Engineering Mechanics - Dynamics Chapter 12

Transcription

1 Engineering Mechanic - Dynaic Chapter 1 ρ 50 ft t 4 3 ft c 3 ft 3 t 1 3 v t + ct a t + ct At t 1 v 1 t 1 + ct 1 v 1 a t1 + ct 1 a n1 ρ a 1 a t1 + a n1 a ft Ditance traveled d 1 t 1 c + 3 t 1 3 d ft Prole 1-10 At a given intant the jet plane ha peed v and acceleration a acting in the direction hown. Deterine the rate of increae in the plane peed and the radiu of curvature ρ of the path. v a 400 ft 70 ft θ 60 deg Rate of increae a t ( a)co( θ) a t 35 ft Radiu of curvature a n ( a)in( θ) v v ρ ρ 639 ft ρ ( a)in θ Prole A particle i oving along a curved path at a contant peed v. The radii of curvature of the path at point P and P' are ρ and ρ', repectively. If it take the particle tie t to go fro P to P', deterine the acceleration of the particle at P and P'. v 60 ft ρ 0 ft ρ' 50 ft t 0 76

2 Engineering Mechanic - Dynaic Chapter 1 a a' v a 180 ft ρ v a' 7 ft ρ' Note that the tie doen t atter here ecaue the peed i contant. *Prole A oat i traveling along a circular path having radiu ρ. Deterine the agnitude of the oat acceleration when the peed i v and the rate of increae in the peed i a t. ρ 0 v 5 a t v a n a n 1.5 ρ a a t + a n a.358 Prole Starting fro ret, a icyclit travel around a horizontal circular path of radiu ρ at a peed v t + c t. Deterine the agnitude of hi velocity and acceleration when he ha traveled a ditance 1. ρ c Gue t 1 1 Given 1 3 t 1 3 c + t 1 t 1 Find( t 1 ) t v 1 t 1 + ct 1 v a t1 t 1 + c a t v 1 a n1 a n ρ a 1 a t1 + a n1 a

3 Engineering Mechanic - Dynaic Chapter 1 Prole The jet plane travel along the vertical paraolic path. When it i at point A it ha peed v which i increaing at the rate a t. Deterine the agnitude of acceleration of the plane when it i at point A. v 00 a t 0.8 d h 5k 10 k yx h x d y' ( x) d dx yx y'' ( x) ρ ( x) d dx y' ( x) ( 1 + y' ( x) ) 3 y'' ( x) v a n a a t + a n a 0.91 ρ ( d) Prole The car travel along the curve having a radiu of R. If it peed i uniforly increaed fro v 1 to v in tie t, deterine the agnitude of it acceleration at the intant it peed i v 3. v 1 15 t 3 78

4 Engineering Mechanic - Dynaic Chapter 1 v 7 R 300 v 3 0 v v 1 v 3 a t a n a a t + a n a 4. t R *Prole The atellite S travel around the earth in a circular path with a contant peed v 1. If the acceleration i a, deterine the altitude h. Aue the earth diaeter to e d. Unit Ued: M 10 3 k v 1 0 M hr a.5 d 1713 k Gue h 1M Given v 1 a h Find( h) h 5.99 M d h + Prole A particle P ove along the curve y x + c with a contant peed v. Deterine the point on the curve where the axiu agnitude of acceleration occur and copute it value. 1 1 c 4 v 5 Maxiu acceleration occur where the radiu of curvature i the allet which occur at x 0. 79

5 Engineering Mechanic - Dynaic Chapter 1 yx x + c y' ( x) ρ ( x) d dx yx y'' ( x) d dx y' ( x) ( 1 + y' ( x) ) 3 ρ in ρ ( 0) ρ in 0.5 y'' ( x) v a ax a ax 50 ρ in Prole The Ferri wheel turn uch that the peed of the paenger i increaed y a t t. If the wheel tart fro ret when θ 0, deterine the agnitude of the velocity and acceleration of the paenger when the wheel turn θ θ 1. 4 ft 3 θ 1 30 deg r 40 ft Guee t 1 1 v 1 1 ft a t1 1 ft Given a t1 t 1 v 1 t 1 rθ 1 6 t 1 3 a t1 v 1 Find( a t1, v 1, t 1 ) t v ft a t1 1.6 ft t 1 v 1 a 1 a t1 + v ft a ft r Prole At a given intant the train engine at E ha peed v and acceleration a acting in the direction hown. Deterine the rate of increae in the train' peed and the radiu of curvature ρ of the path. 80

6 Engineering Mechanic - Dynaic Chapter 1 v 0 a 14 θ 75 deg a t ( a)co θ a t 3.6 a n ( a)in θ a n v ρ ρ a n *Prole 1 11 A package i dropped fro the plane which i flying with a contant horizontal velocity v A. Deterine the noral and tangential coponent of acceleration and the radiu of curvature of the path of otion (a) at the oent the package i releaed at A, where it ha a horizontal velocity v A, and () jut efore it trike the ground at B. v A 150 ft h 1500 ft g 3. ft At A: a An v A g ρ A ρ A 699 ft a An 81

7 Engineering Mechanic - Dynaic Chapter 1 At B: t h v x v A v y gt θ atan v y g v x v v B v x + v y a Bn g co( θ) B ρ B ρ B 8510 ft a Bn Prole The autooile i originally at ret at 0. If it peed i increaed y dv/dt t, deterine the agnitude of it velocity and acceleration when t t ft 4 t 1 18 ρ 40 ft d 300 ft a t1 t 1 a t1 16. ft v 1 3 t 1 3 v ft 1 t ft 1 If ft > d 300 ft then we are on the curved part of the track. v 1 a n1 a n ft ρ a a n1 + a t1 a ft If ft < d 300 ft then we are on the traight part of the track. a n1 0 ft a n1 0 ft a a n1 + a t1 a 16. ft 8

