Exercise (2D motion with acceleration) Relative Trajectories: Monkey and Hunter

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1 Physics 207, Lectue 5, Sept. 17 Goals: Solve poblems with multiple acceleations in 2-2 dimensions (including linea, pojectile and cicula motion) Discen diffeent efeence fames and undestand how they elate to paticle motion in stationay and moving fames Recognize diffeent types of foces and know how they act on an object in a paticle epesentation Identify foces and daw a Fee Body Diagam Assignment: HW3, (Chaptes 4 & 5, due 9/25, Wednesday) Read though Chapte 6, Sections 1-4 Physics 207: Lectue 5, Pg 1 Execise (2D motion with acceleation) Relative Tajectoies: Monkey and Hunte A hunte sees a monkey in a tee, aims his gun at the monkey and fies. At the same instant the monkey lets go. Does the bullet A. go ove the monkey? B. hit the monkey? C. go unde the monkey? Physics 207: Lectue 5, Pg 2 Page 1

2 Schematic of the poblem x B ( t) = d = v 0 cos θ t y B ( t) = h f = v 0 sin θ t ½ g t 2 x M ( t) = d y M ( t) = h ½ g t 2 Does y M ( t) = y B ( t) = h f? (x,y) = (d,h) Monkey Does anyone want to change thei answe? What happens if g=0? How does intoducing g change things? Bullet v 0 θ g h f (x 0,y 0 ) = (0,0) (v x,v y ) = (v 0 cos θ, v 0 sin θ) Physics 207: Lectue 5, Pg 3 Unifom Cicula Motion (UCM) is common so we have specialized tems Ac tavesed s = θ Tangential velocity v t Peiod, T, and fequency, f s Angula position, θ Angula velocity, ω v t θ Peiod (T): The time equied to do one full evolution, 360 o 2π adians Fequency (f): 1/T, numbe of cycles pe unit time Angula velocity o speed ω = 2πf = 2π/T, numbe of adians taced out pe unit time (in UCM aveage and instantaneous will be the same) Physics 207: Lectue 5, Pg 4 Page 2

3 Example Question (note the commonality with linea motion) A hoizontally mounted disk 2 metes in diamete spins at constant angula speed such that it fist undegoes 10 counte clockwise evolutions in 5 seconds and then, again at constant angula speed, 2 counte clockwise evolutions in 5 seconds. 1 What is the peiod of the initial otation? 2 What is initial angula velocity? 3 What is the tangential speed of a point on the im duing this initial peiod? 4 Sketch the angula displacement vesus time plot. 5 What is the aveage angula velocity? 6 If now the tuntable stats fom est and unifomly acceleates thoughout and eaches the same angula displacement in the same time, what must the angula acceleation be? 7 What is the magnitude and diection of the acceleation afte 10 seconds? Physics 207: Lectue 5, Pg 5 Example Question A hoizontal tuntable 2 metes in diamete spins at constant angula speed such that it fist undegoes 10 counte clockwise evolutions in 5 seconds and then, again at constant angula speed, 2 counte clockwise evolutions in 5 seconds. 1 What is the peiod of the tuntable duing the initial otation T (time fo one evolution) = t /# of evolutions/ time = 5 sec / 10 ev = 0.5 s 2 What is initial angula velocity? ω = angula displacement / time = 2 π f = 2 π / T = 12.6 ad / s 3 What is the tangential speed of a point on the im duing this initial peiod? We need moe.. Physics 207: Lectue 5, Pg 6 Page 3

4 Relating otation motion to linea velocity In UCM a paticle moves at constant tangential speed v t aound a cicle of adius (only diection changes). s Distance = tangential velocity time Once aound o, eaanging 2π = v t T (2π/T) = v t ω = v t Definition: If UCM then ω = constant 3 So v T = ω = 4 π ad/s 1 m = 12.6 m/s v t θ 4 A gaph of angula displacement (θ) vs. time Physics 207: Lectue 5, Pg 7 Angula displacement and velocity Ac tavesed s = θ in time t then s = θ so s / t = ( θ / t) in the limit t 0 one gets ds / dt = dθ / dt v t = ω ω = dθ / dt if ω is constant, integating ω = dθ / dt, we obtain: θ = θ ο + ω t Counte-clockwise is positive, clockwise is negative v t s θ Physics 207: Lectue 5, Pg 8 Page 4

5 θ (adians) 30π 20π 10π Sketch of θ vs. time θ = θ ο + ω t θ = 0 + 4π 5 ad θ = θ ο + ω t θ = 20π ad + 4π ad time (seconds) 5 Avg. angula velocity = θ / t = 24 π /10 ad/s Physics 207: Lectue 5, Pg 9 Next pat.. 6 If now the tuntable stats fom est and unifomly acceleates thoughout and eaches the same angula displacement in the same time, what must the tangential acceleation be? Physics 207: Lectue 5, Pg 10 Page 5

