Determine the gravitational attraction between two spheres which are just touching each other. Each sphere has a mass M and radius r.

Transcription

1 Problem 13-1 Determine the gravitational attraction between two phere which are jut touching each other. Each phere ha a ma M and radiu r. r 00 mm M 10 kg G m 3 kg nn N F GM ( r) F 41.7 nn Problem 13- By uing an inclined plane to retard the motion of a falling object, and thu make the obervation more accurate, Galileo wa able to determine experimentally that the ditance through which an object move in free fall i proportional to the quare of the time for travel. Show that thi i the cae, i.e., t by determining the time t B, t C, and t D needed for a block of ma m to lide from ret at A to point B, C, and D, repectively. Neglect the effect of friction. B C D m 4m 9m θ 0 deg g 9.81 m in( θ) a g a gin θ a m 1 at t B B t B 1.09 a 145

2 t C t D C a t C 1.54 D a t D.3 Problem 13-3 A bar B of ma M 1, originally at ret, i being towed over a erie of mall roller. Determine the force in the cable at time t if the motor M i drawing in the cable for a hort time at a rate v kt. How far doe the bar move in time t? Neglect the ma of the cable, pulley, and the roller. kn M N 300 kg t 5 k 0.4 m 3 v kt v 10 m a kt a 4 m T M 1 a T 1. kn d t kt dt d 16.7 m 0 *Problem 13-4 A crate having a ma M fall horizontally off the back of a truck which i traveling with peed v. Determine the coefficient of kinetic friction between the road and the crate if the crate lide a ditance d on the ground with no tumbling along the road before coming to ret. Aume that the initial peed of the crate along the road i v. 146

3 M d 60 kg 45 m v 80 km hr g 9.81 m N C μ k N C Mg 0 N C Mg Ma a μ k g v ad μ k gd v μ k μ k gd Problem 13-5 The crane lift a bin of ma M with an initial acceleration a. Determine the force in each of the upporting cable due to thi motion. M 700 kg b 3 kn 10 3 N a 3 m c 4 T c b + c Mg Ma b + c T M( a + g) T 5.60 kn c 147

4 Problem 13-6 The baggage truck A ha ma m t and i ued to pull the two car, each with ma m c. The tractive force on the truck i F. Determine the initial acceleration of the truck. hat i the acceleration of the truck if the coupling at C uddenly fail? The car wheel are free to roll. Neglect the ma of the wheel. m t 800 kg m c 300 kg F 480 N + Σ F x ma x ; F ( m t + m c )a a F a m m t + m c + Σ F x ma x ; F ( m t + m c )a Fail a Fail F a Fail m m t + m c Problem 13-7 The fuel aembly of ma M for a nuclear reactor i being lifted out from the core of the nuclear reactor uing the pulley ytem hown. It i hoited upward with a contant acceleration uch that 0 and v 0 when t 0 and 1 when t t 1. Determine the tenion in the cable at A during the motion. 148

5 Unit Ued: kn 10 3 N M 1 t kg.5 m 1.5 g 9.81 m a t a 1 a. m t 1 T Mg Ma T Ma ( + g) T kn *Problem 13-8 The crate of ma M i upended from the cable of a crane. Determine the force in the cable at time t if the crate i moving upward with (a) a contant velocity v 1 and (b) a peed of v bt + c. Unit Ued: kn 10 3 N M 00 kg t v 1 m b 0. m 3 c m ( a) a 0 m T a Mg Ma T a Mg ( + a) T a 1.96 kn 149

6 ( b) v bt + c a bt T b Mg ( + a) T b.1 kn Problem 13-9 The elevator E ha a ma M E, and the counterweight at A ha a ma M A. If the motor upplie a contant force F on the cable at B, determine the peed of the elevator at time t tarting from ret. Neglect the ma of the pulley and cable. Unit Ued: kn M E M A 10 3 N 500 kg 150 kg F 5kN t 3 Guee T 1kN a 1 m v 1 m Given T M A g M A a F + T M E g M E a v at T a v Find( T, a, v) T 1.11 kn a.41 m v 7.3 m 150

7 Problem The elevator E ha a ma M E and the counterweight at A ha a ma M A. If the elevator attain a peed v after it rie a ditance h, determine the contant force developed in the cable at B. Neglect the ma of the pulley and cable. Unit Ued: kn M E M A 10 3 N 500 kg 150 kg v 10 m h 40 m Guee T 1kN F 1kN a 1 m Given T M A g M A a F + T M E g M E a v ah F T a Find( F, T, a) a 1.50 m T 1.8 kn F 4.5 kn Problem The water-park ride conit of a led of weight which lide from ret down the incline and then into the pool. If the frictional reitance on the incline i F r1 and in the pool for a hort ditance i F r, determine how fat the led i traveling when. 151

8 800 lb F r1 30 lb F r 80 lb 5ft a 100 ft b g 100 ft 3. ft θ atan b a On the incline in( θ) F r1 F r1 a 1 a 1 g in θ a ft g v 1 a 1 a + b v 1 a 1 a + b v ft In the water F r g a gf r a a 3. ft v v 1 a v v 1 a v ft *Problem 13-1 A car of ma m i traveling at a low velocity v 0. If it i ubjected to the drag reitance of the wind, which i proportional to it velocity, i.e., F D kv determine the ditance and the time the car will travel before it velocity become 0.5 v 0. Aume no other frictional force act on the car. F D ma k v ma 15

9 Find time a d dt v k m v t k m 0 0.5v dt dv v v 0 m t k ln v 0 t 0.5v m 0 k ln( ) t m k d Find ditance a v v d x k m v x 0.5v 0 m k dx m dv x ( 0 v k 0.5v 0) x 0.5 mv 0 k 0 Problem Determine the normal force the crate A of ma M exert on the mooth cart if the cart i given an acceleration a down the plane. Alo, what i the acceleration of the crate? M 10 kg a m θ 30 deg N Mg M( a) in( θ) N M g ( a)in θ N 88.1 N a crate ( a)in θ a crate 1 m Problem Each of the two block ha a ma m. The coefficient of kinetic friction at all urface of contact i μ. If a horizontal force P move the bottom block, determine the acceleration of the bottom block in each cae. 153

10 (a) Block A: ΣF x ma x ; P 3μmg ma A a A P m 3μg ( b) S B + S A L a A a B Block A: ΣF x ma x ; P T 3μmg ma A Block B: ΣF x ma x ; μmg T ma B P Solving imultaenouly a A m μg Problem The driver attempt to tow the crate uing a rope that ha a tenile trength T max. If the crate i originally at ret and ha weight, determine the greatet acceleration it can have if the coefficient of tatic friction between the crate and the road i μ and the coefficient of kinetic friction i μ k. 154

11 T max 00 lb 500 lb μ 0.4 μ k 0.3 g 3. ft θ 30 deg Equilibrium : In order to lide the crate, the towing force mut overcome tatic friction. Initial guee F N 100 lb T 50 lb F Given T co( θ) μ F N 0 F N + T in( θ) N 0 Find( F N, T) T If T lb > T max 00 lb then the truck will not be able to pull the create without breaking the rope. If T lb < T max 00 lb then the truck will be able to pull the create without breaking the rope and we will now calculate the acceleration for thi cae. Initial guee F N 100 lb a 1 ft Require T T max Given T co( θ) μ k F N g a F F N + T in( θ) N 0 Find( F N, a) a a 3.46 ft *Problem An engine of ma M 1 i upended from a preader beam of ma M and hoited by a crane which give it an acceleration a when it ha a velocity v. Determine the force in chain AC and AD during the lift. 155

12 Unit Ued: Mg 10 3 kg kn 10 3 N M 1 M 3.5 Mg 500 kg a 4 m v m θ 60 deg Guee T 1N T' 1N Given T in( θ) ( M 1 + M )g ( M 1 + M )a T' M 1 g M 1 a T T' Find( T, T' ) T AC T AD T T' T AC T AD kn Problem The bullet of ma m i given a velocity due to ga preure caued by the burning of powder within the chamber of the gun. Auming thi preure create a force of F F 0 in(πt / t 0 ) on the bullet, determine the velocity of the bullet at any intant it i in the barrel. hat i the bullet maximum velocity? Alo, determine the poition of the bullet in the barrel a a function of time. 156

13 F 0 inπ t t 0 ma a dv dt F 0 πt in m t 0 t v 1 dv 0 0 F 0 πt in m t 0 dt F 0 t 0 v πm πt 1 co t 0 v max occur when co F 0 t 0 v max πm πt t 0 1, or t t 0 t F 0 t 0 1 d 1 co πt t 0 πm t d 0 0 F 0 t 0 πm t t 0 πt in π t 0 Problem The cylinder of weight at A i hoited uing the motor and the pulley ytem hown. If the peed of point B on the cable i increaed at a contant rate from zero to v B in time t, determine the tenion in the cable at B to caue the motion. 400 lb v B 10 ft t 5 A + B l 157

