DepartamentodeEngenhariaMecânicaÁreaCientíficadeMecânicadosMeiosSólidos

RMM–2015/16

MecânicaAplicada

Cap.131-TrabalhodeumaforçaProblema13.5

Umautomóvelpesando19.125𝑁partedorepousonoponto𝐴sobreum declive de6°e desce em ponto morto por uma distância de150 𝑚até ao ponto𝐵. Os freios são então acionados, fazendo comque o automóvel pare no ponto𝐶 , a21 𝑚 de𝐵 . Sabendo que odeslizamento é iminente durante o período de travagem edesprezandoasresistênciasdoarederolamento,determine:

(a) avelocidadedoautomóvelnoponto𝐵;(b) ocoeficientedeatritoestáticoentreospneuseaestrada.

Problema13.11

Caixassãotransportadasporumacorreiatransportadoracomumavelocidade𝑣!até ao inicio de um plano inclinado fixo em𝐴, ondeelas deslizam e finalmente caem em𝐵 . Sabendo que𝜇! = 0,40 ,determineavelocidadedacorreiatransportadoraparaqueascaixasdeixemoplanoinclinadoem𝐵comumavelocidadede2 𝑚/𝑠.

Problema13.43

Umtrechodapistadeumamontanha-russaconsistededoisarcosdecírculo𝐴𝐵e𝐶𝐷unidosporumtrechoretoBC. O raio de𝐴𝐵é27 𝑚e o raio de𝐶𝐷 é de72 𝑚. Ocarrinhoeosseusocupantes,demassatotalde250 𝑘𝑔,alcançam o ponto𝐴 praticamente sem velocidade eentão caem livremente ao longo da pista. Determine aforça normal exercida pela pista sobre o carro quandoele alcançaoponto𝐵. Ignoreas resistênciasdoar ederolamento.

Problema13.65

Os blocos𝐴 e𝐵 pesam 36𝑁 e 13,5 𝑁 , respectivamente e estãounidos por um sistema de corda e polia, sendo libertados dorepouso na posiçãomostrada na figura, com amola indeformada.Sabendoqueaconstantedamolaé300 𝑁/𝑚,determine:

(a) avelocidadedobloco𝐵apóselesedeslocar15 𝑐𝑚;(b) avelocidademáximadobloco𝐵;(c) odeslocamentomáximodobloco𝐵.

Ignoreoatritoeasmassasdaspoliasedamola.

Problema13.67

O sistema mostrado na figura está em equilíbrio quando𝜙 = 0°.Sabendo que inicialmente 𝜙 = 90° e que o bloco 𝐶 está emmovimento suave e muito lento quando o sistema está naquelaposição, determine a velocidade do bloco quando ele passar pelaposiçãodeequilíbrio𝜙 = 0°.Desprezeamassadabarraedamola.

1Osproblemasapresentadosreferem-seaolivro“MecânicaVetorialparaEngenheiros–Dinâmica,FerdinandP.Beer,E.RussellJohnstonJr.,WilliamE.Clausen,7ªEdMcGraw-Hill”

772 Kinetics of Particles: Energy and Momentum Methods

13.9 A package is projected 10 m up a 15° incline so that it just reaches the top of the incline with zero velocity. Knowing that the coeffi-cient of kinetic friction between the package and the incline is 0.12, determine (a) the initial velocity of the package at A, (b) the velocity of the package as it returns to its original position.

A

CB

10 m

d

15°

Fig. P13.9 and P13.10

30°

B

C

A

7 m

2 m/s

1 m/s

d

Fig. P13.13 and P13.14

15°B

A

v0

20 ft

Fig. P13.11 and P13.12

13.10 A package is projected up a 15° incline at A with an initial velocity of 8 m/s. Knowing that the coefficient of kinetic friction between the package and the incline is 0.12, determine (a) the maximum distance d that the package will move up the incline, (b) the veloc-ity of the package as it returns to its original position.

13.11 Boxes are transported by a conveyor belt with a velocity v0 to a fixed incline at A where they slide and eventually fall off at B. Knowing that mk 5 0.40, determine the velocity of the conveyor belt if the boxes leave the incline at B with a velocity of 8 ft/s.

13.12 Boxes are transported by a conveyor belt with a velocity v0 to a fixed incline at A where they slide and eventually fall off at B. Knowing that mk 5 0.40, determine the velocity of the conveyor belt if the boxes are to have zero velocity at B.