8 Engineering Mechanic - Dynaic Chapter 1 Prole The autooile i originally at ret at 0. If it then tart to increae it peed at dv/dt t, deterine the agnitude of it velocity and acceleration at 1. d 300 ft ρ 40 ft 0.05 ft ft a t t v If t t 1 1 t v v ft 550 ft > d 300 ft the car i on the curved path a t t 1 v If 1 3 t 1 3 v a n ρ 550 ft < d 300 ft the car i on the traight path a a t + a n a ft a t t 1 a n 0 ft a a t + a n a ft Prole The truck travel in a circular path having a radiu ρ at a peed v 0. For a hort ditance fro 0, it peed i increaed y a t. Deterine it peed and the agnitude of it acceleration when it ha oved a ditance 1. ρ v

9 Engineering Mechanic - Dynaic Chapter 1 a t v 1 1 v dv d v 0 0 v 1 v 0 1 v 1 v v v 1 a t1 1 a n1 a 1 a t1 + a n1 a ρ *Prole The particle travel with a contant peed v along the curve. Deterine the particle acceleration when it i located at point x x 1. v 300 k x 1 00 yx y' ( x) k x d dx yx y'' ( x) d dx y' ( x) ρ ( x) ( 1 + y' ( x) ) 3 y'' ( x) θ ( x) atan( y' ( x) ) θ 1 θ x 1 θ deg a v in θ a ρ ( x 1 ) co( θ 1 ) 88 a 3 84

10 Engineering Mechanic - Dynaic Chapter 1 Prole Car ove around the traffic circle which i in the hape of an ellipe. If the peed liit i poted at v, deterine the axiu acceleration experienced y the paenger. v a 60 k hr Maxiu acceleration occur where the radiu of curvature i the allet. In thi cae that happen when y 0. y x( y) a 1 x' ( y) d dy x( y) x'' ( y) d dy x' ( y) ρ ( y) ( 1 + x' ( y) ) 3 ρ in ρ ( 0) ρ in x'' ( y) v a ax a ax 10.4 ρ in Prole Car ove around the traffic circle which i in the hape of an ellipe. If the peed liit i poted at v, deterine the iniu acceleration experienced y the paenger. v 60 k hr a

11 Engineering Mechanic - Dynaic Chapter 1 Miniu acceleration occur where the radiu of curvature i the larget. In thi cae that happen when x 0. x yx 1 y' ( x) a d dx yx y'' ( x) d dx y' ( x) ρ ( x) ( 1 + y' ( x) ) 3 ρ ax ρ ( 0) ρ ax 90 y'' ( x) v a in a in 3.09 ρ ax Prole The car B turn uch that it peed i increaed y dv B /dt e ct. If the car tart fro ret when θ 0, deterine the agnitude of it velocity and acceleration when the ar AB rotate to θ θ 1. Neglect the ize of the car. 0.5 c 1 1 θ 1 30 deg ρ 5 a Bt e ct v B ρθ c ect 1 c e ct c t c Gue t

12 Engineering Mechanic - Dynaic Chapter 1 Given ρθ 1 c e ct 1 c t 1 c t 1 Find( t 1 ) t 1.13 v B1 c ect 1 1 v B a Bt1 e ct 1 v B1 a Bn1 a B1 a Bt1 + a Bn1 ρ a Bt a Bn1.708 a B *Prole 1-10 The car B turn uch that it peed i increaed y dv B /dt e ct. If the car tart fro ret when θ 0, deterine the agnitude of it velocity and acceleration when t t 1. Neglect the ize of the car. Alo, through what angle θ ha it traveled? 0.5 c 1 1 t 1 ρ 5 a Bt e ct v B c ect 1 ρθ c e ct c t c v B1 c ect 1 1 v B a Bt1 e ct 1 v B1 a Bn1 a B1 a Bt1 + a Bn1 ρ 87

13 Engineering Mechanic - Dynaic Chapter 1 a Bt a Bn1.041 a B θ 1 ρ c e ct 1 c t 1 c θ deg Prole 1 11 The otorcycle i traveling at v 0 when it i at A. If the peed i then increaed at dv/dt a t, deterine it peed and acceleration at the intant t t 1. k a t 0.1 v 0 1 t 1 5 yx kx y' ( x) kx y'' ( x) k ρ ( x) ( 1 + y' ( x) ) 3 y'' ( x) 1 v 1 v 0 + a t t 1 1 v 0 t 1 a + t t 1 v x 1 Gue x 1 1 Given y' ( x) dx x 1 Find x 1 0 a 1t v 1 a t a 1n a 1 a 1t + a 1n a ρ ( x 1 ) Prole 1-1 The all i ejected horizontally fro the tue with peed v A. Find the equation of the path y f (x), and then find the all velocity and the noral and tangential coponent of acceleration when t t 1. 88

14 Engineering Mechanic - Dynaic Chapter 1 v A 8 t g 9.81 x v A t t x y v A g t y g x paraola v A when t t 1 v y v x v A v y g t 1 θ atan θ deg v x a n g co( θ) a n a t g in( θ) a t.875 Prole 1 13 The car travel around the circular track having a radiu r uch that when it i at point A it ha a velocity v 1 which i increaing at the rate dv/dt kt. Deterine the agnitude of it velocity and acceleration when it ha traveled one-third the way around the track. k r 300 v 1 5 a t () t kt k vt () v 1 + t p () t v 1 t + k 6 t3 89

15 Engineering Mechanic - Dynaic Chapter 1 Gue t 1 1 Given p ( t 1 ) v 1 vt ( 1 ) v 1 πr t 1 Find( t 1 ) t v 1 a t1 a t ( t 1 ) a n1 a 1 a t1 + a n1 r 43.0 a *Prole 1 14 The car travel around the portion of a circular track having a radiu r uch that when it i at point A it ha a velocity v 1 which i increaing at the rate of dv/dt k. Deterine the agnitude of it velocity and acceleration when it ha traveled three-fourth the way around the track. k 0.00 r 500 ft v 1 ft 3 p1 4 πr a d t v v k p d p Gue v 1 1 ft Given a t1 v 1 p1 v dv k p d p v 1 Find v v 1 k p1 a n1 a 1 a t1 + a n1 v ft r a 1.7 ft Prole 1-15 The two particle A and B tart at the origin O and travel in oppoite direction along the circular path at contant peed v A and v B repectively. Deterine at t t 1, (a) the diplaceent along the path of each particle, () the poition vector to each particle, and (c) the hortet ditance etween the particle. 90