14 a B v B t a A a B T a A g T 1 a A T 06 lb g Problem A uitcae of weight lide from ret a ditance d down the mooth ramp. Determine the point where it trike the ground at C. How long doe it take to go from A to C? 40 lb θ 30 deg d h 0 ft g 3. ft 4ft in( θ) v B g a a gin( θ) ft a 16.1 ad v B ft t AB v B t AB a Guee t BC 1 R 1ft Given g t BC v B in( θ)t BC + h 0 R v B co( θ)t BC t BC R Find t BC, R t BC 0.41 R ft t AB + t BC

15 *Problem 13-0 A uitcae of weight lide from ret a ditance d down the rough ramp. The coefficient of kinetic friction along ramp AB i μ k. The uitcae ha an initial velocity down the ramp v 0. Determine the point where it trike the ground at C. How long doe it take to go from A to C? 40 lb d 0 ft h 4ft μ k 0. θ 30 deg v 0 g 10 ft 3. ft F N co( θ) 0 F N co( θ) μ k co( θ) in θ a g g( in( θ) μ k co( θ) ) a ft v B ad + v 0 v B.83 ft a v B v 0 t AB t AB 1.19 a Guee t BC 1 R 1ft Given t BC R g t BC v B in( θ)t BC + h 0 R v B co( θ)t BC Find t BC, R t BC 0.57 R ft t AB + t BC

16 Problem 13-1 The winding drum D i drawing in the cable at an accelerated rate a. Determine the cable tenion if the upended crate ha ma M. Unit Ued: kn 1000 N a 5 m M 800 kg g 9.81 m a L A + B a B a B.5 T Mg M a B T m Mg a B T 4.94 kn Problem 13- At a given intant block A of weight A i moving downward with a peed v 1. Determine it peed at the later time t. Block B ha weight B, and the coefficient of kinetic friction between it and the horizontal plane i μ k. Neglect the ma of the pulley and cord. A B 5lb v 1 4 ft 6lb t μ k 0.3 B + A L Guee a A 1 ft a B 1 ft T 1lb F N 1lb Given F N B 0 a B + a A 0 160

17 F N T a A a B B T μ k F N g a B T A Find F N, T, a A, a B F N T A g a A lb a A a B ft v v 1 + a A t v 44.6 ft Problem 13-3 A force F i applied to the cord. Determine how high the block A of weight rie in time t tarting from ret. Neglect the weight of the pulley and cord. F 15 lb t 30 lb g 3. ft 4F a g a g ( 4F ) a 3. ft d 1 at d 64.4 ft *Problem 13-4 At a given intant block A of weight A i moving downward with peed v A0. Determine it peed at a later time t. Block B ha a weight B and the coefficient of kinetic friction between it and the 161

18 horizontal plane i μ k. Neglect the ma of the pulley and cord. A 10 lb v A0 6 ft t B 4lb μ k 0. g 3. ft L B + A Guee a A 1 ft a B 1 ft T 1lb Given T μ k B B g a B T A 0 a B + a A A g a A T a A a B Find T, a A, a B T lb a A a B ft v A v A0 + a A t v A 6.8 ft Problem 13-5 A freight elevator, including it load, ha ma M e. It i prevented from rotating due to the track and wheel mounted along it ide. If the motor M develop a contant tenion T in it attached cable, determine the velocity of the elevator when it ha moved upward at a ditance d tarting from ret. Neglect the ma of the pulley and cable. 16

19 Unit Ued: kn M e 10 3 N 500 kg T d 1.50 kn 3m g 9.81 m 4T M e g M e a a 4 v T M e ad v 3.6 m g Problem 13-6 At the intant hown the block A of weight A i moving down the plane at v 0 while being attached to the block B of weight B. If the coefficient of kinetic friction i μ k, determine the acceleration of A and the ditance A lide before it top. Neglect the ma of the pulley and cable. A 100 lb B 50 lb v 0 5 ft μ k 0. a 3 b 4 θ atan a b Rope contraint 163

20 A + C L 1 D + D B L C + D + d d' Guee a A 1 ft a B 1 ft a C 1 ft a D 1 ft T A 1lb T B 1lb N A 1lb Given a A + a C 0 a D a B 0 a C + a D 0 T B B T A N A B g a B A in( θ) A + μ k N A g a A A co( θ) 0 T A T B 0 a A a B a C a D T A T B N A Find a A, a B, a C, a D, T A, T B, N A T A T B N A lb a A a B a C a D ft v 0 ft d A a A 1.87 a A d A 9.71 ft 164

21 Problem 13-7 The afe S ha weight and i upported by the rope and pulley arrangement hown. If the end of the rope i given to a boy B of weight b, determine hi acceleration if in the confuion he doen t let go of the rope. Neglect the ma of the pulley and rope. 00 lb b 90 lb g 3. ft L + b Initial guee: a b 1 ft a 1 ft T 1lb Given 0 a + a b T g a T b b g a b a b a T Find( a b, a, T) T lb a 1.15 ft ft a b.3 Negative mean up *Problem 13-8 The mine car of ma m car i hoited up the incline uing the cable and motor M. For a hort time, the force in the cable i F bt. If the car ha an initial velocity v 0 when t 0, determine it velocity when t t

22 m car 400 kg b 300 N v 0 t 1 m g 9.81 m c 8 d 15 bt m car g c c + d m car a a v 1 b m car t gc c + d b gct 1 m car 3 c + d + v 0 v m t 1 3 Problem 13-9 The mine car of ma m car i hoited up the incline uing the cable and motor M. For a hort time, the force in the cable i F bt. If the car ha an initial velocity v 0 when t 0, determine the ditance it move up the plane when t t

23 m car 400 kg b 300 N v 0 t 1 m g 9.81 m c 8 d 15 bt m car g c c + d m car a a b m car t gc c + d v 1 b m car t 3 3 gct + v 0 c + d 4 b t 1 gc t 1 m car 1 c + d + v 0 t m Problem The tanker ha a weight and i traveling forward at peed v 0 in till water when the engine are hut off. If the drag reitance of the water i proportional to the peed of the tanker at any intant and can be approximated by F D cv, determine the time needed for the tanker peed to become v 1. Given the initial velocity v 0 through what ditance mut the tanker travel before it top? lb c lb ft v 0 3 ft v ft av c g v 167

24 v 1 1 t dv t 43.1 av v 0 0 v d dv d ft av v 0 Problem The pring mechanim i ued a a hock aborber for railroad car. Determine the maximum compreion of pring HI if the fixed bumper R of a railroad car of ma M, rolling freely at peed v trike the plate P. Bar AB lide along the guide path CE and DF. The end of all pring are attached to their repective member and are originally untretched. Unit Ued: kn M 10 3 N Mg 10 3 kg 5Mg k 80 kn m v m k' 160 kn m The pring tretch or compre an equal amount x. Thu, k' + k d ( k' + k)x Ma a x v v M d x Gue d 1m Given 0 v d k' + k v dv x dx d Find( d) M 0 d 0.50 m *Problem 13-3 The collar C of ma m c i free to lide along the mooth haft AB. Determine the acceleration of collar C if (a) the haft i fixed from moving, (b) collar A, which i fixed to haft AB, move 168

25 downward at contant velocity along the vertical rod, and (c) collar A i ubjected to downward acceleration a A. In all cae, the collar move in the plane. m c kg a A m g 9.81 m θ 45 deg (a) m c g co( θ) m c a a a a g co θ a a m (b) m c g co( θ) m c a b a b g co θ a b m (c) m c ( g a A )co( θ) m c a crel a crel ( g a A )co( θ) in θ m a c a crel + a co( θ) A a c a c 7.08 m Problem The collar C of ma m c i free to lide along the mooth haft AB. Determine the acceleration of collar C if collar A i ubjected to an upward acceleration a. The collar move in the plane. m C kg a 4 m g 9.81 m θ 45 deg The collar accelerate along the rod and the rod accelerate upward. 169

26 m C a CA a m C g co θ co θ a CA ( g + a)co θ a C a CA in θ m a a CA co( θ) C + a.905 a C m Problem The boy ha weight and hang uniformly from the bar. Determine the force in each of hi arm at time t t 1 if the bar i moving upward with (a) a contant velocity v 0 and (b) a peed v bt 80 lb t 1 v 0 b 3 ft 4 ft 3 (a) T a 0 T a T a 40 lb (b) bt 1 T b T b g + g bt 1 T b lb Problem The block A of ma m A ret on the plate B of ma m B in the poition hown. Neglecting the ma of the rope and pulley, and uing the coefficient of kinetic friction indicated, determine the time needed for block A to lide a ditance ' on the plate when the ytem i releaed from ret. m A m B 10 kg 50 kg ' 0.5 m μ AB 0. μ BC 0.1 θ 30 deg g 9.81 m 170