6 Then angula velocity is no longe constant so dω/dt 0 Define tangential acceleation as a t = dv t /dt = dω/dt So s = s 0 + (ds/dt) 0 t + ½ a t t 2 and s = θ We can elate a t to dω/dt Well, if ω is linealy inceasing 1 θ = θ o + ω o t + t 2 2 a t ω = ω o + t Many analogies to linea motion but it isn t one-to-one Note: Even if the angula velocity is constant, thee is always a adial acceleation. a t Physics 207: Lectue 5, Pg 11 Tangential acceleation? 6 If now the tuntable stats fom est and unifomly acceleates thoughout and eaches the same angula displacement in the same time, what must the tangential acceleation be? 1 θ = θ o + ω o t + t 2 2 (fom plot, afte 10 seconds) 24 π ad = 0 ad + 0 ad/s t + ½ (a t /) t 2 a t v t s θ 48 π ad 1m / 100 s 2 = a t 7 What is the magnitude and diection of the acceleation afte 10 seconds? Physics 207: Lectue 5, Pg 12 Page 6

7 Cicula motion also has a adial (pependicula) component Unifom cicula motion involves only changes in the diection of the velocity vecto, thus acceleation is pependicula to the tajectoy at any point, acceleation is only in the adial diection. Quantitatively (see text) v t Centipetal Acceleation a a = v t2 / Cicula motion involves continuous adial acceleation Physics 207: Lectue 5, Pg 13 Non-unifom Cicula Motion Fo an object moving along a cuved tajectoy, with non-unifom speed a = a + a t (adial and tangential) a t a = v 2 a d v a t = dt Physics 207: Lectue 5, Pg 14 Page 7

8 Tangential acceleation? 7 What is the magnitude and diection of the acceleation afte 10 seconds? s a t = 0.48 π m / s 2 v t a t and ω = ω o + t = 4.8 π m/s = v t a = v 2 t / = 23 π 2 m/s 2 Tangential acceleation is too small to plot! θ Physics 207: Lectue 5, Pg 15 Angula motion, signs Also note: if the angula displacement, velocity and/o accelaations ae counte clockwise then this is said to be positive. Clockwise is negative Physics 207: Lectue 5, Pg 16 Page 8

9 Execise A Ladybug sits at the oute edge of a mey-go-ound, and a June bug sits halfway between the oute one and the axis of otation. The mey-go-ound makes a complete evolution once each second. What is the June bug s angula velocity? A. half the Ladybug s. B. the same as the Ladybug s. C. twice the Ladybug s. D. impossible to detemine. J L Physics 207: Lectue 5, Pg 17 Cicula Motion UCM enables high acceleations (g s) in a small space Comment: In automobile accidents involving otation sevee injuy o death can occu even at modest speeds. [In physics speed doesn t kill.acceleation does (i.e., the sudden change in velocity).] Physics 207: Lectue 5, Pg 18 Page 9

10 Mass-based sepaation with a centifuge Befoe Afte How many g s? a =v t2 / and f = 10 4 pm is typical with = 0.1 m and v t = ω = 2π f ca g s Physics 207: Lectue 5, Pg 19 bb5 1 g Standing g s s with espect to humans 1.2 g Nomal elevato acceleation (up) g Walking down stais. 2-3 g Hopping down stais. 1.5 g Commecial ailine duing takeoff un. 2 g Commecial ailine at otation 3.5 g Maximum acceleation in amusement pak ides (design guidelines). 4 g Indy cas in the second tun at Disney Wold (side and down foce). 4+ g Caie-based aicaft launch. 10 g Theshold fo blackout duing violent maneuves in high pefomance aicaft. 11 g Alan Shepad in his histoic sub obital Mecuy flight expeience a maximum foce of 11 g. 20 g Colonel Stapp s expeiments on acceleation in ocket sleds indicated that in the g ange thee was the possibility of injuy because of ogans moving inside the body. Beyond 20 g they concluded that thee was the potential fo death due to intenal injuies. Thei expeiments wee limited to 20 g. 30 g The design maximum fo sleds used to test dummies with commecial estaint and ai bag systems is 30 g. Physics 207: Lectue 5, Pg 20 Page 10