16 Engineering Mechanic - Dynaic Chapter 1 v A 0.7 v B 1.5 t 1 ρ 5 (a) The diplaceent along the path A v A t 1 A 1.4 B v B t 1 B 3 () The poition vector to each particle θ A θ B A ρ in θ A 1.38 r A r A ρ ρ ρco( θ A ) B ρ in θ B.83 r B r B ρ ρ ρco( θ B ) (c) The hortet ditance etween the particle d r B r A d 4.6 Prole 1-16 The two particle A and B tart at the origin O and travel in oppoite direction along the circular path at contant peed v A and v B repectively. Deterine the tie when they collide and the agnitude of the acceleration of B jut efore thi happen. v A 0.7 v B 1.5 ρ 5 91

17 Engineering Mechanic - Dynaic Chapter 1 ( v A + v B )t πρ t πρ v A + v B t 14.8 v B a B ρ a B 0.45 Prole 1-17 The race car ha an initial peed v A at A. If it increae it peed along the circular track at the rate a t, deterine the tie needed for the car to travel ditance 1. v A ρ 150 a t d v v d v v dv d v v A 0 v A d v dt v A + 9

18 Engineering Mechanic - Dynaic Chapter v A + t d 1 dt t v A + d t *Prole 1-18 A oy it on a erry-go-round o that he i alway located a ditance r fro the center of rotation. The erry-go-round i originally at ret, and then due to rotation the oy peed i increaed at the rate a t. Deterine the tie needed for hi acceleration to ecoe a. r 8ft a t ft a 4 ft a n a a t v a n r t v t.63 a t Prole 1 19 A particle ove along the curve y in(cx) with a contant peed v. Deterine the noral and tangential coponent of it velocity and acceleration at any intant. v 1 c 1 y in( cx) y' c co( cx) y'' c in( cx) ρ ( 1 + y' ) 3 y'' ( cco( cx) ) c in( cx) a n v cin( cx) a t 0 v t 0 v n ( cco( cx) ) Prole The otion of a particle along a fixed path i defined y the paraetric equation r, θ ct 93

19 Engineering Mechanic - Dynaic Chapter 1 p g p y p q and z dt. Deterine the unit vector that pecifie the direction of the inoral axi to the oculating plane with repect to a et of fixed x, y, z coordinate axe when t t 1. Hint: Forulate the particle velocity v p and acceleration a p in ter of their i, j, k coponent. Note that x rco θ and y rin θ. The inoral i parallel to v p a p. Why? 8ft c 4 rad d 6 ft t 1 r p1 co ct 1 c in ct 1 c co ct 1 in( ct 1 ) v p1 cco( ct 1 ) a p1 dt 1 dt 1 c in( ct 1 ) Since v p and a p are in the noral plane and the inoral direction i perpendicular to thi plane then we can ue the cro product to define the inoral direction. d u v p1 a p1 u v p1 a p Prole Particle A and B are traveling counter-clockwie around a circular track at contant peed v 0. If at the intant hown the peed of A i increaed y dv A /dt A, deterine the ditance eaured counterclockwie along the track fro B to A when t t 1. What i the agnitude of the acceleration of each particle at thi intant? v t 1 1 r 5 θ 10 deg Ditance a At dv A v A A d A v A v A dv A v 0 94 A A d A 0

20 Engineering Mechanic - Dynaic Chapter 1 v A v 0 A v A v 0 + A d A dt Gue A1 1 Given A1 Find( A1 ) A t 1 A1 1 dt v 0 + A d A B1 v 0 t 1 B1 8 AB A1 + rθ B1 AB a A ( A1 ) v 0 + A1 + a A r v 0 a B a B 1.8 r Prole 1-13 Particle A and B are traveling around a circular track at peed v 0 at the intant hown. If the peed of B i increaed y dv B /dt a Bt, and at the ae intant A ha an increae in peed dv A /dt t, deterine how long it take for a colliion to occur. What i the agnitude of the acceleration of each particle jut efore the colliion occur? v 0 8 r 5 a Bt 4 θ 10 deg v B a Bt t + v 0 B a Bt t + v 0 t a At t v A t + v 0 A 6 t3 + v 0 t Aue that B catche A Gue t

21 Engineering Mechanic - Dynaic Chapter 1 Given a Bt t 1 + v 0 t 1 6 t v 0 t 1 + rθ t 1 Find( t 1 ) t Aue that A catche B Gue t 13 Given a Bt t + v 0 t + r( π θ) Take the aller tie t in t 1, t t t 3 + v 0 t t Find( t ) t a A ( t) t + v 0 + a Bt t + v 0 a B a Bt + r r a A a B Prole The truck travel at peed v 0 along a circular road that ha radiu ρ. For a hort ditance fro 0, it peed i then increaed y dv/dt. Deterine it peed and the agnitude of it acceleration when it ha oved a ditance 1. v 0 4 ρ d a t v d v v 1 1 v dv d v 0 0 v 1 v 0 1 v 1 v v

22 Engineering Mechanic - Dynaic Chapter 1 v 1 a t 1 a n a a t + a n a ρ Prole A go-cart ove along a circular track of radiu ρ uch that it peed for a hort period of tie, 0 < t < t, i v 1 e ct. Deterine the agnitude of it acceleration when t t. How far 1 ha it traveled in t t? Ue Sipon rule with n tep to evaluate the integral. ρ 100 ft t ft c 1 t n 50 t t v 1 e ct a t cte ct v a n a a t + a n a 35.0 ft ρ t 1 e ct dt 67.1 ft 0 Prole A particle P travel along an elliptical piral path uch that it poition vector r i defined y r (a co t i + c in dt j + et k). When t t 1, deterine the coordinate direction angle α, β, and γ, which the inoral axi to the oculating plane ake with the x, y, and z axe. Hint: Solve for the velocity v p and acceleration a p of the particle in ter of their i, j, k coponent. The inoral i parallel to v p a p. Why? 97