27 A Guee + B L a A 1 m a B 1 m T 1N N A 1N N B 1N Given a A N A N B + a B 0 m A g co( θ) 0 m B g co( θ) 0 N A T μ AB N A T m A g in( θ) + μ AB N A μ BC N B m A a A + m B g in θ m B a B a A a B T N A N B Find a A, a B, T, N A, N B T N A N B a A a B N m a BA a B a A a BA m t ' t a BA *Problem Determine the acceleration of block A when the ytem i releaed from ret. The coefficient of kinetic friction and the weight of each block are indicated. Neglect the ma of the pulley and cord. A 80 lb B 0 lb θ 60 deg μ k

28 g 3. ft A + B L Guee a A T Given 1 ft a B 1 ft 1lb N A 1lb T N A A in( θ) A co( θ) 0 T B a A + a B 0 + μ k N A B g a B A g a A a A a B T N A Find( a A, a B, T, N A ) a A 4.8 ft Problem The conveyor belt i moving at peed v. If the coefficient of tatic friction between the conveyor and the package B of ma M i μ, determine the hortet time the belt can top o that the package doe not lide on the belt. v M 4 m 10 kg μ 0. g 9.81 m μ Mg Ma a μ g a 1.96 m t v t.039 a 17

29 Problem An electron of ma m i dicharged with an initial horizontal velocity of v 0. If it i ubjected to two field of force for which F x F 0 and F y 0.3F 0 where F 0 i contant, determine the equation of the path, and the peed of the electron at any time t. F 0 ma x 0.3F 0 ma y F 0 a x a y 0.3 F 0 m m F 0 v x m t + v 0 v y 0.3 F 0 m t F 0 t x + v 0 t y 0.3 F 0 t t m m 0 y m 3F 0 10 y Thu x 3 + v 0 0 y m 3F 0 v v x + v y v F 0 m t + v F 0 m t *Problem The conveyor belt deliver each crate of ma M to the ramp at A uch that the crate peed i v A directed down along the ramp. If the coefficient of kinetic friction between each crate and the ramp i μ k, determine the peed at which each crate lide off the ramp at B. Aume that no tipping occur. M v A 1 kg.5 m d 3m μ k 0.3 θ 30 deg g 9.81 m 173

30 N C Mgco( θ) 0 Mgin( θ) μ k N C N C Mgco θ N C μ k Ma a gin θ a.356 m M v B v A + ad v B m *Problem A parachutit having a ma m open hi parachute from an at-ret poition at a very high altitude. If the atmopheric drag reitance i F D kv, where k i a contant, determine hi velocity when he ha fallen for a time t. hat i hi velocity when he land on the ground? Thi velocity i referred to a the terminal velocity, which i found by letting the time of fall t. kv mg m a a g v t 0 k m v dv dt g 1 k m v dv t m gk atanh v k gm mg v k tanh gk m t hen t v m g k Problem Block B ret on a mooth urface. If the coefficient of tatic and kinetic friction between A and B are μ and μ k repectively, determine the acceleration of each block if omeone puhe horizontally on block A with a force of (a) F F a and (b) F F b. μ 0.4 F a 6lb μ k 0.3 F b 50 lb A 0 lb B 30 lb 174

31 g 3. ft Guee F A 1lb F max 1lb a A 1 ft a B 1 ft ( a) F F a Firt aume no lip Given F F A a A A g a B A F A g a B a B F max μ A F A F max a A a B Find F A, F max, a A, a B If F A lb< F max 8lbthen our a A 3.86 aumption i correct and a B 3.86 ft ( b) F F b Firt aume no lip Given F F A F A F max a A a B a A A g a B A F A g a B a B F max μ A Find F A, F max, a A, a B Since F A 30 lb> F max 8lbthen our aumption i not correct. Now we know that it lip Given F A μ k A F F A F A a A a B Find F A, a A, a B A g a B A F A g a A a B ft a B 175

32 Problem 13-4 Block A and B each have a ma M. Determine the larget horizontal force P which can be applied to B o that A will not move relative to B. All urface are mooth. Require a A a B a Block A: + ΣF y 0; Nco ( θ ) Mg 0 Block B: ΣF x Ma x ; Nin( θ) Ma a gtan( θ) ΣF x Ma x ; P Nin( θ) Ma P Mgtan θ Problem Block A and B each have ma m. Determine the larget horizontal force P which can be applied to B o that A will not lip up B. The coefficient of tatic friction between A and B i μ. Neglect any friction between B and C. Require a A a B a Block A: ΣF y 0; ΣF x ma x ; μ Nin( θ) Nco θ μ Nco( θ) mg 0 Nin θ + ma N co θ mg μ in( θ) Block B: a g in θ + co θ ΣF x ma x ; P μ Nco( θ) μ co( θ) μ in( θ) Nin( θ) ma 176

33 P p μ in( θ) + μ co( θ) co( θ) μ in( θ) μ mgco θ co θ mg in θ mg in θ + co θ μ co( θ) μ in( θ) *Problem Each of the three plate ha ma M. If the coefficient of tatic and kinetic friction at each urface of contact are μ and μ k repectively, determine the acceleration of each plate when the three horizontal force are applied. M 10 kg μ 0.3 μ k 0. F B F C F D 15 N 100 N 18 N g 9.81 m Cae 1: Aume that no lipping occur anywhere. F ABmax μ ( 3Mg) F BCmax μ ( Mg) F CDmax μ ( Mg) Guee F AB 1N F BC 1N F CD 1N Given F D + F CD 0 F C F CD F BC 0 F B F AB + F BC 0 F AB F BC F CD Find F AB, F BC, F CD F AB F BC F CD N F ABmax F BCmax F CDmax N If F AB F CD 67 N < F ABmax 88.9 N and F BC 8 N > F BCmax N and 18 N < F CDmax 9.43 N then nothing move and there i no acceleration. 177

34 Cae : If F AB 67 N < F ABmax 88.9 N and F BC 8 N > F BCmax N and F CD 18 N < F CDmax 9.43 N then lipping occur between B and C. e will aume that no lipping occur at the other urface. Set F BC μ k ( Mg) a B 0 a C a D a Guee F AB 1N F CD 1N a 1 m Given F D + F CD Ma F C F CD F BC Ma F B F AB + F BC 0 F AB F CD a Find F AB, F CD, a F AB F CD N 4.4 a.138 m a C a a D a If F AB 4.4 N < F ABmax 88.9 N and F CD N > F CDmax 9.43 N then we have the correct anwer and the acceleration are a B 0, a C.138 m, a D.138 m Cae 3: If F AB 4.4 N < F ABmax 88.9 N and F CD N > F CDmax 9.43 N then lipping occur between C and D a well a between B and C. e will aume that no lipping occur at the other urface. Set F BC μ k ( Mg) F CD μ k ( Mg) Guee F AB 1N a C 1 m a D 1 m Given F D + F CD Ma D F C F CD F BC Ma C F B F AB + F BC 0 F AB a C a D Find F AB, a C, a D F AB 4.4 N a C a D m 0.16 If F AB 4.4 N < F ABmax 88.9 N then we have the correct anwer and the acceleration are a B 0, a C m, a D 0.16 m There are other permutation of thi problem depending on the number that one chooe. Problem Crate B ha a ma m and i releaed from ret when it i on top of cart A, which ha a ma 3m. Determine the tenion in cord CD needed to hold the cart from moving while B i 178

35 liding down A. Neglect friction. g Block B: N B N B mgco( θ) 0 mgco( θ) Cart: T + N B in( θ) 0 T T mgin( θ)co( θ) mg in( θ) Problem The tractor i ued to lift load B of ma M with the rope of length h, and the boom, and pulley ytem. If the tractor i traveling to the right at contant peed v, determine the tenion in the rope when A d. hen A 0, B 0 Unit ued: kn M v d 10 3 N 150 kg 4 m h 1 m 5m g 9.81 m v A v A d Guee T 1kN B 1m a B 1 m v B 1 m Given h B + A + h h A v A v B + 0 A + h 179

36 T B v B a B v A a B + A + h A va 3 0 T Mg Ma B ( A + h ) Find T, B, v B, a B B 1m a B m T 1.69 kn v B m Problem The tractor i ued to lift load B of ma M with the rope of length h, and the boom, and pulley ytem. If the tractor i traveling to the right with an acceleration a and ha peed v at the intant A d, determine the tenion in the rope. hen A 0, B 0. Unit ued: kn 10 3 N d 5m h 1 m M 150 kg g 9.81 m v 4 m a 3 m a A a v A v A d Guee T 1kN B 1m a B 1 m v B 1 m 180