11 A bad day at the lab. In 1998, a Conell campus laboatoy was seiously damaged when the oto of an ultacentifuge failed while in use. Desciption of the Conell Accident -- On Decembe 16, 1998, milk samples wee unning in a Beckman. L2-65B ultacentifuge using a lage aluminum oto. The oto had been used fo this pocedue many times befoe. Appoximately one hou into the opeation, the oto failed due to excessive mechanical stess caused by the g-foces of the high otation speed. The subsequent explosion completely destoyed the centifuge. The safety shielding in the unit did not contain all the metal fagments. The half inch thick sliding steel doo on top of the unit buckled allowing fagments, including the steel oto top, to escape. Fagments uined a neaby efigeato and an ulta-cold feeze in addition to making holes in the walls and ceiling. The unit itself was popelled sideways and damaged cabinets and shelving that contained ove a hunded containes of chemicals. Sliding cabinet doos pevented the containes fom falling to the floo and beaking. A shock wave fom the accident shatteed all fou windows in the oom. The shock wave also destoyed the contol system fo an incubato and shook an inteio wall. Physics 207: Lectue 5, Pg 21 Relative motion and fames of efeence Refeence fame S is stationay Refeence fame S is moving at v o This also means that S moves at v o elative to S Define time t = 0 as that time when the oigins coincide Physics 207: Lectue 5, Pg 22 Page 11

12 Relative Velocity The positions, and, as seen fom the two efeence fames ae elated though the velocity, v o, whee v o is velocity of the efeence fame elative to = v o t The deivative of the position equation will give the velocity equation v = v v o These ae called the Galilean tansfomation equations Refeence fames that move with constant velocity (i.e., at constant speed in a staight line) ae defined to be inetial efeence fames (IRF); anyone in an IRF sees the same acceleation of a paticle moving along a tajectoy. a = a (dv o / dt = 0) Physics 207: Lectue 5, Pg 23 Cental concept fo poblem solving: x and y components of motion teated independently. Example: Man on cat tosses a ball staight up in the ai. You can view the tajectoy fom two efeence fames: Refeence fame on the moving cat. y(t) motion govened by 1) a = -g y 2) v y = v 0y g t 3) y = y 0 + v 0y g t 2 /2 x motion: x = v x t Refeence fame on the gound. Net motion: R = x(t) i + y(t) j (vecto) Physics 207: Lectue 5, Pg 24 Page 12

13 Example (with fames of efeence) Vecto addition An expeimental aicaft can fly at full thottle in still ai at 200 m/s. The pilot has the nose of the plane pointed west (at full thottle) but, unknown to the pilot, the plane is actually flying though a stong wind blowing fom the nothwest at 140 m/s. Just then the engine fails and the plane stats to fall at 5 m/s 2. What is the magnitude and diections of the esulting B x velocity (elative to the gound) By the instant the engine fails? B y Calculate: A + B A A x + B x = x 0.71 and A y + B y = x 0.71 x Physics 207: Lectue 5, Pg 25 Home Execise, Relative Motion You ae swimming acoss a 50 m wide ive in which the cuent moves at 1 m/s with espect to the shoe. You swimming speed is 2 m/s with espect to the wate. You swim acoss in such a way that you path is a staight pependicula line acoss the ive. How many seconds does it take you to get acoss? a) 50 2 = 25 s b) c) 50 1 = 50 s 50 3 = 29 s 50m 2m/s 1m/s d) 50 2 = 35 s Physics 207: Lectue 5, Pg 26 Page 13

14 Home Execise Choose x axis along ivebank and y axis acoss ive The time taken to swim staight acoss is (distance acoss) / (v y ) Since you swim staight acoss, you must be tilted in the wate so that you x component of velocity with espect to the wate exactly cancels the velocity of the wate in the x diection: v y 1 m/s y x y x 2m/s ive s fame = 3 m/s 1m/s Physics 207: Lectue 5, Pg 27 Home Execise 2 Whee do you land if the ive flows at 2 m/s while swimming at the same heading in the ive (i.e., θ = asin ½)? The time taken to swim staight acoss = (distance acoss) / (v y ) time = 50 m / ( 2 m/s cos θ) = 50/3 ½ seconds Dist in ive = v x t = -2 m/s sin θ t = -2 (50/3½) 1/2 m = -29 m (upsteam) Dist ive flows = v t = 2 m/s t = -2 (50/3½) m = 58 m Final position = -29 m + 58 m = 29 m down the shoe. v y 1 m/s y x y x 2 m/s θ ive s fame = 3 m/s 2 m/s Physics 207: Lectue 5, Pg 28 Page 14

15 What causes motion? (Actually changes in motion) What ae foces? What kinds of foces ae thee? How ae foces and motion elated? Physics 207: Lectue 5, Pg 29 Physics 207, Lectue 5, Sept. 17 Assignment: HW3, (Chaptes 4 & 5, due 9/25, Wednesday) Read Chapte 5 though Chapte 6, Sections 1-4 Physics 207: Lectue 5, Pg 30 Page 15