23 Engineering Mechanic - Dynaic Chapter 1 a d e c 1.5 t 1 8 t t 1 r p v p ( a)co( t) c in( dt) et a in( t) a cdco( dt) a p e co( t) c d in( dt) v p a p u u v p a p α β γ aco( u ) α β γ deg *Prole The tie rate of change of acceleration i referred to a the jerk, which i often ued a a ean of eauring paenger dicofort. Calculate thi vector, a', in ter of it cylindrical coponent, uing Eq ur + ( rθ'' + r'θ' )u θ + z''u z ( rθ'θ'' ) ur + ( r'' rθ' ) u'r a r'' rθ' a' r''' r'θ'... + ( r'θ'' + rθ''' + r''θ' + r'θ'' )u θ + ( rθ'' + r'θ'' )u' θ + z'''u z + z''u' z But u r θ'u θ u' θ θ' u r u' z 0 Sutituting and coining ter yield ur ( rθ''' + 3r'θ'' + 3r''θ' rθ' 3 ) uθ a' r''' 3r'θ' 3rθ'θ'' + + ( z''' )u z 98

24 Engineering Mechanic - Dynaic Chapter 1 Prole If a particle poition i decried y the polar coordinate r a(1 + in t) and θ ce dt, deterine the radial and tangential coponent of the particle velocity and acceleration when t t 1. a c rad d 1 1 t 1 When t t 1 r a( 1 + in( t) ) r' a co( t) r'' a in( t) θ ce dt θ' cde dt θ'' c d e dt v r r' v r 1.66 vθ rθ' vθ.07 a r r'' rθ' a r 4.0 aθ rθ'' + r'θ' aθ.97 Prole The lotted fork i rotating aout O at a contant rate θ'. Deterine the radial and tranvere coponent of the velocity and acceleration of the pin A at the intant θ θ 1. The path i defined y the piral groove r + cθ, where θ i in radian. θ' 3 rad 5in c 1 π in θ 1 π rad θ θ 1 99

25 Engineering Mechanic - Dynaic Chapter 1 r + cθ r' cθ' r'' 0 in θ'' 0 rad v r r' vθ rθ' a r r'' rθ' aθ rθ'' + r'θ' v r in vθ 1 in in a r 63 aθ 5.73 in Prole The lotted fork i rotating aout O at the rate θ ' which i increaing at θ '' when θ θ 1. Deterine the radial and tranvere coponent of the velocity and acceleration of the pin A at thi intant. The path i defined y the piral groove r (5 + θ /π) in., where θ i in radian. θ' 3 rad θ'' rad 5in c 1 π in θ 1 π rad θ θ 1 r + cθ r' cθ' r'' cθ'' v r r' vθ rθ' a r r'' rθ' aθ rθ'' + r'θ' v r in vθ 1 in in a r aθ in *Prole If a particle ove along a path uch that r aco(t) and θ ct, plot the path r f(θ) and deterine the particle radial and tranvere coponent of velocity and acceleration. 100

26 Engineering Mechanic - Dynaic Chapter 1 a ft 1 1 c 0.5 rad The plot θ t r ( a)co θ c c θ 0, 0.01 π.. π r θ a co θ c 1 ft Ditance in ft r( θ) θ Angle in radian r ( a)co( t) r' a in( t) r'' a co( t) θ ct θ' c θ'' 0 co t v r r' a in( t) a r r'' rθ' a + c vθ rθ' acco( t) aθ rθ'' + r'θ' acin( t) Prole If a particle poition i decried y the polar coordinate r ainθ and θ ct, deterine the radial and tangential coponent of it velocity and acceleration when t t 1. a rad c 4 rad t 1 1 t t 1 r ( a)in( ct) r' a c co( ct) r'' a c in( ct) θ ct θ' c θ'' 0 rad v r r' v r.38 vθ rθ' vθ

27 Engineering Mechanic - Dynaic Chapter 1 a r r'' rθ' a r aθ rθ'' + r'θ' aθ Prole 1-14 A particle i oving along a circular path having a radiu r. It poition a a function of tie i given y θ t. Deterine the agnitude of the particle acceleration when θ θ 1. The particle tart fro ret when θ 0. r 400 rad θ 1 30 deg t θ 1 t 0.51 θ t θ' t θ'' a ( rθ' ) + ( rθ'' ) a.317 Prole A particle ove in the x - y plane uch that it poition i defined y r ati + t j. Deterine the radial and tangential coponent of the particle velocity and acceleration when t t 1. a ft 4 ft t 1 t t 1 Rectangular x at v x a a x 0 ft y t v y t a y Polar θ atan y θ deg x 10

28 Engineering Mechanic - Dynaic Chapter 1 v r v y in( θ) v x co θ + v r ft v y co( θ) vθ v x in θ + vθ 1.94 ft a r a x co( θ) + a y in( θ) a r ft aθ a x in( θ) + a y co( θ) aθ 1.94 ft *Prole A truck i traveling along the horizontal circular curve of radiu r with a contant peed v. Deterine the angular rate of rotation θ' of the radial line r and the agnitude of the truck acceleration. r 60 v 0 θ' a v θ' rad r rθ' a Prole A truck i traveling along the horizontal circular curve of radiu r with peed v which i increaing at the rate v'. Deterine the truck radial and tranvere coponent of acceleration. 103

29 Engineering Mechanic - Dynaic Chapter 1 r v 60 0 v' 3 v a r a r r aθ v' aθ 3 Prole A particle i oving along a circular path having radiu r uch that it poition a a function of tie i given y θ c in t. Deterine the acceleration of the particle at θ θ 1. The particle tart fro ret at θ 0. r 6in c 1 rad 3 1 θ 1 30 deg t 1 ain θ 1 t c θ c in( t) θ' cco( t) θ'' c in( t) a ( rθ' ) + ( rθ'' ) a in Prole The lotted link i pinned at O, and a a reult of the contant angular velocity θ' it drive the peg P for a hort ditance along the piral guide r aθ. Deterine the radial and tranvere coponent of the velocity and acceleration of P at the intant θ θ

