3. Vectors. Vectors: quantities which indicate both magnitude and direction. Examples: displacemement, velocity, acceleration

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1 Rutgers University Deprtment of Physics & Astronomy 01:750:271 Honors Physics I Lecture 3 Pge 1 of 57

2 3. Vectors Vectors: quntities which indicte both mgnitude nd direction. Exmples: displcemement, velocity, ccelertion Sclrs: quntities which indicte only mgnitude. Pge 2 of 57 Exmples: time, speed, mss

3 Vectors re represented by rrows: (i) The length of the rrow signifies mgnitude. (ii) The hed of the rrow signifies direction. Displcement vector for prticle trvelling from A to B on stright pth Note: All three vectors re identicl becuse they hve the sme direction nd mgnitude. Pge 3 of 57 A shift preserving both direction nd mgnitude does not chnge the vector. (Trnsltion.)

4 Displcement vector for prticle trvelling on curved pth. Note: independent of the pth from A to B. Pge 4 of 57 Nottion:, b, c,... or, b, c,... The mgnitude of vector : or

5 Adding vectors geometriclly Wht is the sum of two vectors? b + =? Step 1. Drw the vectors hed to til Pge 5 of 57 b

6 Step 2. The vector sum of nd b is the vector c pointing from the til of to the hed of b. b Mthemticl formul: + b = c c Pge 6 of 57

7 Commuttivity: + b = b + b b c Pge 7 of 57

8 Associtivity: ( + b ) + c = + ( b + c ) b c ( + b ) + c Pge 8 of 57 b c + ( b + c )

9 Inverse: + ( ) = 0 - Note: hs the sme mgnitude s, but it points in opposite direction Vector subtrction: b = + ( b ). Pge 9 of 57 -b - b b + b

10 Multiplying vectors by sclrs If vector, s 0 number then s = vector with mgnitude s = s Exmple nd direction 1.75 sme s if s > 0 opposite to if s < 0 Note: 0 = 0. Pge 10 of

11 i-clicker Which of the following sttements is flse for the three vectors below? A) + b + c = 0 b B) c + b = C) c < + b c D) c = + b E) None of the bove. Pge 11 of 57

12 Answer Which of the following sttements is flse for the three vectors below? A) + b + c = 0 c b B) c + b = C) c < + b D) c = + b E) None of the bove. Tringle inequlity: c < + b since, b, c not coliner. Pge 12 of 57

13 i-clicker Which of the following reltions is true for the three vectors below? b A) + b + c = 0 B) + b c = 0 c C) + b 2 c = 0 D) + b + 2 c = 0 Pge 13 of 57 E) None of the bove.

14 i-clicker Which of the following reltions is true for the three vectors below? b c A) + b + c = 0 2c B) + b c = 0 C) + b 2 c = 0 D) + b + 2 c = 0 Pge 14 of 57 E) None of the bove.

15 i-clicker Which of the following reltions is true for the three vectors below? b A) + b + c = 0 B) + b c = 0 C) + b 2 c = 0 Pge 15 of 57 D) + b + 2 c = 0 c E) None of the bove.

16 i-clicker Which of the following reltions is true for the three vectors below? b c -2 c A) + b + c = 0 B) + b c = 0 C) + b 2 c = 0 D) + b + 2 c = 0 E) None of the bove. Pge 16 of 57

17 Components of vectors Axis = line equipped with preferred direction, lso clled orienttion. Exmple: one dimensionl motion positive direction x O = origin: x = 0 Pge 17 of 57

18 Projection: suppose nd given xis re in the sme plne 90 o 90 proj projection of long given xis o xis Note: proj is vector long the given xis. proj is not the component of long the given xis. (s stted in the textbook.) Pge 18 of 57

19 = = The component of long given xis is number = proj if proj points in the positive direction proj if proj points in the negtive direction Pge 19 of o xis 90 o xis 90 o proj = proj > 0 90 o proj = - proj < 0

20 = Right tringle rule = cos θ 90 o q proj = proj = xis cos q θ = ngle between the xis nd the vector (counterclockwise) Pge 20 of 57

21 = proj = - = cos ( q-p ) =- cos q 90 o proj proj q = cos q xis θ = ngle between the xis nd the vector (counterclockwise) Pge 21 of 57

22 Summry: The projection of is the vector proj. The component of is the number = cos θ Pge 22 of 57

23 Right hnded coordinte system: three mutully orthogonl xes meeting t point O clled origin. o 90 = p/ 2 z y 90 o = p/ 2 o 90 = p/ 2 O = origin x The x nd y xes re in the pge. The z-xis sticks out of the pge. x, y, z: coordintes Pge 23 of 57