37 Given h B + A + h h A v A v B + A + h 0 v A + A a A a B + A + h T Mg Ma B A va 3 0 ( A + h ) T B v B a B Find T, B, v B, a B B 1m a B.03 m v B m T 1.80 kn *Problem Block B ha a ma m and i hoited uing the cord and pulley ytem hown. Determine the magnitude of force F a a function of the block vertical poition y o that when F i applied the block rie with a contant acceleration a B. Neglect the ma of the cord and pulley. mg F co θ ma B where co ( θ) F y y + d F m a B + g 4y y y + d mg ma B 4y + d Problem Block A ha ma m A and i attached to a pring having a tiffne k and untretched length l 0. If another block B, having ma m B i preed againt A o that the pring deform a ditance d, 181

38 determine the ditance both block lide on the mooth urface before they begin to eparate. hat i their velocity at thi intant? Block A: k( x d) N m A a A Block B: N m B a B Since a A a B a, a N kd ( x) m A + m B km B ( d x) m A + m B Separation occur when N 0 or x d v d kd ( x) v dv 0 m A + m B 0 v dx k d dd v m A + m B kd m A + m B Problem Block A ha a ma m A and i attached to a pring having a tiffne k and untretched length l 0. If another block B, having a ma m B i preed againt A o that the pring deform a ditance d, how that for eparation to occur it i neceary that d > μ k g(m A +m B )/k, where μ k i the coefficient of kinetic friction between the block and the ground. Alo, what i the ditance the block lide on the urface before they eparate? Block A: k( x d) N μ k m A g m A a A Block B: N μ k m B g m B a B 18

39 Since a A a B a a kd ( x) m A + m B μ k g N km B ( d x) m A + m B N 0, then x d for eparation. At the moment of eparation: v d v dv 0 0 kd ( x) dx μ k g m A + m B dx v kd μ k gm ( A + m B )d m A + m B Require v > 0, o that kd μ k g μ k gm ( A + m B )d > 0 d > ( m A + m B ) Q.E.D k Problem The block A ha ma m A and ret on the pan B, which ha ma m B Both are upported by a pring having a tiffne k that i attached to the bottom of the pan and to the ground. Determine the ditance d the pan hould be puhed down from the equilibrium poition and then releaed from ret o that eparation of the block will take place from the urface of the pan at the intant the pring become untretched. ( m A + m B )g For Equilibrium ky eq ( m A + m B )g 0 y eq k m A g + N m A a Block: 183

40 Block and Pan ( m A + m B )g + k( y eq + y) ( m A + m B )a m A + m B Thu, ( m A + m B ) g + k + y m A + m B k Set y -d, N 0 Thu d y eq g ( m A + m B )g k m A g + N m A *Problem 13-5 Determine the ma of the un, knowing that the ditance from the earth to the un i R. Hint: Ue Eq to repreent the force of gravity acting on the earth. R km G N m kg Σ F n ma n ; v v t G M e M R πr v m 1yr v M e M v R M kg R G Problem The helicopter of ma M i traveling at a contant peed v along the horizontal curved path while banking at angle θ. Determine the force acting normal to the blade, i.e., in the y' direction, and the radiu of curvature of the path. 184

41 Unit Ued: kn 10 3 N v 40 m M kg θ 30 deg g 9.81 m Guee F N 1kN ρ 1m Given F N co( θ) Mg 0 F N in θ M v ρ F N ρ Find F N, ρ F N kn ρ 8 m Problem The helicopter of ma M i traveling at a contant peed v along the horizontal curved path having a radiu of curvature ρ. Determine the force the blade exert on the frame and the bank angle θ. Unit Ued: kn 10 3 N v 33 m M kg ρ 300 m g 9.81 m 185

42 Guee F N 1kN θ 1 deg Given F N θ F N co( θ) Mg 0 F N in θ M v ρ Find F N, θ F N kn θ 0 deg Problem The plane i traveling at a contant peed v along the curve y bx + c. If the pilot ha weight, determine the normal and tangential component of the force the eat exert on the pilot when the plane i at it lowet point. b c 5000 ft 1 ft v 180 lb 800 ft x 0ft y bx + c y' bx y'' b ρ ( 1 + y' ) 3 y'' F n v F n v + F n 33 lb g ρ g ρ a t 0 F t g a t F t 0 186

43 *Problem The jet plane i traveling at a contant peed of v along the curve y bx + c. If the pilot ha a weight, determine the normal and tangential component of the force the eat exert on the pilot when y y 1. b lb ft c 5000 ft v 1000 ft g 3. ft y ft yx bx + c y' ( x) bx y'' ( x) b ρ ( x) ( 1 + y' ( x) ) 3 y'' ( x) Guee x 1 1ft F n 1lb θ 1 deg F t 1lb Given y 1 yx ( 1 ) tan( θ) y' ( x 1 ) F n co( θ) v F g ρ ( x 1 ) t in( θ) 0 x 1 θ F n F t Find x 1, θ, F n, F t x ft θ 3.3 deg F n F t 87.1 lb

44 Problem The wrecking ball of ma M i upended from the crane by a cable having a negligible ma. If the ball ha peed v at the intant it i at it lowet point θ, determine the tenion in the cable at thi intant. Alo, determine the angle θ to which the ball wing before it top. Unit Ued: kn 10 3 N M 600 kg v r 8 m 1 m g 9.81 m At the lowet point T Mg M v T Mg + M v T kn r r At ome arbitrary angle M g in( θ) Ma t a t g in( θ) v r 0 θ v dv rgin( θ) dθ v 0 dv dθ v v rg( co( θ) 1) θ aco1 θ 43.3 deg rg Problem Prove that if the block i releaed from ret at point B of a mooth path of arbitrary hape, the peed it attain when it reache point A i equal to the peed it attain when it fall freely through a ditance h; i.e., v gh. 188

45 ΣF t ma t ; ( mg)in( θ) ma t a t g in θ νdν a t d gin( θ)d However dy d in(θ) dy d in( θ) v h v dv 0 0 g dy v gh v gh Q.E.D Problem The led and rider have a total ma M and tart from ret at A(b, 0). If the led decend the mooth lope, which may be approximated by a parabola, determine the normal force that the ground exert on the led at the intant it arrive at point B. Neglect the ize of the led and rider. Hint: Ue the reult of Prob Unit Ued: kn 10 3 N a M v m b 10 m c 5m 80 kg g 9.81 m gc yx c x b c y' ( x) ρ ( x) x c b y'' ( x) ( 1 + y' ( x) ) 3 y'' ( x) c b 189

46 N b Mg M v ρ v N b Mg+ M N b 1.57 kn ρ ( 0m) *Problem The led and rider have a total ma M and tart from ret at A(b, 0). If the led decend the mooth lope which may be approximated by a parabola, determine the normal force that the ground exert on the led at the intant it arrive at point C. Neglect the ize of the led and rider. Hint: Ue the reult of Prob Unit Ued: kn 10 3 N a M m b 10 m c 5m 80 kg g 9.81 m yx c x b c y' ( x) c b y'' ( x) x c b ρ ( x) ( 1 + y' ( x) ) 3 y'' ( x) v g( y( a) ) θ atan( y' ( a) ) N C Mgco( θ) M v v N C M gco( θ) + N C 1.48 kn ρ ρ ( a) Problem At the intant θ θ 1 the boy center of ma G ha a downward peed v G. Determine the rate of increae in hi peed and the tenion in each of the two upporting cord of the wing at thi 190

47 intant.the boy ha a weight. Neglect hi ize and the ma of the eat and cord. θ 1 60 lb 60 deg l 10 ft v G 15 ft g 3. ft co( θ 1 ) a t g co( θ 1 ) g a t a t 16.1 ft T in( θ 1 ) g v l T 1 v G + in( θ 1 ) T 46.9 lb g l Problem 13-6 At the intant θ θ 1 the boy center of ma G i momentarily at ret. Determine hi peed and the tenion in each of the two upporting cord of the wing when θ θ. The boy ha a weight. Neglect hi ize and the ma of the eat and cord. 60 lb g 3. ft θ 1 60 deg θ 90 deg l 10 ft 191

48 co( θ) a t a t g co( θ) g v v θ gl θ ft co( θ) dθ T in( θ ) g v l T in ( θ v ) + T 38.0 lb gl Problem If the cret of the hill ha a radiu of curvature ρ, determine the maximum contant peed at which the car can travel over it without leaving the urface of the road. Neglect the ize of the car in the calculation. The car ha weight. ρ 00 ft 3500 lb g m Limiting cae i N 0. ΣF n ma n ; v v gρ v 80.5 ft g ρ *Problem The airplane, traveling at contant peed v i executing a horizontal turn. If the plane i banked at angle θ when the pilot experience only a normal force on the eat of the plane, determine the radiu of curvature ρ of the turn. Alo, what i the normal force of the eat on the pilot if he ha ma M? Unit Ued: kn 10 3 N 19