30 Engineering Mechanic - Dynaic Chapter 1 θ' a 3 rad π θ 1 3 rad θ θ 1 r aθ r' aθ' r'' 0 v r r' v r 1. vθ rθ' vθ 1.57 a r r'' rθ' a r 3.77 aθ r'θ' aθ 7. *Prole The lotted link i pinned at O, and a a reult of the angular velocity θ' and the angular acceleration θ'' it drive the peg P for a hort ditance along the piral guide r aθ. Deterine the radial and tranvere coponent of the velocity and acceleration of P at the intant θ θ 1. θ' θ'' 3 rad π θ 1 3 rad a rad 0.5 θ θ 1 105

31 Engineering Mechanic - Dynaic Chapter 1 r aθ r' aθ' r'' aθ'' v r r' v r 1. vθ rθ' vθ 1.57 a r r'' rθ' a r 0.57 aθ rθ'' + r'θ' aθ Prole The lotted link i pinned at O, and a a reult of the contant angular velocity θ' it drive the peg P for a hort ditance along the piral guide r aθ where θ i in radian. Deterine the velocity and acceleration of the particle at the intant it leave the lot in the link, i.e., when r. θ' a 3 rad θ a r aθ r' aθ' r'' 0 v r r' vθ rθ' a r r'' rθ' aθ r'θ' v v r + vθ a a r + aθ v 1.91 a Prole A train i traveling along the circular curve of radiu r. At the intant hown, it angular rate of rotation i θ', which i decreaing at θ''. Deterine the agnitude of the train velocity and acceleration at thi intant. 106

32 Engineering Mechanic - Dynaic Chapter 1 r θ' 600 ft 0.0 rad θ'' rad v rθ' v 1 ft a ( rθ' ) + ( rθ'' ) a ft Prole A particle travel along a portion of the four-leaf roe defined y the equation r a co(θ). If the angular velocity of the radial coordinate line i θ' ct, deterine the radial and tranvere coponent of the particle velocity and acceleration at the intant θ θ 1. When t 0, θ 0. a 5 c 3 rad 3 θ 1 30 deg θ () t c 3 t3 θ' () t ct θ'' () t ct rt () ( a)co θ () t r' () t 1 3 3θ 1 When θ θ 1 t 1 c v r r' ( t 1 ) d dt rt () r'' () t d dt r' () t v r

33 Engineering Mechanic - Dynaic Chapter 1 vθ rt ( 1 )θ'( t 1 ) vθ 4.87 a r r'' ( t 1 ) rt ( 1 )θ' t 1 a r 89.4 aθ rt ( 1 )θ'' ( t 1 ) + r' ( t 1 )θ'( t 1 ) aθ 53.7 *Prole 1-15 At the intant hown, the waterprinkler i rotating with an angular peed θ' and an angular acceleration θ''. If the nozzle lie in the vertical plane and water i flowing through it at a contant rate r', deterine the agnitude of the velocity and acceleration of a water particle a it exit the open end, r. θ' rad θ'' 3 rad r' 3 r 0. v r' + rθ' v 3.07 a ( rθ' ) + ( rθ'' + r'θ' ) a 1.65 Prole The oy lide down the lide at a contant peed v. If the lide i in the for of a helix, defined y the equation r contant and z (hθ )/(π), deterine the oy angular velocity aout the z axi, θ' and the agnitude of hi acceleration. v r h

34 Engineering Mechanic - Dynaic Chapter 1 z h π θ z' h π θ' v z' + rθ' h + r θ' π θ' v θ' rad h + r π a rθ' a.55 Prole A caeraan tanding at A i following the oveent of a race car, B, which i traveling along a traight track at a contant peed v. Deterine the angular rate at which he ut turn in order to keep the caera directed on the car at the intant θ θ 1. v 80 ft θ 1 60 deg a 100 ft θ θ 1 a rin( θ) 0 r' in θ + x Gue rco( θ) rθ' co( θ) v x v r' co θ rθ' in( θ) r 1ft r' 1 ft θ' 1 rad Given a rin( θ) 109

35 Engineering Mechanic - Dynaic Chapter 1 0 r' in θ + v rθ' co( θ) r' co θ rθ' in( θ) r r' θ' Find r, r', θ' ft r ft r' 40 θ' 0.6 rad Prole For a hort ditance the train travel along a track having the hape of a piral, r a/θ. If it aintain a contant peed v, deterine the radial and tranvere coponent of it velocity when θ θ 1. a 1000 v 0 θ 1 9 π 4 rad θ θ 1 a a r r' θ θ θ' v r' + r θ' vθ a a θ' a 1 + θ r r' θ θ θ' a θ 4 + a θ θ' v r r' v r.80 vθ rθ' vθ *Prole For a hort ditance the train travel along a track having the hape of a piral, r a / θ. If the angular rate θ' i contant, deterine the radial and tranvere coponent of it velocity and acceleration when θ θ 1. a 1000 θ' 0. rad θ 1 9 π 4 θ θ 1 r a r' θ a θ θ' r'' a θ 3 θ' 110

36 Engineering Mechanic - Dynaic Chapter 1 v r r' v r vθ rθ' vθ 8.3 a r r'' rθ' a r 5.43 aθ r'θ' aθ Prole The ar of the root ha a variale length o that r reain contant and it grip. A ove along the path z a inθ. If θ ct, deterine the agnitude of the grip velocity and acceleration when t t 1. r 3ft c 0.5 rad a 3ft t t t 1 θ ct r r z ain( ct) θ' c r' 0 ft z' a c co( ct) θ'' 0 rad r'' 0 ft z'' a c in( ct) v r' + rθ' + z' v ft rθ'' r'θ' a r'' rθ' + ( + ) + z'' a ft Prole For a hort tie the ar of the root i extending o that r' reain contant, z t and θ ct. Deterine the agnitude of the velocity and acceleration of the grip A when t t 1 and r r

