 = 0 To an obseve inside the ship an inetial foce equal to ma 0 is acting towad the floo. (A dopped ball falls.)
c) Centifugal foce: the inetial foce in a otating efeence fame Conside a cup of wate swung aound in a vetical cicle. What keeps the wate in the cup when it is upside-down? In an inetial fame, only the nomal foce and gavity act to give centipetal foce which poduces centipetal acceleation (no such thing as centifugal foce) Answe: Inetia. The cup pulls the wate down faste than its natual falling ate. Gavity plus the nomal foce povide centipetal foce that poduces cicula motion: F N + mg = mv2 F N mg
mv 2 / What keeps wate in the cup fom the pespective of the cup-dwelle? Answe: Invent centifugal foce to balance eal foces and ensue zeo acceleation. So F = 0 F N + mg mv2 = 0 FN mg
What happens if a ball is dopped? mv 2 / If the foce holding it F h is emoved, it should fall adially (down to the cupdwelle), but in an inetial fame it moves tangentially. How can both be tue? F h mg F N
An object eleased moves on the tangent, accoding to Newton s fist law. In the otating efeence fame it acceleates adially outwad (down to the obseve), apat fom a small cuvatue which deceases fo lage cylinde adii.
an object launched adially (jumping) moves with constant velocity at a small angle to the tangent moves up and down in the otating fame (except fo small displacement due to finite adius) An acceleating efeence fame can duplicate the effect of gavity (not only when in contact).
31
inetial fame otating fame
7) Vetical cicula motion Not usually unifom motion since the speed is changing Net foce not always towad the cente Component of acceleation towad the cente (the centipetal component) is still v 2 /, so F c = mv2
F N 3 + mg = mv 2 3 F N 4 = mv 2 4 F N 2 = mv 2 2 F N1 mg = mv 2 1
F N 3 + mg = mv 2 3 Minimum speed at top: F N 3 = 0 mg = mv 2 3 v 3 = g Fo = 6 m, v = 7.6 m/s o 27 km/h
In an automatic clothes dye, a hollow cylinde moves the clothes on a vetical cicle (adius = 0.32 m), as the dawing shows. The appliance is designed so that the clothes tumble gently as they dy. This means that when a piece of clothing eaches an angle of above the hoizontal, it loses contact with the wall of the cylinde and falls onto the clothes below. How many evolutions pe second should the cylinde make in ode that the clothes lose contact with the wall when theta = 70.0º? = 0.32 m θ = 70º