49 v 50 m θ 15 deg M 70 kg g m + ΣF b ma b ; N p in( θ) Mg 0 N p M g N p.654 kn in θ ΣF n ma n ; N P co( θ) M v v ρ M ρ 68.3 m ρ N p co θ Problem The man ha weight and lie againt the cuhion for which the coefficient of tatic friction i μ. Determine the reultant normal and frictional force the cuhion exert on him if, due to rotation about the z axi, he ha contant peed v. Neglect the ize of the man. 150 lb μ 0.5 v 0 ft θ 60 deg d 8ft Aume no lipping occur Guee F N 1lb F 1lb Given F N in θ + F co( θ) v F N co θ g d + F in( θ) 0 F N F Find( F N, F) F N F lb F max μ F N F max lb Since F lb < F max lb then our aumption i correct and there i no lipping. 193

50 Problem The man ha weight and lie againt the cuhion for which the coefficient of tatic friction i μ. If he rotate about the z axi with a contant peed v, determine the mallet angle θ of the cuhion at which he will begin to lip off. 150 lb μ 0.5 v 30 ft d 8ft Aume verge of lipping Guee F N 1lb θ 0 deg Given F N θ F N in θ μ F N co( θ) v F N co θ g d Find F N, θ F N lb θ deg μ F N in( θ) 0 Problem Determine the contant peed of the paenger on the amuement-park ride if it i oberved that the upporting cable are directed at angle q from the vertical. Each chair including it paenger ha a ma m c. Alo, what are the component of force in the n, t, and b direction which the chair exert on a paenger of ma m p during the motion? θ 30 deg d 4m m c m p 80 kg b 6m 50kg g 9.81 m The initial guee: T 100 N v 10 m 194

51 Given v T in( θ) m c d + T co( θ) m c g 0 T v bin θ Find( T, v) T N v 6.30 m m p v ΣF n ma n ; F n d + bin( θ) F n 83 N ΣF t ma t ; F t 0N F t 0 ΣF b ma b ; F b m p g 0 F b m p g F b 491 N *Problem The nowmobile of ma M with paenger i traveling down the hill at a contant peed v. Determine the reultant normal force and the reultant frictional force exerted on the track at the intant it reache point A. Neglect the ize of the nowmobile. Unit Ued: kn 10 3 N M 00 kg v 6 m a b 5m 10 m g 9.81 m yx 3 x a b y' ( x) 3 x a y'' ( x) 6 b 3 ρ ( x) a b 3 x ( 1 + y' ( x) ) 3 y'' ( x) 195

52 θ atan( y' ( b) ) Guee N S 1N F 1N Given N S Mgco( θ) F M v ρ ( b) Mgin( θ) 0 N S F Find( N S, F) N S F kn Problem The nowmobile of ma M with paenger i traveling down the hill uch that when it i at point A, it i traveling at peed v and increaing it peed at v'. Determine the reultant normal force and the reultant frictional force exerted on the track at thi intant. Neglect the ize of the nowmobile. Unit Ued: kn M 10 3 N 00 kg a 5m v 4 m b 10 m g 9.81 m v' m 3 x yx a y' ( x) 3 b x a y'' ( x) 6 b 3 ρ ( x) θ atan( y' ( b) ) a b 3 x ( 1 + y' ( x) ) 3 y'' ( x) Guee N S 1N F 1N Given N S Mgco( θ) M v ρ ( b) 196

53 F Mgin( θ) Mv' N S F Find( N S, F) N S F kn Problem A collar having a ma M and negligible ize lide over the urface of a horizontal circular rod for which the coefficient of kinetic friction i μ k. If the collar i given a peed v 1 and then releaed at θ 0 deg, determine how far, d, it lide on the rod before coming to ret. M 0.75 kg r 100 mm μ k 0.3 g 9.81 m v 1 4 m N Cz N Cn Mg 0 M v r N C N Cz + N Cn F C μ k N C M a t a t ( v) μ k g v 4 + r 0 v d dv d m a t ( v) v 1 Problem The roller coater car and paenger have a total weight and tarting from ret at A travel down the track that ha the hape hown. Determine the normal force of the track on the car when the car i at point B, it ha a velocity of v. Neglect friction and the ize of the car and paenger. 600 lb 197

54 v 15 ft a b 0 ft 40 ft yx b co πx y' ( x) a y'' ( x) d dx y' ( x) ρ ( x) d dx yx ( 1 + y' ( x) ) 3 y'' ( x) At B θ atan( y' ( a) ) F N co( θ) v g ρ F N v co( θ) + F N 18.0 lb g ρ ( a) *Problem 13-7 The mooth block B, having ma M, i attached to the vertex A of the right circular cone uing a light cord. The cone i rotating at a contant angular rate about the z axi uch that the block attain peed v. At thi peed, determine the tenion in the cord and the reaction which the cone exert on the block. Neglect the ize of the block. 198

55 M 0. kg v 0.5 m a 300 mm b 400 mm c 00 mm g 9.81 m Guee T 1N N B 1N Given T N B Set θ atan a θ deg b c ρ a + b a ρ 10 mm T in( θ) N B co( θ) M v T co( θ) + N B in( θ) Mg 0 ρ Find( T, N B ) T N B 1.8 N Problem The pendulum bob B of ma M i releaed from ret when θ 0. Determine the initial tenion in the cord and alo at the intant the bob reache point D, θ θ 1. Neglect the ize of the bob. M 5kg θ 1 45 deg L m g 9.81 m Initially, v 0 o a n 0 T 0 At D we have Mgco( θ 1 ) Ma t a t g co( θ 1 ) a t m T D Mgin( θ 1 ) Mv L Now find the velocity v Gue v 1 m 199

56 Given v θ 1 v dv 0 0 g co( θ)l dθ v Find( v) v 5.68 m M v T D Mgin θ 1 + T D N L Problem A ball having a ma M and negligible ize move within a mooth vertical circular lot. If it i releaed from ret at θ 1, determine the force of the lot on the ball when the ball arrive at point A and B. M kg θ 90 deg θ 1 10 deg r 0.8 m g 9.81 m Mgin θ a t g in θ Ma t At A θ A 90 deg v A N A N A g θ A θ 1 Mgco( θ A ) in( θ)r dθ v A M r M v A Mgco θ A N A 38.6 r N At B θ B 180 deg θ 1 v B N B N B g θ B θ 1 Mgco( θ B ) in( θ)r M dθ v B r M v B Mgco θ B N B 96.6 r N 00

57 Problem The rotational peed of the dik i controlled by a mooth contact arm AB of ma M which i pring-mounted on the dik.hen the dik i at ret, the center of ma G of the arm i located ditance d from the center O, and the preet compreion in the pring i a. If the initial gap between B and the contact at C i b, determine the (controlling) peed v G of the arm ma center, G, which will cloe the gap. The dik rotate in the horizontal plane. The pring ha a tiffne k and it end are attached to the contact arm at D and to the dik at E. M 30 gm a 0 mm b 10 mm d 150 mm k 50 N m F ka ( + b) F 1.5 N v G a n d + b F M v G d + b 1 v G M Mk( a+ b) ( d + b) v G.83 m *Problem The pool S of ma M fit looely on the inclined rod for which the coefficient of tatic friction i μ. If the pool i located a ditance d from A, determine the maximum contant peed the pool can have o that it doe not lip up the rod. M kg e 3 μ 0. f 4 d 0.5 m g 9.81 m ρ d f e + f Guee N 1N v 1 m 01

58 Given N e e + f f μ N e + f M v ρ N v N f e + f μ N e + Mg e + f 0 Find N, v N 1.36 N v m Problem The box of ma M ha a peed v 0 when it i at A on the mooth ramp. If the urface i in the hape of a parabola, determine the normal force on the box at the intant x x 1. Alo, what i the rate of increae in it peed at thi intant? M 35 kg a 4m v 0 m b m x 1 3m yx a bx y' ( x) bx y'' ( x) b ρ ( x) ( 1 + y' ( x) ) 3 y'' ( x) θ ( x) atan( y' ( x) ) Find the velocity ( ) v 1 v 0 + g y0m yx 1 v m Guee F N 1N v' 1 m Given F N Mgco( θ ( x 1 )) M v 1 M g in θ x ρ ( x 1 ) 1 ( ) Mv' 0