37 Engineering Mechanic - Dynaic Chapter 1 r' c 1.5 ft 4 ft 0.5 rad t 1 r 1 3 3ft t t 1 r r 1 θ ct z t θ' c z' t z'' v r' + rθ' + z' v 4.1 ft a ( rθ' ) + ( r'θ' ) + z'' a ft Prole The rod OA rotate counterclockwie with a contant angular velocity of θ'. Two pin-connected lider lock, located at B, ove freely on OA and the curved rod whoe hape i a liaçon decried y the equation r (c co(θ)). Deterine the peed of the lider lock at the intant θ θ 1. θ' 5 rad 100 c θ 1 10 deg θ θ 1 ( ) r c co θ r' in( θ)θ' 11

38 Engineering Mechanic - Dynaic Chapter 1 v r' + rθ' v 1.33 *Prole The rod OA rotate counterclockwie with a contant angular velocity of θ'. Two pin-connected lider lock, located at B, ove freely on OA and the curved rod whoe hape i a liaçon decried y the equation r (c co(θ)). Deterine the acceleration of the lider lock at the intant θ θ 1. θ' 5 rad 100 c θ 1 10 deg θ θ 1 ( ) r c co θ r' in( θ)θ' r'' co( θ)θ' r'θ' a r'' rθ' + a 8.66 Prole The earchlight on the oat anchored a ditance d fro hore i turned on the autooile, which i traveling along the traight road at a contant peed v. Deterine the angular rate of rotation of the light when the autooile i r r 1 fro the oat. d 000 ft v r 1 80 ft 3000 ft 113

39 Engineering Mechanic - Dynaic Chapter 1 r r 1 θ ain d r θ deg θ' θ' v in( θ) r rad Prole 1-16 The earchlight on the oat anchored a ditance d fro hore i turned on the autooile, which i traveling along the traight road at peed v and acceleration a. Deterine the required angular acceleration θ'' of the light when the autooile i r r 1 fro the oat. d 000 ft v 80 ft a 15 ft r ft r r 1 θ ain d θ deg r θ' v in θ θ' rad r r' ft v co( θ) r'

40 Engineering Mechanic - Dynaic Chapter 1 θ'' a in( θ) r'θ' r θ'' rad Prole For a hort tie the ucket of the ackhoe trace the path of the cardioid r a(1 coθ). Deterine the agnitude of the velocity and acceleration of the ucket at θ θ 1 if the oo i rotating with an angular velocity θ' and an angular acceleration θ'' at the intant hown. a 5 ft θ' rad θ 1 10 deg θ'' 0. rad θ θ 1 r a( 1 co( θ) ) r' a in( θ)θ' r'' a in( θ)θ'' + a co( θ)θ' v r' + rθ' v 86.6 ft rθ'' r'θ' a r'' rθ' + ( + ) a 66 ft *Prole A car i traveling along the circular curve having a radiu r. At the intance hown, it angular rate of rotation i θ', which i decreaing at the rate θ''. Deterine the radial and tranvere coponent of the car' velocity and acceleration at thi intant. r 400 ft θ' 0.05 rad θ'' rad 115

41 Engineering Mechanic - Dynaic Chapter 1 v r rθ' v r vθ 0 a r rθ'' a r aθ rθ' aθ Prole The echani of a achine i contructed o that for a hort tie the roller at A follow the urface of the ca decried y the equation r a + coθ. If θ' and θ'' are given, deterine the agnitude of the roller velocity and acceleration at the intant θ θ 1. Neglect the ize of the roller. Alo deterine the velocity coponent v Ax and v Ay of the roller at thi intant. The rod to which the roller i attached reain vertical and can lide up or down along the guide while the guide ove horizontally to the left. θ' 0.5 rad θ 1 30 deg a 0.3 θ'' 0 rad 0. θ θ 1 r a + co θ r' r'' in( θ)θ' in( θ)θ'' co( θ)θ' v r' + rθ' v 0.4 rθ'' r'θ' a r'' rθ' + ( + ) a

42 Engineering Mechanic - Dynaic Chapter 1 v Ax v Ay rθ' in( θ) r' co θ + v Ax 0.16 rθ' co( θ) r' in θ + v Ay 0.18 Prole The roller coater i traveling down along the piral rap with a contant peed v. If the track decend a ditance h for every full revolution, deterine the agnitude of the roller coater acceleration a it ove along the track, r of radiu. Hint: For part of the olution, note that the tangent to the rap at any point i at an angle φ tan -1 (h/πr) fro the horizontal. Ue thi to deterine the velocity coponent v θ and v z which in turn are ued to deterine θ and z. v 6 h 10 r 5 h φ atan φ deg πr θ' v co φ a rθ' r a Prole A caeraan tanding at A i following the oveent of a race car, B, which i traveling around a curved track at contant peed v B. Deterine the angular rate at which the an ut turn in order to keep the caera directed on the car at the intant θ θ 1. v B 30 θ 1 30 deg a 0 0 θ θ 1 117

43 Engineering Mechanic - Dynaic Chapter 1 Gue Given r 1 r' 1 θ' 1 rad φ 0 deg φ' rad in( φ) rin θ r' in( θ) + rco( θ)θ' co( φ)φ' a + co( φ) rco θ r' co θ v B φ' rin θ θ' in( φ)φ' r r' θ' φ φ' Find( r, r', θ', φ, φ' ) r r' 15 φ 60 deg φ' 1.5 rad θ' 0.75 rad *Prole The pin follow the path decried y the equation r a + coθ. At the intant θ θ 1. the angular velocity and angular acceleration are θ' and θ''. Deterine the agnitude of the pin velocity and acceleration at thi intant. Neglect the ize of the pin. a θ 1 θ' θ'' 30 deg 0.7 rad 0.5 rad θ θ 1 r a + co( θ) r' in( θ)θ' r'' co( θ)θ' in( θ)θ'' 118