13.13 Packages are thrown down an incline at A with a velocity of 1 m/s. The packages slide along the surface ABC to a conveyor belt which moves with a velocity of 2 m/s. Knowing that mk 5 0.25 between the packages and the surface ABC, determine the distance d if the packages are to arrive at C with a velocity of 2 m/s.

13.14 Packages are thrown down an incline at A with a velocity of 1 m/s. The packages slide along the surface ABC to a conveyor belt which moves with a velocity of 2 m/s. Knowing that d 5 7.5 m and mk 5 0.25 between the packages and all surfaces, determine (a) the speed of the package at C, (b) the distance a package will slide on the con-veyor belt before it comes to rest relative to the belt.

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779Problems 13.39 The sphere at A is given a downward velocity v0 and swings in a vertical circle of radius l and center O. Determine the smallest velocity v0 for which the sphere will reach point B as it swings about point O (a) if AO is a rope, (b) if AO is a slender rod of negligible mass.

13.40 The sphere at A is given a downward velocity v0 of magnitude 5 m/s and swings in a vertical plane at the end of a rope of length l 5 2 m attached to a support at O. Determine the angle u at which the rope will break, knowing that it can withstand a maxi-mum tension equal to twice the weight of the sphere.

13.41 A section of track for a roller coaster consists of two circular arcs AB and CD joined by a straight portion BC. The radius of AB is 90 ft and the radius of CD is 240 ft. The car and its occupants, of total weight 560 lb reach point A with practically no velocity and then drop freely along the track. Determine the normal force exerted by the track on the car as the car reaches point B. Ignore air resistance and rolling resistance.

A

B

Ol

v0

q

Fig. P13.39 and P13.40

13.42 A section of track for a roller coaster consists of two circular arcs AB and CD joined by a straight portion BC. The radius of AB is 90 ft and the radius of CD is 240 ft. The car and its occupants, of total weight 560 lb, reach point A with practically no velocity and then drop freely along the track. Determine the maximum and minimum values of the normal force exerted by the track on the car as the car travels from A to D. Ignore air resistance and rolling resistance.

13.43 A small sphere B of mass m is released from rest in the position shown and swings freely in a vertical plane, first about O and then about the peg A after the cord comes in contact with the peg. Determine the tension in the cord (a) just before the sphere comes in contact with the peg, (b) just after it comes in contact with the peg.

A

BO30°

q

0.4 m

0.8 m

Fig. P13.43

90 ft

60 ft

B

C

D

A

r = 240 ft40°

Fig. P13.41 and P13.42

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797Problems 13.67 The system shown is in equilibrium when f 5 0. Knowing that initially f 5 90° and that block C is given a slight nudge when the system is in that position, determine the speed of the block as it passes through the equilibrium position f 5 0. Neglect the weight of the rod.

A

B

D

f

0.3 ft

25 lb

k = 600 lb/ft

2.1 ft

1.1 ft

C

Fig. P13.67

A

BC

1.2 m

30°

r = 0.8 m

Fig. P13.70 and P13.71

8 m

20°

Cable

2 m/s

50 kg

Fig. P13.68

13.68 A spring is used to stop a 50-kg package which is moving down a 20° incline. The spring has a constant k 5 30 kN/m and is held by cables so that it is initially compressed 50 mm. Knowing that the velocity of the package is 2 m/s when it is 8 m from the spring and neglecting friction, determine the maximum additional deforma-tion of the spring in bringing the package to rest.

13.69 Solve Prob. 13.68 assuming the kinetic coefficient of friction between the package and the incline is 0.2.

13.70 A 300-g pellet is released from rest at A and slides with friction along the surface shown. Determine the force exerted on the pel-let by the surface (a) just before the pellet reaches B, (b) immedi-ately after it has passed through B.

x

y

z

C

B

A

D

3 in.

r = 6 in.

12 in.

Fig. P13.72

13.71 A 300-g pellet is released from rest at A and slides without friction along the surface shown. Determine the force exerted on the pel-let by the surface (a) just before the pellet reaches C, (b) imme-diately after it has passed through C.

13.72 A 1.2-lb collar can slide without friction along the semicircular rod BCD. The spring is of constant 1.8 lb/in and its undeformed length is 8 in. Knowing that the collar is released from rest at B, deter-mine (a) the speed of the collar as it passes through C, (b) the force exerted by the rod on the collar at C.

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6m

𝑟 = 72 𝑚 27 𝑚

18 𝑚

𝑘 = 8 𝑘𝑁/𝑚 300 𝑚𝑚

600 𝑚𝑚

100 𝑚𝑚

10 𝑘𝑔