59 F N v' Find F N, v' F N N v' 5.44 m Problem The man ha ma M and it a ditance d from the center of the rotating platform. Due to the rotation hi peed i increaed from ret by the rate v'. If the coefficient of tatic friction between hi clothe and the platform i μ, determine the time required to caue him to lip. M 80 kg μ 0.3 d 3m D 10 m v' 0.4 m g 9.81 m Gue t 1 Given μ Mg ( Mv' ) M ( v' t) + d t Find() t t Problem The collar A, having a ma M, i attached to a pring having a tiffne k. hen rod BC rotate about the vertical axi, the collar lide outward along the mooth rod DE. If the pring i untretched when x 0, determine the contant peed of the collar in order that x x 1. Alo, what i the normal force of the rod on the collar? Neglect the ize of the collar. M 0.75 kg k 00 N m x mm 03

60 Guee g 9.81 m N b 1N N t 1N v 1 m Given N b Mg 0 N t 0 kx 1 M v x 1 N b N t v Find N b, N t, v N b N t 7.36 N 0 N b N t 7.36 N v m *Problem The block ha weight and it i free to move along the mooth lot in the rotating dik. The pring ha tiffne k and an untretched length δ. Determine the force of the pring on the block and the tangential component of force which the lot exert on the ide of the block, when the block i at ret with repect to the dik and i traveling with contant peed v. k δ lb.5 lb ft 1.5 ft v 1 ft v ΣF n ma n ; F k( ρ δ) g ρ Chooing the poitive root, 1 ρ kg kgδ + ( k g δ + 4 kgv ) ρ.617 ft F k ρ δ F lb ΣF t ma t ; ΣF t ma t ; F t 0 Problem If the bicycle and rider have total weight, determine the reultant normal force acting on the 04

61 bicycle when it i at point A while it i freely coating at peed v A. Alo, compute the increae in the bicyclit peed at thi point. Neglect the reitance due to the wind and the ize of the bicycle and rider. 180 lb d 5ft v A 6 ft h 0 ft g 3. ft yx h co π x h y' ( x) d dx yx y'' ( x) d dx y' ( x) At A x d θ atan( y' ( x) ) ρ ( 1 + y' ( x) ) 3 y'' ( x) Guee F N 1lb v' 1 ft Given F N co( θ) F N v' v A g ρ in( θ) g v' Find( F N, v' ) F N lb v' 9.36 ft Problem 13-8 The package of weight ride on the urface of the conveyor belt. If the belt tart from ret and increae to a contant peed v 1 in time t 1, determine the maximum angle θ o that none of the package lip on the inclined urface AB of the belt. The coefficient of tatic friction between the belt and a package i μ. At what angle φ do the package firt begin to lip off the urface of the belt after the belt i moving at it contant peed of v 1? Neglect the ize of the package. 05

62 v 1 5lb t 1 r 6in ft μ 0.3 Guee a v 1 t 1 N 1 1lb N 1lb θ 1 deg φ 1 deg Given N 1 N co( θ) 0 N 1 N θ φ co( φ) μ N 1 in θ v 1 μ N in( φ) 0 g r Find N 1, N, θ, φ N 1 N 4.83 lb g a θ deg φ 1.61 Problem A particle having ma M move along a path defined by the equation r a + bt, θ ct + d and z e + ft 3. Determine the r, θ, and z component of force which the path exert on the particle when t t 1. M 1.5 kg a 4m b 3 m c 1 rad d rad e 6m f 1 m 3 t 1 g 9.81 m t t 1 r a + bt r' b r'' 0 m θ ct + d θ' ct θ'' c z e + ft 3 z' 3 ft z'' 6 ft 06

63 F r M r'' rθ' F r 40 N Fθ Mrθ'' + r'θ' Fθ 66.0 N F z M z'' + M g F z 3.85 N *Problem The path of motion of a particle of weight in the horizontal plane i decribed in term of polar coordinate a r at + b and θ ct + dt. Determine the magnitude of the unbalanced force acting on the particle when t t 1. 5lb a ft b 1ft c 0.5 rad d 1 rad t 1 g 3. ft t t 1 r at + b r' a r'' 0 ft θ ct + dt θ' ct + d θ'' c a r r'' rθ' a r 5 ft aθ rθ'' + r'θ' aθ 9 ft F a r + aθ F lb g Problem The pring-held follower AB ha weight and move back and forth a it end roll on the contoured urface of the cam, where the radiu i r and z ain(θ). If the cam i rotating at a contant rate θ', determine the force at the end A of the follower when θ θ 1. In thi poition the pring i compreed δ 1. Neglect friction at the bearing C. 07

64 r 0.75 lb δ ft 0. ft θ 1 45 deg a 0.1 ft k 1 lb ft θ' 6 rad g 9.81 m θ θ 1 z ( a)in θ z' ( a) co( θ)θ' z'' 4( a) in( θ)θ' F a kδ 1 z'' g z'' F a kδ 1 + F a lb g Problem The pring-held follower AB ha weight and move back and forth a it end roll on the contoured urface of the cam, where the radiu i r and z a in(θ). If the cam i rotating at a contant rate of θ', determine the maximum and minimum force the follower exert on the cam if the pring i compreed δ 1 when θ 45 o lb r a 0. ft 0.1 ft θ' 6 rad δ 1 0. ft k 1 lb ft g 3. ft hen θ 45 deg z ( a)coθ z 0m So in other poition the pring i compree a ditance δ 1 + z z ( a)in( θ) z' ( a) co( θ)θ' z'' 4( a) in( θ)θ' 08

65 F a k δ 1 + z z'' F a k δ 1 + ( a)in( θ) g g 4( a) in( θ)θ' The maximum value occur when in(θ) -1 and the minimum occur when in(θ) 1 F amin k( δ 1 a) + 4aθ' F amin lb g F amax k( δ 1 + a) 4aθ' F amax 3.65 lb g Problem The pool of ma M lide along the rotating rod. At the intant hown, the angular rate of rotation of the rod i θ', which i increaing at θ''. At thi ame intant, the pool i moving outward along the rod at r' which i increaing at r'' at r. Determine the radial frictional force and the normal force of the rod on the pool at thi intant. M θ' 4kg r 0.5 m 6 rad r' 3 m θ'' rad r'' 1 m g 9.81 m a r r'' rθ' aθ rθ'' + r'θ' F r F z Ma r Fθ Maθ Mg F r 68.0 N Fθ + F z N *Problem The boy of ma M i liding down the piral lide at a contant peed uch that hi poition, meaured from the top of the chute, ha component r r 0, θ bt and z ct. Determine the component of force F r, F θ and F z which the lide exert on him at the intant t t 1. Neglect 09

66 the ize of the boy. M r 0 b 40 kg 1.5 m 0.7 rad m c 0.5 t 1 g 9.81 m r r 0 r' 0 m r'' 0 m θ bt θ' b θ'' 0 rad z ct z' c z'' 0 m F r M r'' rθ' F r 9.4 Fθ Mrθ'' + r'θ' Fθ 0 N F z Mg M z'' F z M( g + z'' ) F z 39 N Problem The girl ha a ma M. She i eated on the hore of the merry-go-round which undergoe contant rotational motion θ'. If the path of the hore i defined by r r 0, z b in(θ), determine the maximum and minimum force F z the hore exert on her during the motion. 10

67 M θ' r 0 50 kg 1.5 rad 4m b z z' z'' 0.5 m bin( θ) b co( θ)θ' b in( θ)θ' F z Mg M z'' F z Mg bin( θ)θ' F zmax Mg+ bθ' F zmax 547 N F zmin Mg bθ' F zmin 434 N Problem The particle of weight i guided along the circular path uing the lotted arm guide. If the arm ha angular velocity θ' and angular acceleration θ'' at the intant θ θ 1, determine the force of the guide on the particle. Motion occur in the horizontal plane. θ 1 30 deg 0.5 lb a 0.5 ft θ' 4 rad b 0.5 ft θ'' 8 rad g 3. ft θ θ 1 ( a)in θ φ ain a b in( θ) φ 30 deg b in( φ) 11

68 ( a)co( θ)θ' bco( φ)φ' φ' ( a)co( θ)θ'' ( a)in( θ)θ' ( a)co θ b co( φ) θ' φ' 4 rad b co( φ)φ'' b in( φ)φ' φ'' ( a)co( θ)θ'' ( a)in( θ)θ' + b in( φ)φ' φ'' 8 rad b co( φ) r ( a)co( θ) + b co( φ) r' ( a) in( θ)θ' bin( φ)φ' r'' ( a) in( θ)θ'' ( a)co( θ)θ' F N co φ F M r'' rθ' F N in( φ) b in( φ)φ'' b co( φ)φ' r'' rθ' F N F g co( φ) N lb rθ'' ( + r'θ' ) F F N in( φ) g + g ( rθ'' + r'θ' ) F lb Problem The particle ha ma M and i confined to move along the mooth horizontal lot due to the rotation of the arm OA. Determine the force of the rod on the particle and the normal force of the lot on the particle when θ θ 1. The rod i rotating with a contant angular velocity θ'. Aume the particle contact only one ide of the lot at any intant. M θ 1 θ' θ'' 0.5 kg 30 deg rad 0 rad h 0.5 m g 9.81 m θ θ 1 h rco θ r h r m co( θ) 1