24 The components of long the three xes y trnsltion , 2, 3 : the projections of on the x, y, z xes. x (vectors) Pge 24 of 57 z x, y, z : the components of long the x, y, z xes (numbers)

25 Plnr vectors in x, y plne Pge 25 of 57

26 The right tringle rules for plnr vectors x = cos θ y = sin θ = 2 x + 2 y Pge 26 of 57 tn θ = y x (if x 0).

27 Pge 27 of 57

28 i-clicker A vector is contined in the (y, z) plne such tht the ngle between nd the y xis is φ. Wht re the components of? z f y x A) x = cos φ, y = sin φ, z = 0 B) x = cos φ, y = 0, z = sin φ C) x = 0, y = sin φ, z = cos φ D) x = 0, y = cos φ, z = sin φ E) x = sin φ, y = 0, z = cos φ Pge 28 of 57

29 Answer A vector is contined in the (y, z) plne such tht the ngle between nd the y xis is φ. Wht re the components of? z f z y y x A) x = cos φ, y = sin φ, z = 0 B) x = cos φ, y = 0, z = sin φ C) x = 0, y = sin φ, z = cos φ D) x = 0, y = cos φ, z = sin φ E) x = sin φ, y = 0, z = cos φ Pge 29 of 57

30 Unit vectors Unit vector = vector of mgnitude 1 pointing in the positive direction long n xis xis u u =1 Pge 30 of 57

31 Unit vectors for right hnded coordinte system If hs components x, y, z, its projections re 1 = x î 2 = y ĵ 3 = z k Pge 31 of 57 = x î + y ĵ + z k

32 Two vectors re equl if nd only if their components re equl. = b x = b x, y = b y, z = b z. Pge 32 of 57

33 i-clicker Which of the following expressions is correct for the vector shown below? y A) = cosφî + sinφĵ B) = sinφî + cosφĵ f x C) = sinφî + cosφĵ D) = cosφî sinφĵ E) None of the bove. Pge 33 of 57

34 Answer Which of the following expressions is correct for the vector shown below? y A) = cosφî + sinφĵ B) = sinφî + cosφĵ C) = sinφî + cosφĵ Pge 34 of 57 x f y x D) = cosφî sinφĵ E) None of the bove.

35 Adding vectors by components For ny two vectors: = x î + y ĵ + z k b = bx î + b y ĵ + b z k we hve: + b = ( x + b x )î + ( y + b y )ĵ + ( z + b z ) k Pge 35 of 57 b = ( x b x )î + ( y b y )ĵ + ( z b z ) k More generlly, if s, t re sclrs, s + t b = (s x + tb x )î + (s y + tb y )ĵ + (s z + tb z ) k

36 The sclr product Assocites to ny two vectors, b the number b = b cos φ = b commuttive Pge 36 of 57 Order is irrelevnt!

37 Sclr product in unit vector nottion î î = ĵ ĵ = k k = 1 î ĵ = ĵ k = k î = 0 = x î + y ĵ + z k b = bx î + b y ĵ + b z k Pge 37 of 57 b = ( x î + y ĵ + z k) ( bx î + b y ĵ + b z k) = x b x + y b y + z b z Sclr product nd componets = ( î)î + ( ĵ)ĵ + ( k) k x = î, y = ĵ, z = k

38 Exmple: the mgnitude of vector sum nd difference + b 2 = ( + b) ( + b) = + 2 b + b b = 2 + b b cos(θ) b 2 = ( b) ( b) = 2 b + b b = 2 + b 2 2 b cos(θ) Pge 38 of 57

39 Vectors nd the lws of physics Reltions mong vectors do not depend on the choice of coordinte system. Reltions in physics re lso independent of the choice of coordinte system. = 2 x + 2 y = ( x )2 + ( y )2 θ = θ + φ Pge 39 of 57

40 Position nd Displcement In 3D the position of prticle is given by position vector r = xî + yĵ + z k strting t the origin O. x, y, z re the components of r, lso clled the coordintes of the prticle. The pth of the prticle is generlly curve. Pge 40 of 57

41 Displcement: the chnge of the position vector r over time intervl t. y r 1 D r r 2 x r 1 = x 1 î + y 1 ĵ + z 1 k r 2 = x 2 î + y 2 ĵ + z 2 k t time t1 t time t2 Pge 41 of 57 z r = (x 2 î + y 2 ĵ + z 2 k) (x1 î + y 1 ĵ + z 1 k) = (x 2 x 1 )î + (y 2 y 1 )ĵ + (z 2 z 1 ) k. = x î + y ĵ + z k.