44 Engineering Mechanic - Dynaic Chapter 1 v r' + rθ' v 0.37 rθ'' r'θ' a r'' rθ' + ( + ) a 0.78 Prole For a hort tie the poition of the roller-coater car along it path i defined y the equation r r 0, θ at, and z coθ. Deterine the agnitude of the car velocity and acceleration when t t 1. r 0 5 a 0.3 rad 8 t 1 4 t t 1 r r 0 θ at z co θ θ' a z' in( θ)θ' z'' co( θ)θ' v ( rθ' ) + z' v 7.86 a z'' rθ' + a.65 Prole The all waher i liding down the cord OA. When it i at the idpoint, it peed i v and it acceleration i a'. Expre the velocity and acceleration of the waher at thi point in ter of it cylindrical coponent. v 00 a'

45 Engineering Mechanic - Dynaic Chapter 1 a c 700 v r v z v a + a + + c v r vθ 0 v c a + + c v z a r a r a co( α) a' a + a + + c a r aθ 0 v c a z a + + c a z Prole A doule collar C i pin-connected together uch that one collar lide over a fixed rod and the other lide over a rotating rod. If the geoetry of the fixed rod for a hort ditance can e defined y a lenicate, r (a co θ), deterine the collar radial and tranvere coponent of velocity and acceleration at the intant θ 0 a hown. Rod OA i rotating at a contant rate of θ'. a 4ft θ' 6 rad θ 0 deg r aco θ r a co( θ) rr' a in( θ)θ' r' a in( θ)θ' r 10

46 Engineering Mechanic - Dynaic Chapter 1 rr'' + r' a co( θ)θ' r'' v r r' v r 0 a co( θ)θ' r' r vθ rθ' vθ 1 ft a r r'' rθ' a r 16 aθ r'θ' aθ 0 ft ft *Prole 1-17 If the end of the cale at A i pulled down with peed v, deterine the peed at which lock B rie. v v A v L B + A 0 v B + v A v B v A v B 1 Prole If the end of the cale at A i pulled down with peed v, deterine the peed at which lock B rie. v 11

47 Engineering Mechanic - Dynaic Chapter 1 v A v L 1 A + C v A 0 v A + v C v C B L B C + 0 v B v C v C v B v B 0.5 Prole Deterine the contant peed at which the cale at A ut e drawn in y the otor in order to hoit the load at B a ditance d in a tie t. d 15 ft t 5 L 4 B + A 0 4v B + v A v A 4v B v A 4 v A 1 ft d t Prole Deterine the tie needed for the load at B to attain peed v, tarting fro ret, if the cale i drawn into the otor with acceleration a. v 8 1

48 Engineering Mechanic - Dynaic Chapter 1 a 0. v B v L 4 B + A 0 4v B + v A v A 1 v B 4 4 at 4v B t t 160 a *Prole If the hydraulic cylinder at H draw rod BC in y a ditance d, deterine how far the lider at A ove. d 8in Δ H d L A + H 0 Δ A + Δ H Δ A Δ H Δ A 16 in Prole The crate i eing lifted up the inclined plane uing the otor M and the rope and pulley arrangeent hown. Deterine the peed at which the cale ut e taken up y the otor in order to ove the crate up the plane with contant peed v. v 4 ft 13

49 Engineering Mechanic - Dynaic Chapter 1 v A v L A + A P 0 3v A v P v P 3v A v P 1 ft Prole Deterine the diplaceent of the lock at B if A i pulled down a ditance d. d 4ft Δ A d + B L 1 A + C L B C 0 Δ A + Δ C 0 Δ B Δ C Δ C Δ C Δ A Δ B Δ B ft Prole The hoit i ued to lift the load at D. If the end A of the chain i travelling downward at v A and the end B i travelling upward at v B, deterine the velocity of the load at D. v A 5 ft v B ft 14

50 Engineering Mechanic - Dynaic Chapter 1 L B + A + D 0 v B + v A + v D v B v A ft v D v D 1.5 Poitive ean down, Negative ean up *Prole The pulley arrangeent hown i deigned for hoiting aterial. If BC reain fixed while the plunger P i puhed downward with peed v, deterine the peed of the load at A. v 4 ft v P v L 6 P + A 0 6v P + v A v A 6v P v A 4 ft Prole If lock A i oving downward with peed v A while C i oving up at peed v C, deterine the peed of lock B. v A 4 ft 15

51 Engineering Mechanic - Dynaic Chapter 1 v C ft S A + S B + S C L Taking tie derivative: v A + v B + v C 0 v C + v A v B v B 1 ft Poitive ean down, negative ean up. Prole 1-18 If lock A i oving downward at peed v A while lock C i oving down at peed v C, deterine the relative velocity of lock B with repect to C. v A 6 ft v C 18 ft S A + S B + S C L Taking tie derivative v A + v B + v C 0 v A + v C ft v B v B 1 ft v BC v B v C v BC 30 Poitive ean down, negative ean up Prole The otor draw in the cale at C with a contant velocity v C. The otor draw in the cale at D with a contant acceleration of a D. If v D 0 when t 0, deterine (a) the tie needed for lock A to rie a ditance h, and () the relative velocity of lock A with repect to lock B when thi occur. 16

52 Engineering Mechanic - Dynaic Chapter 1 v C 4 a D 8 h 3 L 1 D + A 0 v D + v A 0 a D + a A L B + B C 0 v B v C 0 a B a C a A a D v A a A t t A h a A t h t 1.5 a A v A 1 a A t v B v C v AB v A v B v AB.90 *Prole If lock A of the pulley yte i oving downward with peed v A while lock C i oving up at v C deterine the peed of lock B. v A 4 ft v C ft 17