69 0 r' co( θ) rin( θ)θ' r' 0 r'' co θ r'' r' in θ θ' rco( θ)θ' rin θ co( θ) θ' r' m rin( θ)θ'' r'θ' tan θ + + rtan( θ)θ'' r'' m rθ' ( F N Mg)co( θ) M r'' rθ' r'' rθ' F N Mg+ M F co( θ) N N F + ( F N Mg)in( θ) M( rθ'' + r'θ' ) F ( F N Mg)in( θ) + Mrθ'' ( + r'θ' ) F N *Problem 13-9 The particle ha ma M and i confined to move along the mooth horizontal lot due to the rotation of the arm OA. Determine the force of the rod on the particle and the normal force of the lot on the particle when θ θ 1. The rod i rotating with angular velocity θ' and angular acceleration θ''. Aume the particle contact only one ide of the lot at any intant. M 0.5 kg θ 1 30 deg θ' rad h 0.5 m θ'' 3 rad g 9.81 m θ θ 1 h rco θ r 0 r' co( θ) rin( θ)θ' r' 0 r'' co θ r' in θ θ' h r m co( θ) θ' rin θ r' m co( θ) rco( θ)θ' rin( θ)θ'' 13

70 r'' r'θ' tan θ + + rtan( θ)θ'' r'' m rθ' ( F N Mg)co( θ) M r'' rθ' r'' rθ' F N Mg+ M F N N co θ F + ( F N Mg)in( θ) M rθ'' + r'θ' F ( F N Mg)in( θ) + Mrθ'' ( + r'θ' ) F.93 N Problem A mooth can C, having a ma M, i lifted from a feed at A to a ramp at B by a rotating rod. If the rod maintain a contant angular velocity of θ', determine the force which the rod exert on the can at the intant θ θ 1. Neglect the effect of friction in the calculation and the ize of the can o that r b coθ. The ramp from A to B i circular, having a radiu of b. M 3kg θ 1 30 deg θ' 0.5 rad b 600 mm θ θ 1 b co( θ) r r' b in( θ)θ' r'' b co( θ)θ' Guee F N 1N F 1N Given F N F Mgin( θ) F N in( θ) F N co θ M r'' rθ' F + Mgco θ M( r'θ' ) Find F N, F F N N F N Problem The collar of weight lide along the mooth horizontal piral rod r bθ, where θ i in 14

71 radian. If it angular rate of rotation θ' i contant, determine the tangential force P needed to caue the motion and the normal force that the rod exert on the pool at the intant θ θ 1. lb θ 1 θ' 90 deg 4 rad b ft θ θ 1 r bθ r' bθ' ψ atan rθ' r' Guee N B 1lb P 1lb Given N B in ψ P in ψ + + P co( ψ) N B co( ψ) g rθ' g r'θ' N B P Find N B, P ψ 57.5 deg N B lb P lb Problem The collar of weight lide along the mooth vertical piral rod r bθ, where θ i in radian. If it angular rate of rotation θ' i contant, determine the tangential force P needed to caue the motion and the normal force that the rod exert on the pool at the intant θ θ 1. 15

72 lb θ 1 90 deg θ' b 4 rad ft θ θ 1 r bθ r' bθ' ψ atan rθ' r' Guee N B 1lb P 1lb Given N B in ψ P in ψ + N B P P co( ψ) + N B co( ψ) g g r'θ' rθ' Find N B, P ψ 57.5 deg N B lb P.751 lb *Problem The forked rod i ued to move the mooth particle of weight around the horizontal path in the hape of a limacon r a + bcoθ. If θ ct, determine the force which the rod exert on the particle at the intant t t 1. The fork and path contact the particle on only one ide. a b lb ft 1ft c 0.5 rad 16

73 t 1 1 g 3. ft t t 1 θ ct θ' ct θ'' c Find the angel ψ uing rectangular coordinate. The path i tangent to the velocity therefore. rco( θ) b co( θ) x ( a)co θ + x' ( a) in θ y rin( θ) 1 ( a)in( θ) + b in ( θ ) y' ( a)co θ + ψ θ atan y' ψ deg x' Now do the dynamic uing polar coordinate b co θ b co ( θ ) θ' in( θ) θ' r a + bco( θ) r' b in( θ)θ' r'' b co( θ)θ' b in( θ)θ'' Guee F 1lb F N 1lb Given F F N co( ψ) F F N g ( rθ'' + r'θ' ) F N in( ψ) Find F, F N F N 0.67 lb F lb g r'' rθ' Problem The mooth particle ha ma M. It i attached to an elatic cord extending from O to P and due to the lotted arm guide move along the horizontal circular path r b inθ. If the cord ha tiffne k and untretched length δ determine the force of the guide on the particle when θ θ 1. The guide ha a contant angular velocity θ'. M b k 80 gm 0.8 m 30 N m δ 0.5 m θ 1 60 deg 17

74 θ' 5 rad θ'' 0 rad θ θ 1 r bin( θ) r' b co( θ)θ' r'' b co( θ)θ'' b in( θ)θ' Guee N P 1N F 1N Given kr ( δ) N P in θ M r'' rθ' N P co( θ) F Mrθ'' + r'θ' F N P Find F, N P N P 1.14 N F 7.67 N Problem The mooth particle ha ma M. It i attached to an elatic cord extending from O to P and due to the lotted arm guide move along the horizontal circular path r b inθ. If the cord ha tiffne k and untretched length δ determine the force of the guide on the particle when θ θ 1. The guide ha angular velocity θ' and angular acceleration θ'' at thi intant. M b k 80 gm 0.8 m 30 N m δ 0.5 m θ 1 θ' 60 deg 5 rad θ'' rad θ θ 1 r bin( θ) r' b co( θ)θ' r'' b co( θ)θ'' b in( θ)θ' Guee N P 1N F 1N 18

75 Given kr ( δ) N P in θ M r'' rθ' N P co( θ) F Mrθ'' + r'θ' F N P Find F, N P N P 1.14 N F N Problem Determine the normal and frictional driving force that the partial piral track exert on the motorcycle of ma M at the intant θ,θ', and θ''. Neglect the ize of the motorcycle. Unit Ued: kn 10 3 N M 00 kg b 5m θ 5 π rad 3 θ' 0.4 rad θ'' 0.8 rad r bθ r' bθ' r'' bθ'' ψ atan rθ' ψ deg r' Guee F N 1N F 1N Given F N in ψ F co( ψ) + Mgin θ M r'' rθ' F in( ψ) F N co ψ + Mgco θ Mrθ'' + r'θ' F N F Find( F N, F) F N F kn 19

76 *Problem Uing a forked rod, a mooth cylinder C having a ma M i forced to move along the vertical lotted path r aθ. If the angular poition of the arm i θ bt, determine the force of the rod on the cylinder and the normal force of the lot on the cylinder at the intant t. The cylinder i in contact with only one edge of the rod and lot at any intant. M 0.5 kg a 0.5 m b t 1 t t 1 Find the angle ψ uing rectangular component. The velocity i parallel to the track therefore rco( θ) co ( bt ) in ( bt ) x abt x' ( abt)co bt rin( θ) y abt y' ( abt)in bt bt ψ atan y' + π ψ deg x' Now do the dynamic uing polar coordinate ( ab t 3 ) in ( bt ) ( ab t 3 ) + co ( bt ) θ bt θ' bt θ'' b r aθ r' aθ' r'' aθ'' Guee F 1N N C 1N Given N C in( ψ) Mgin( θ) M r'' rθ' F N C F N C co( ψ) Find( F, N C ) Mgco( θ) F N C Mrθ'' + r'θ' N Problem The ball ha ma M and a negligible ize. It i originally traveling around the horizontal circular path of radiu r 0 uch that the angular rate of rotation i θ' 0. If the attached cord ABC i drawn down through the hole at contant peed v, determine the tenion the cord exert on the ball at the intant r r 1. Alo, compute the angular velocity of the ball at thi intant. Neglect the effect of 0

77 friction between the ball and horizontal plane. Hint: Firt how that the equation of motion in the θ direction yield a θ rθ'' + r'θ' (1/r)(d(r θ')/dt) 0.hen integrated, r θ' c where the contant c i determined from the problem data. M r 0 θ' 0 kg 0.5 m 1 rad v r 1 0. m 0.5 m ΣF θ Ma θ ; 0 Mrθ'' + r'θ' M 1 r d dt r θ' r 0 Thu c r 0 θ'0 r 1 θ'1 θ' 1 θ' 0 θ' 1 4 rad r 1 r r 1 r' v r'' 0 m θ' θ' 1 T M r'' rθ' T 8N Problem The mooth urface of the vertical cam i defined in part by the curve r (a coθ + b). If the forked rod i rotating with a contant angular velocity θ', determine the force the cam and the rod exert on the roller of ma M at angle θ. The attached pring ha a tiffne k and an untretched length l. a 0. m k 30 N θ 30 deg m b 0.3 m l 0.1 m θ' 4 rad g 9.81 m M kg θ'' 0 rad r ( a)co( θ) + b 1