42 i-clicker A prticle trvels long 3D pth s shown in the figure strting from the initil position r 1 = (1 m)î nd ending with the finl position r 2 = (0.5 m)î. Wht is the displcement vector? y 0.7m 0.5m 1m x 0.3m z A) r = ( 1 m)î + (0.7 m)ĵ B) r = (0.7 m)ĵ (0.3 m) k C) r = ( 0.5 m)î D) r = 0 E) r = (0.5 m)î Pge 42 of 57

43 Answer A prticle trvels long 3D pth s shown in the figure strting from the initil position r 1 = (1 m)î nd ending with the finl position r 2 = (0.5 m)î. Wht is the displcement vector? 0.7m z y 0.3m 0.5m Dr=(-0.5 1m m) i ^ x A) r = ( 1 m)î + (0.7 m)ĵ B) r = (0.7 m)ĵ (0.3 m) k C) r = ( 0.5 m)î D) r = 0 E) r = (0.5 m)î Pge 43 of 57

44 Averge nd Instntneous Velocity Averge velocity over time intervl t Displcement vector v vg = = r time intervl t r 1 D r r 2 x y v vg Note: Since t > 0 v vg is lwys prllel with r nd points in the sme direction. Pge 44 of 57 z

45 Instntneous velocity t time t r v = lim t 0 t = d r ( ) ( ) dx dy dt = î + ĵ + dt dt ( dz dt ) k y r 1 v Dr r 2 tngent to pth x Note: v is lwys tngent to the pth of the prticle t its current position. Pge 45 of 57 z

46 Components of v: v x = dx dt, v y = dy dt, v z = dz dt Pge 46 of 57

47 Exmple: circulr trjectory r v Pge 47 of 57 v x = vsin(θ), v y = vcos(θ)

48 Averge nd Instntneous Accelertion Averge ccelertion over time intervl t vg = v t y v 1 vg Note: Pge 48 of 57 v 2 Dv x Since t > 0 vg is lwys prllel with v nd points in the sme direction. z

49 Instntneous ccelertion t time t v = lim t 0 t = d v ( ) ( ) ( ) dt = dvx dvy dvz î + ĵ + k dt dt dt z y v 1 v 2 Dv x Accelertion is vector which mesures the chnge of the velocity vector. Components of : x = dv x dt, y = dv y dt, z = dv z dt Pge 49 of 57

50 The ccelertion is non-zero if either the mgnitude of v chnges. or the direction of v chnges. Pge 50 of 57 The ccelertion is 0 if nd only if both the mgnitude nd the direction of v re constnt.

51 i-clicker A prticle moves on pth s shown below such tht the mgnitude of its velocity vector is constnt. Where is the instntneous ccelertion zero? z A y v A B v B C x v C A) Everywhere B) At A C) At C D) At B E) Nowhere Pge 51 of 57

52 Answer A prticle moves on pth s shown below such tht the mgnitude of its velocity vector is constnt. Where is the instntneous ccelertion zero? z A y v A B v B liner segment C x v C A) Everywhere B) At A C) At C D) At B E) Nowhere v constnt v constnt on liner segment; v not constnt on curved segments; chnges direction. Pge 52 of 57

53 i-clicker A prticle moves on helix s shown below such tht the y-component of its velocity is constnt. Which of the following sttements is flse? v j ^ y y vy constnt A) x 0 B) z 0 Pge 53 of 57 C) y = 0 D) y 0 E) None of the bove. z x

54 Answer A prticle moves on helix s shown below such tht the y-component of its velocity is constnt. Which of the following sttements is flse? y vy constnt A) x 0 B) z 0 Pge 54 of 57 v C) y = 0 v v ^ y j D) y 0 E) None of the bove. z x

55 Exmple A bus is moving horizontlly with constnt speed v = 10 m/s. Inside the bus bll flls distnce h = 1 m in uniform grvittionl field with zero initil velocity. Assuming ir resistnce negligible, wht is the horizontl displcement of the bll during the fll reltive to the ground? Wht is the velocity vector of the bll when it hits the floor mesured by ground bsed observer? Pge 55 of 57

56 Flling time: y = h gt 2 /2 t = Horizontl displcement: Finl velocity vector: x = v t = v v y = gt, 2h/g v x = v v f = vî 2ghĵ 2h/g Pge 56 of 57

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