53 Engineering Mechanic - Dynaic Chapter 1 S A v A + S B + S C L v C v A + v B + v C 0 v B v B 0 Prole If the point A on the cale i oving upward at v A, deterine the peed of lock B. v A 14 + ( D E ) L 1 D A 0 v D v A v E + ( C E ) L D E 0 v D + v C v E C L 3 C D + + E 0 v C v D + v E Guee v C 1 v D 1 v E 1 Given 0 v D v A v E 0 v D + v C v E 0 v C v D + v E v C v D v E Find v C, v D, v E v C v D v E 10 6 v B v C v B Poitive ean down, Negative ean up 18

54 Engineering Mechanic - Dynaic Chapter 1 Prole The cylinder C i eing lifted uing the cale and pulley yte hown. If point A on the cale i eing drawn toward the dru with peed of v A, deterine the peed of the cylinder. v A L C + C A 0 3v C v A v A v C 3 v C Poitive ean down, negative ean up. Prole The cord i attached to the pin at C and pae over the two pulley at A and D. The pulley at A i attached to the ooth collar that travel along the vertical rod. Deterine the velocity and acceleration of the end of the cord at B if at the intant A the collar i oving upward at peed v, which i decreaing at rate a. a 3ft v A 5 ft 4ft a A ft L a + A + B A Guee v B 1 ft a B 1 ft Given 0 A v A a + A + v B 19

55 Engineering Mechanic - Dynaic Chapter 1 0 A a A v A + a A + A va a ( + A ) 3 + a B v B a B Find( v B, a B ) v B 8 ft ft a B 6.8 *Prole The cord of length L i attached to the pin at C and pae over the two pulley at A and D. The pulley at A i attached to the ooth collar that travel along the vertical rod. When B, the end of the cord at B i pulled downward with a velocity v B and i given an acceleration a B. Deterine the velocity and acceleration of the collar A at thi intant. L 16 ft a 3ft v B 4 ft 6ft a B 3 ft B Guee v A Given 1 ft a A 1 ft A 1ft L a + A + B 0 A v A a + A + v B 0 A a A v A + a A + A va a ( + A ) 3 + a B A v A a A ft ft Find( A, v A, a A ) A 4ft v A.50 a A

56 Engineering Mechanic - Dynaic Chapter 1 Prole The crate C i eing lifted y oving the roller at A downward with contant peed v A along the guide. Deterine the velocity and acceleration of the crate at the intant 1. When the roller i at B, the crate ret on the ground. Neglect the ize of the pulley in the calculation. Hint: Relate the coordinate x C and x A uing the prole geoetry, then take the firt and econd tie derivative. v A 1 1 d 4 e 4 x C e 1 L d + e Guee v C 1 a C 1 x A 1 Given L x C + x A + d x A v A 0 v C + x A + d x A v C a C x A va 0 a C x A + d 3 v A + x A + d Find( x A, v C, a C ) x A 3 v C 1. a C 0.51 Prole The girl at C tand near the edge of the pier and pull in the rope horizontally at contant peed v C. Deterine how fat the oat approache the pier at the intant the rope length AB i d. v C 6 ft h 8ft d 50 ft 131

57 Engineering Mechanic - Dynaic Chapter 1 x B d h L x C h x B v B + + x B 0 v C + h + x B v B h + x B ft v C v B Poitive ean to the right, negative to the left. x B Prole The an pull the oy up to the tree li C y walking ackward. If he tart fro ret when x A 0 and ove ackward with contant acceleration a A, deterine the peed of the oy at the intant y B y B1. Neglect the ize of the li. When x A 0, y B h o that A and B are coincident, i.e., the rope i h long. a A 0. y B1 4 h 8 y B y B1 Guee x A 1 v A 1 v B 1 Given h x A + h x A v A + y B 0 x A + h + v B v A a A x A x A v A v B Find( x A, v A, v B ) x A v A v B 1.41 Poitive ean down, negative ean up *Prole 1-19 Collar A and B are connected to the cord that pae over the all pulley at C. When A i located at D, B i a ditance d 1 to the left of D. If A ove at a contant peed v A, to the right, deterine the peed of B when A i ditance d to the right of D. 13

58 Engineering Mechanic - Dynaic Chapter 1 h 10 ft d 1 4 ft d 4ft v A ft L h + d 1 + h A d B + h L A + h B L A + h h B ft B v B B + h A v A A v A B + h ft A + h v B B A + h v B Poitive ean to the left, negative to the right. Prole If lock B i oving down with a velocity v B and ha an acceleration a B, deterine the velocity and acceleration of lock A in ter of the paraeter hown. L B + A + h A v A 0 v B + A + h v A v B A A + h A va 0 a B 3 ( A + h ) + v A + A a A A + h 133

59 Engineering Mechanic - Dynaic Chapter 1 A v A A h + v A a B A + h a A A + h a B a A A A A v B h A 3 Prole Vertical otion of the load i produced y oveent of the piton at A on the oo. Deterine the ditance the piton or pulley at C ut ove to the left in order to lift the load a ditance h. The cale i attached at B, pae over the pulley at C, then D, E, F, and again around E, and i attached at G. h ft Δ F h L C + F Δ C Δ F Δ C Δ F Δ C ft Prole The otion of the collar at A i controlled y a otor at B uch that when the collar i at A, it i oving upward at v A and lowing down at a A. Deterine the velocity and acceleration of the cale a it i drawn into the otor B at thi intant. d 4ft A 3ft v A ft a A 1 ft L A + d + B Guee v B 1 ft a B 1 ft 134

60 Engineering Mechanic - Dynaic Chapter 1 v B A v A A + d v A + A a A a B A + d + A va ( A + d ) 3 v B 1. ft ft a B 1.11 *Prole The roller at A i oving upward with a velocity v A and ha an acceleration a A at A. Deterine the velocity and acceleration of lock B at thi intant. A v A 4ft a A 4 ft 3 ft d 3ft l B + A + d A v A 0 v B + A + d v B A v A ft A + d v B.4 v A A a A A va ft a B + a B A + d A + d 3 Prole Two plane, A and B, are flying at the ae altitude. If their velocitie are v A and v B uch that the angle etween their traight-line coure i θ, deterine the velocity of plane B with repect to plane A. 135