78 r' r'' ( a) in( θ)θ' ( a) co( θ)θ' ( a)in( θ)θ'' ψ atan rθ' r' + π Guee F N 1N F 1N Given Mgin( θ) FN in ψ kr ( l) M r'' rθ' F F N co( ψ) Mgco( θ) Mrθ'' + r'θ' F F N Find( F, F N ) F F N N Problem The collar ha ma M and travel along the mooth horizontal rod defined by the equiangular piral r ae θ. Determine the tangential force F and the normal force N C acting on the collar when θ θ 1 if the force F maintain a contant angular motion θ'. M kg a 1m θ 1 90 deg θ' rad θ θ 1 θ' θ' θ'' 0 rad r e θ ae θ r' aθ'e θ r'' a θ'' + θ' Find the angle ψ uing rectangular coordinate. The velocity i parallel to the path therefore x rco( θ) rin( θ) x' x' r' co θ rθ' in( θ) + rθ' co( θ) y y' r' in θ ψ atan y' θ + π ψ deg Now do the dynamic uing polar coordinate Guee F 1N N C 1N

79 Given F N C N C co( ψ) F co ψ M r'' rθ' N C in( ψ) F in ψ + Mrθ'' + r'θ' Find( F, N C ) F N C N *Problem The mooth urface of the vertical cam i defined in part by the curve r (a coθ + b). The forked rod i rotating with an angular acceleration θ'', and at angle θ the angular velocity i θ'. Determine the force the cam and the rod exert on the roller of ma M at thi intant. The attached pring ha a tiffne k and an untretched length l. a 0. m k 100 N m θ 45 deg b 0.3 m l 0.1 m θ' 6 rad g 9.81 m M kg θ'' rad r aco( θ) + b r' ( a) in( θ)θ' r'' ( a) co( θ)θ' ( a)in( θ)θ'' ψ atan rθ' r' + π Guee F N 1N F 1N Given Mgin( θ) F N in ψ kr ( l) M r'' rθ' F F N co( ψ) Mgco( θ) Mrθ'' + r'θ' F F N Find( F, F N ) F F N N Problem The pilot of an airplane execute a vertical loop which in part follow the path of a four-leaved roe, r a coθ. If hi peed at A i a contant v p, determine the vertical reaction the eat of the plane exert on the pilot when the plane i at A. Hi weight i. 3

80 a 600 ft 130 lb v p 80 ft g 3. ft θ 90 deg r Guee ( a)co( θ) r' 1 ft r'' 1 ft θ' 1 rad θ'' 1 rad Given Note that v p i contant o dv p /dt 0 r' ( a) in( θ)θ' r'' ( a) in( θ)θ'' ( a)co( θ)4θ' v p r' + ( rθ' ) 0 r' r'' + rθ' rθ'' + r'θ' r' + ( rθ' ) r' r'' θ' θ'' F N Find( r', r'', θ', θ'' ) r' ft ft r'' 4.7 θ' rad θ'' rad M( r'' rθ' ) F N ( g r'' ) rθ' F N 85.3 lb Problem Uing air preure, the ball of ma M i forced to move through the tube lying in the horizontal plane and having the hape of a logarithmic piral. If the tangential force exerted on the ball due to the air i 4

81 F, determine the rate of increae in the ball peed at the intant θ θ 1.hat direction doe it act in? M θ kg a 0. m b 0.1 π F 6N tan( ψ) r d dθ r ae bθ abe bθ 1 b F ψ atan 1 ψ deg b Mv' v' F v' 1 m M Problem Uing air preure, the ball of ma M i forced to move through the tube lying in the vertical plane and having the hape of a logarithmic piral. If the tangential force exerted on the ball due to the air i F, determine the rate of increae in the ball peed at the intant θ θ 1. hat direction doe it act in? M 0.5 kg a 0. m b 0.1 π F 6N θ 1 tan( ψ) r d dθ r ae bθ abe bθ 1 b ψ atan 1 ψ deg b F Mgco( ψ) Mv' v' F M g co( ψ) v' m 5

82 *Problem The arm i rotating at the rate θ' when the angular acceleration i θ'' and the angle i θ 0. Determine the normal force it mut exert on the particle of ma M if the particle i confined to move along the lotted path defined by the horizontal hyperbolic piral rθ b. θ' θ'' θ 0 M 5 rad rad 90 deg 0.5 kg b 0. m θ θ 0 r tan( ψ) θ' θ'' b b b r' θ θ r'' θ + b r θ θ ψ atan( θ) ψ deg d dθ r b θ Guee N P 1N F 1N Given N P in ψ M r'' rθ' F + N P co ψ Mrθ'' + r'θ' b θ 3 θ' N P F Find( N P, F) N P F N Problem The collar, which ha weight, lide along the mooth rod lying in the horizontal plane and having the hape of a parabola r a/( 1 coθ). If the collar' angular rate i θ', determine the tangential retarding force P needed to caue the motion and the normal force that the collar exert on the rod at the intatnt θ θ 1. 3lb a 4ft θ' 4 rad θ'' 0 rad 6

83 θ 1 90 deg g 3. ft M θ θ 1 a r r' 1 co θ g a in( θ) ( ) θ' 1 co θ r'' a in( θ) ( ) θ'' a + 1 co θ co( θ)θ' ( ) 1 co θ + a in( θ) θ' ( ) 3 1 co θ Find the angle ψ uing rectangular coordinate. The velocity i parallel to the path x rco( θ) rθ' in( θ) x' r' co θ y rin θ y' r' in θ + rθ' co( θ) x'' y'' r'' co θ r'θ' in( θ) r'θ' co( θ) rθ'' in( θ) rθ' co( θ) r'' in θ + + rθ'' in θ rθ' in θ ψ atan y' ψ 45 deg Guee P 1lb H 1lb x' Given H in( ψ) P co ψ + M x'' P in ψ H co( ψ) M y'' P H Find( P, H) P H lb Problem The tube rotate in the horizontal plane at a contant rate θ'. If a ball B of ma M tart at the origin O with an initial radial velocity r' 0 and move outward through the tube, determine the radial and tranvere component of the ball velocity at the intant it leave the outer end at C. Hint: Show that the equation of motion in the r direction i r'' rθ' 0. The olution i of the form r Ae -θ't + Be θ't. Evaluate the integration contant A and B, and determine the time t at r 1. Proceed to obtain v r and v θ. 7

84 θ' M 4 rad r' m 0. kg r m 0 M r'' rθ' rt () Ae θ't + Be θ' t r' () t θ' Ae θ't Be θ' t Gue A 1m B 1m t 1 Given 0 A + B r' 0 θ' ( A B) r 1 Ae θ't + Be θ' t A A B Find( A, B, t) B t m t rt () Ae θ't + Be θ' t r' () t θ' ( Ae θ't Be θ' t ) v r r' ( t 1 ) vθ rt ( 1 )θ' v r vθ.5 m Problem A pool of ma M lide down along a mooth rod. If the rod ha a contant angular rate of rotation θ ' in the vertical plane, how that the equation of motion for the pool are r'' rθ' ginθ 0 and Mθ'r' + N Mgcoθ 0 where N i the magnitude of the normal force of the rod on the pool. Uing the method of differential equation, it can be hown that the olution of the firt of thee equation i r C 1 e θ't + C e θ't (g/θ' )in(θ't). If r, r' and θ are zero when t 0, evaluate the contant C 1 and C and determine r at the intant θ θ 1. 8

85 M 0. kg θ' rad π θ 1 4 θ'' 0 rad g 9.81 m ΣF r Ma r ; ΣF θ Ma θ ; M r'' rθ' Mgin θ r'' rθ' Mgco( θ) N g in( θ) 0 [1] Mrθ'' + r'θ' Mθ'r' + N Mgco θ 0 The olution of the differential equation (Eq.[1] i given by r r' C 1 e θ' t + C e θ't g θ' θ' C 1 e θ' t + θ'c e θ't in( θ't) g θ' co ( θ't ) At t 0 r 0 0 C 1 + C r' 0 0 θ' C 1 + θ'c (Q.E.D) g θ' g g Thu C 1 4θ' C 4θ' t θ 1 t 0.39 θ' r C 1 e θ' t + C e θ't g θ' in( θ't) r m *Problem The rocket i in circular orbit about the earth at altitude h. Determine the minimum increment in peed it mut have in order to ecape the earth' gravitational field. 9